Monoidal categorification of cluster algebras

Abstract

We prove that the quantum cluster algebra structure of a unipotent quantum coordinate ring $A_q(\mathfrak{n}(w))$, associated with a symmetric Kac–Moody algebra and its Weyl group element $w$, admits a monoidal categorification via the representations of symmetric Khovanov–Lauda–Rouquier algebras. In order to achieve this goal, we give a formulation of monoidal categorifications of quantum cluster algebras and provide a criterion for a monoidal category of finite-dimensional graded $R$-modules to become a monoidal categorification, where $R$ is a symmetric Khovanov–Lauda–Rouquier algebra. Roughly speaking, this criterion asserts that a quantum monoidal seed can be mutated successively in all the directions, once the first-step mutations are possible. Then, we show the existence of a quantum monoidal seed of $A_q(\mathfrak{n}(w))$ which admits the first-step mutations in all the directions. As a consequence, we prove the conjecture that any cluster monomial is a member of the upper global basis up to a power of $q^{1/2}$. In the course of our investigation, we also give a proof of a conjecture of Leclerc on the product of upper global basis elements.

Introduction

The purpose of this paper is to provide a monoidal categorification of the quantum cluster algebra structure on the unipotent quantum coordinate ring $A_q(\mathfrak{n}(w))$, which is associated with a symmetric Kac–Moody algebra $\mathfrak{g}$ and a Weyl group element $w$.

The notion of cluster algebras was introduced by Fomin and Zelevinsky in 6 for studying total positivity and upper global bases. Since their introduction, a lot of connections and applications have been discovered in various fields of mathematics including representation theory, Teichmüller theory, tropical geometry, integrable systems, and Poisson geometry.

A cluster algebra is a $\mathbb{Z}$-subalgebra of a rational function field given by a set of generators, called the cluster variables. These generators are grouped into overlapping subsets, called the clusters, and the clusters are defined inductively by a procedure called mutation from the initial cluster $\{ X_i\}_{1 \le i \le r}$, which is controlled by an exchange matrix $\widetilde{B}$. We call a monomial of cluster variables in each cluster a cluster monomial.

Fomin and Zelevinsky proved that every cluster variable is a Laurent polynomial of the initial cluster $\{ X_i\}_{1 \le i \le r}$, and they conjectured that this Laurent polynomial has positive coefficients 6. This positivity conjecture was proved by Lee and Schiffler in the skew-symmetric cluster algebra case in 30. The linearly independence conjecture on cluster monomials was proved in the skew-symmetric cluster algebra case in 4.

The notion of quantum cluster algebras, introduced by Berenstein and Zelevinsky in 3, can be considered as a $q$-analogue of cluster algebras. The commutation relation among the cluster variables is determined by a skew-symmetric matrix $L$. As in the cluster algebra case, every cluster variable belongs to $\mathbb{Z}[q^{\pm 1/2}][X_i^{\pm 1}]_{1 \le i \le r}$ 3 and is expected to be an element of $\mathbb{Z}_{\ge 0}[q^{\pm 1/2}][X_i^{\pm 1}]_{1 \le i \le r}$, which is referred to as the quantum positivity conjecture (cf. 5, Conjecture 4.7). In 24, Kimura and Qin proved the quantum positivity conjecture for quantum cluster algebras containing acyclic seed and specific coefficients.

The unipotent quantum coordinate rings $A_q(\mathfrak{n})$ and $A_q(\mathfrak{n}(w))$ are examples of quantum cluster algebras arising from Lie theory. The algebra $A_q(\mathfrak{n})$ is a $q$-deformation of the coordinate ring $\mathbb{C}[N]$ of the unipotent subgroup and is isomorphic to the negative half $U^-_q(\mathfrak{g})$ of the quantum group as $\mathbb{Q}(q)$-algebras. The algebra $A_q(\mathfrak{n}(w))$ is a $\mathbb{Q}(q)$-subalgebra of $A_q(\mathfrak{n})$ generated by a set of the dual Poincaré–Birkhoff–Witt (PBW) basis elements associated with a Weyl group element $w$. The unipotent quantum coordinate ring $A_q(\mathfrak{n})$ has a very interesting basis, the so-called upper global basis (dual canonical basis) $\mathbf{B}^\mathrm{up}$, which is dual to the lower global basis (canonical basis) 1631. The upper global basis has been studied emphasizing its multiplicative structure. For example, Berenstein and Zelevinsky 2 conjectured that, in the case $\mathfrak{g}$ is of type $A_n$, the product $b_1 b_2$ of two elements $b_1$ and $b_2$ in $\mathbf{B}^\mathrm{up}$ is again an element of $\mathbf{B}^\mathrm{up}$ up to a multiple of a power of $q$ if and only if they are $q$-commuting; i.e., $b_1b_2=q^m b_2b_1$ for some $m\in \mathbb{Z}$. This conjecture turned out to be not true in general, because Leclerc 29 found examples of an imaginary element $b \in \mathbf{B}^\mathrm{up}$ such that $b^2$ does not belong to $\mathbf{B}^\mathrm{up}$. Nevertheless, the idea of considering subsets of $\mathbf{B}^\mathrm{up}$ whose elements are $q$-commuting with each other and studying the relations between those subsets has survived, and it became one of the motivations of the study of (quantum) cluster algebras.

In a series of papers 8911, Geiß, Leclerc, and Schröer showed that the unipotent quantum coordinate ring $A_q(\mathfrak{n}(w))$ has a skew-symmetric quantum cluster algebra structure whose initial cluster consists of the so-called unipotent quantum minors. In 23, Kimura proved that $A_q(\mathfrak{n}(w))$ is compatible with the upper global basis ${\mathbf{B}}^{{\mathrm{up}}}$ of $A_q(\mathfrak{n})$; i.e., the set ${\mathbf{B}}^{{\mathrm{up}}}(w) \mathbin{:=}A_q(\mathfrak{n}(w)) \cap {\mathbf{B}}^{{\mathrm{up}}}$ is a basis of $A_q(\mathfrak{n}(w))$. Thus, with a result of 4, one can expect that every cluster monomial of $A_q(\mathfrak{n}(w))$ is contained in the upper global basis ${\mathbf{B}}^{{\mathrm{up}}}(w)$, which is named the quantization conjecture by Kimura 23.

Conjecture (11, Conjecture 12.9, 23, Conjecture 1.1(2)).

When $\mathfrak{g}$ is a symmetric Kac–Moody algebra, every quantum cluster monomial in $A_{q^{1/2}}(\mathfrak{n}(w))\mathbin{:=}\mathbb{Q}(q^{1/2})\otimes _{\mathbb{Q}(q)}A_q(\mathfrak{n}(w))$ belongs to the upper global basis $\mathbf{B}^\mathrm{up}$ up to a power of $q^{1/2}$.

It can be regarded as a reformulation of Berenstein–Zelevinsky’s ideas on the multiplicative properties of ${\mathbf{B}}^\mathrm{up}$. There are some partial results of this conjecture. It is proved for $\mathfrak{g}= A_2$, $A_3$, $A_4$ and $A_q(\mathfrak{n}(w))=A_q(\mathfrak{n})$ in 2 and 7, Section 12. When $\mathfrak{g}=A^{(1)}_1$, $A_n$ and $w$ is a square of a Coxeter element, it is shown in 26 and 27 that the cluster variables belong to the upper global basis. When $\mathfrak{g}$ is symmetric and $w$ is a square of a Coxeter element, the conjecture is proved in 24. Notably, Qin provided recently a proof of the conjecture for a large class with a condition on the Weyl group element $w$ 37. Note that Nakajima proposed a geometric approach of this conjecture via quiver varieties 35.

In this paper, we prove the above conjecture completely by showing that there exists a monoidal categorification of $A_{q^{1/2}}(\mathfrak{n}(w))$.

In 12, Hernandez and Leclerc introduced the notion of monoidal categorification of cluster algebras. A simple object $S$ of a monoidal category $\mathcal{C}$ is real if $S \mathop{\otimes }S$ is simple, and it is prime if there exists no nontrivial factorization $S \simeq S_1 \mathop{\otimes }S_2$. They say that $\mathcal{C}$ is a monoidal categorification of a cluster algebra $A$ if the Grothendieck ring of $\mathcal{C}$ is isomorphic to $A$ and if

(M1)

the cluster monomials of $A$ are the classes of real simple objects of $\mathcal{C}$,

(M2)

the cluster variables of $A$ are the classes of real simple prime objects of $\mathcal{C}$.

(Note that the above version is weaker than the original definition of the monoidal categorification in 12.) They proved that certain categories of modules over symmetric quantum affine algebras $U_q'(\mathfrak{g})$ give monoidal categorifications of some cluster algebras. Nakajima extended this result to the cases of the cluster algebras of types $A,D,E$ 36 (see also 13). It is worthwhile to remark that once a cluster algebra $A$ has a monoidal categorification, the positivity of cluster variables of $A$ and the linear independence of cluster monomials of $A$ follow (see 12, Proposition 2.2).

In this paper, we refine Hernandez–Leclerc’s notion of monoidal categorifications including the quantum cluster algebra case. Let us briefly explain it. Let $\mathcal{C}$ be an abelian monoidal category equipped with an auto-equivalence $q$ and a tensor product which is compatible with a decomposition $\mathcal{C}=\mathop{\textstyle \bigoplus }\limits \nolimits _{\beta \in \mathsf{Q}}\mathcal{C}_\beta$. Fix a finite index set ${J}={J}_\mathrm{ex}\sqcup {J}_\mathrm{fr}$ with a decomposition into the exchangeable part and the frozen part. Let $\mathscr{S}$ be a quadruple $(\{M_i\}_{i \in {J}}, L,\widetilde{B}, D)$ of a family of simple objects $\{M_i\}_{i \in {J}}$ in $\mathscr{C}$, an integer-valued skew-symmetric ${J}\times {J}$-matrix $L=(\lambda _{i,j})$, an integer-valued ${J}\times {J}_\mathrm{ex}$-matrix $\widetilde{B}= (b_{i,j})$ with a skew-symmetric principal part, and a family of elements $D=\{d_i\}_{i \in {J}}$ in $\mathsf{Q}$. If this datum satisfies the conditions in Definition 6.2.1 below, then it is called a quantum monoidal seed in $\mathcal{C}$. For each $k\in {J}_\mathrm{ex}$, we have mutations $\mu _k(L),\;\mu _k(\widetilde{B})$, and $\mu _k(D)$ of $L$, ${\widetilde{B}}$, and $D$, respectively. We say that a quantum monoidal seed $\mathscr{S} =(\{M_i\}_{i\in {J}}, L,\widetilde{B}, D)$ admits a mutation in direction $k\in {J}_\mathrm{ex}$ if there exists a simple object $M_k' \in \mathcal{C}_{\mu _k(D)_k}$ which fits into two short exact sequences 0.2 below in $\mathcal{C}$ reflecting the mutation rule in quantum cluster algebras, and thus obtained quadruple $\mu _k(\mathscr{S})\mathbin{:=}(\{M_i\}_{i\neq k}\cup \{M_k'\},\mu _k(L), \mu _k(\widetilde{B}), \mu _k(D))$ is again a quantum monoidal seed in $\mathcal{C}$. We call $\mu _k(\mathscr{S})$ the mutation of $\mathscr{S}$ in direction $k\in {J}_\mathrm{ex}$.

Now the category $\mathcal{C}$ is called a monoidal categorification of a quantum cluster algebra $A$ over $\mathbb{Z}[q^{\pm 1/2}]$ if

(0.1)

(i)

the Grothendieck ring $\mathbb{Z}[q^{\pm 1/2}]\mathop{\otimes }_{\mathbb{Z}[q^{\pm 1}]} K(\mathcal{C})$ is isomorphic to $A$,

(ii)

there exists a quantum monoidal seed $\mathscr{S} =(\{M_i\}_{i\in {J}}, L,\widetilde{B}, D)$ in $\mathcal{C}$ such that $[\mathscr{S}]\mathbin{:=}(\{q^{m_i}[M_i]\}_{i\in {J}}, L, \widetilde{B})$ is a quantum seed of $A$ for some $m_i \in \frac{1}{2}\mathbb{Z}$,

(iii)

$\mathscr{S}$ admits successive mutations in all directions in ${J}_\mathrm{ex}$.

The existence of monoidal category $\mathcal{C}$ which provides a monoidal categorification of quantum cluster algebra $A$ implies the following:

(QM1)

Every quantum cluster monomial corresponds to the isomorphism class of a real simple object of $\mathcal{C}$. In particular, the set of quantum cluster monomials is $\mathbb{Z}[q^{\pm 1/2}]$-linearly independent.

(QM2)

The quantum positivity conjecture holds for $A$.

In the case of unipotent quantum coordinate ring $A_q(\mathfrak{n})$, there is a natural candidate for monoidal categorification, the category of finite-dimensional graded modules over a Khovanov–Lauda–Rouquier algebras (2122, 38). The Khovanov–Lauda–Rouquier algebras (abbreviated by KLR algebras), introduced by Khovanov–Lauda 2122 and Rouquier 38 independently, are a family of $\mathbb{Z}$-graded algebras which categorifies the negative half $U_q^-(\mathfrak{g})$ of a symmetrizable quantum group $U_q(\mathfrak{g})$. More precisely, there exists a family of algebras $\{ R(-\beta ) \}_{\beta \in \mathsf{Q}^-}$ such that the Grothendieck ring of $R \text{-gmod}\mathbin{:=}\bigoplus _{\beta \in \mathsf{Q}^-}R(-\beta )\text{-gmod}$, the direct sum of the categories of finite-dimensional graded $R(-\beta )$-modules, is isomorphic to the integral form $A_q(\mathfrak{n})_{\mathbb{Z}[q^{\pm 1}]}$ of $A_q(\mathfrak{n}) \simeq U_q^-(\mathfrak{g})$. Here the tensor functor $\mathop{\otimes }$ of the monoidal category $R \text{-gmod}$ is given by the convolution product $\circ$, and the action of $q$ is given by the grading shift functor. In 3940, Varagnolo–Vasserot and Rouquier proved that the upper global basis ${\mathbf{B}}^{\mathrm{up}}$ of $A_q(\mathfrak{n})$ corresponds to the set of the isomorphism classes of all self-dual simple modules of $R \text{-gmod}$ under the assumption that $R$ is associated with a symmetric quantum group $U_q(\mathfrak{g})$ and the base field is of characteristic $0$.

Combining works of 112340, the unipotent quantum coordinate ring $A_q(\mathfrak{n}(w))$ associated with a symmetric quantum group $U_q(\mathfrak{g})$ and a Weyl group element $w$ is isomorphic to the Grothendieck group of a monoidal abelian full subcategory $\mathcal{C}_w$ of $R \text{-gmod}$ whose base field $\mathbf{k}$ is of characteristic $0$, satisfying the following properties: (i) $\mathcal{C}_w$ is stable under extensions and grading shift functor, (ii) the composition factors of $M \in \mathcal{C}_w$ are contained in ${\mathbf{B}}^{{\mathrm{up}}}(w)$ (see Definition 11.2.1). In particular, the first condition in 0.1 holds. However, it is not evident that the second and the third conditions in 0.1 on quantum monoidal seeds are satisfied. The purpose of this paper is to ensure that those conditions hold in $\mathcal{C}_w$.

In order to establish it, in the first part of the paper, we start with a continuation of the work of 15 about the convolution products, heads, and socles of graded modules over symmetric KLR algebras. One of the main results in 15 is that the convolution product $M \circ N$ of a real simple $R(\beta )$-module $M$ and a simple $R(\gamma )$-module $N$ has a unique simple quotient and a unique simple submodule. Moreover, if $M \circ N \simeq N \circ M$ up to a grading shift, then $M \circ N$ is simple. In such a case we say that $M$ and $N$ commute. The main tool of 15 was the R-matrix $\mathbf{r}_{\!_{M,N}}$, constructed in 14, which is a homogeneous homomorphism from $M \circ N$ to $N \circ M$ of degree $\Lambda (M,N)$. In this work, we define some integers encoding necessary information on $M\circ N$,

$$\begin{align*} \widetilde{\Lambda }(M,N) \mathbin{:=}\frac{1}{2}\bigl (\Lambda (M,N)+(\beta ,\gamma )\bigr ), \quad \operatorname {\mathfrak{d}}(M,N)\mathbin{:=}\frac{1}{2}\bigl (\Lambda (M,N)+\Lambda (N,M)\bigr ), \end{align*}$$

and study the representation theoretic meaning of the integers $\Lambda (M,N)$, $\widetilde{\Lambda }(M,N)$, and $\operatorname {\mathfrak{d}}(M,N)$.

We then prove Leclerc’s first conjecture 29 on the multiplicative structure of elements in ${\mathbf{B}}^{\mathrm{up}}$, when the generalized Cartan matrix is symmetric (Theorem 4.1.1 and Theorem 4.2.1). Theorem 4.2.1 is due to McNamara 34, Lemma 7.5, and the authors thank him for informing us of his result.

We say that $b \in {\mathbf{B}}^{\mathrm{up}}$ is real if $b^2 \in q^\mathbb{Z}\,{\mathbf{B}}^{\mathrm{up}}\mathbin{:=}\bigsqcup _{n\in \mathbb{Z}}q^n{\mathbf{B}}^{\mathrm{up}}$.

Theorem (29, Conjecture 1).

Let $b_1$ and $b_2$ be elements in ${\mathbf{B}}^{\mathrm{up}}$ such that one of them is real and $b_1b_2 \not \in q^\mathbb{Z}{\mathbf{B}}^{\mathrm{up}}$. Then the expansion of $b_1b_2$ with respect to ${\mathbf{B}}^{\mathrm{up}}$ is of the form

$$\begin{equation*} b_1b_2= q^m b' + q^s b'' + \sum _{c \ne b',b''}\gamma ^c_{b_1,b_2}(q)c, \end{equation*}$$

where $b' \ne b''$, $m,s \in \mathbb{Z}$, $m < s$, and

$$\begin{equation*} \gamma ^c_{b_1,b_2}(q) \in q^{m+1}\mathbb{Z}[q] \cap q^{s-1}\mathbb{Z}[q^{-1}]. \end{equation*}$$

More precisely, we prove that $q^mb'$ and $q^sb''$ correspond to the simple head and the simple socle of $M\circ N$, respectively, when $b_1$ corresponds to a simple module $M$ and $b_2$ corresponds to a simple module $N$.

Next, we move to provide an algebraic framework for monoidal categorification of quantum cluster algebras. In order to simplify the conditions of quantum monoidal seeds and their mutations, we introduce the notion of admissible pairs in $\mathcal{C}_w$. A pair $(\{M_i\}_{i \in {J}}, \widetilde{B})$ is called admissible in $\mathcal{C}_w$ if (i) $\{M_i\}_{i \in {J}}$ is a commuting family of self-dual real simple objects of $\mathcal{C}_w$, (ii) $\widetilde{B}$ is an integer-valued ${J}\times {J}_\mathrm{ex}$-matrix with a skew-symmetric principal part, and (iii) for each $k \in {J}$, there exists a self-dual simple object $M'_k$ in $\mathcal{C}_w$ such that $M_k'$ commutes with $M_i$ for all $i\in {J}\setminus \{k\}$ and there are exact sequences in $\mathcal{C}_w$

$$\begin{eqnarray} &&\begin{array}{l} 0 \to q \mathop{\textstyle \bigodot }\limits _{b_{i,k} >0} M_i^{\text{$\odot $}b_{i,k}} \to q^{\widetilde{\Lambda }(M_k,M_k')} M_k \circ M_k' \to \mathop{\textstyle \bigodot }\limits _{b_{i,k} <0} M_i^{\text{$\odot $}(-b_{i,k})} \to 0,\\0 \to q \mathop{\textstyle \bigodot }\limits _{b_{i,k} <0} M_i^{\text{$\odot $}(- b_{i,k})} \to q^{\widetilde{\Lambda }(M_k',M_k)} M'_k \circ M_k \to \mathop{\textstyle \bigodot }\limits _{b_{i,k} >0} M_i^{\text{$\odot $}b_{i,k}} \to 0, \end{array} {\tag{0.2}} \end{eqnarray}$$

where $\widetilde{\Lambda }(M_k,M_k')$ and $\widetilde{\Lambda }(M_k',M_k)$ are prescribed integers and $\mathop{\textstyle \bigodot }\limits$ is a convolution product up to a power of $q$.

For an admissible pair $(\{M_i\}_{i \in {J}}, \widetilde{B})$, let $\Lambda =(\Lambda _{i,j})_{i,j \in {J}}$ be the skew-symmetric matrix where $\Lambda _{i,j}$ is the homogeneous degree of $\mathbf{r}_{\!_{{M_i},{M_j}}}$, the R-matrix between $M_i$ and $M_j$, and let $D=\{d_i\}_{ i \in {J}}$ be the family of elements in $\mathsf{Q}$ given by $M_i \in R(-d_i) \text{-gmod}$.

Then, together with the result of 11, our main theorem in the first part of the paper reads as follows.

Main Theorem 1 (Theorem 7.1.3 and Corollary 7.1.4).

If there exists an admissible pair $(\{M_i\}_{i\in K},\widetilde{B})$ in $\mathcal{C}_w$ such that $[\mathscr{S}]\mathbin{:=}\bigl (\{q^{-(\operatorname {wt}(M_i), \operatorname {wt}(M_i))/4}[M_i]\}_{i\in {J}}, -\Lambda , \widetilde{B}, D\bigr )$ is an initial seed of $A_{q^{1/2}}(\mathfrak{n}(w))$, then $\mathcal{C}_w$ is a monoidal categorification of $A_{q^{1/2}}(\mathfrak{n}(w))$.

The second part of this paper (Sections 8–11) is mainly devoted to showing that there exists an admissible pair in $\mathcal{C}_w$ for every symmetric Kac–Moody algebra $\mathfrak{g}$ and its Weyl group element $w$. In 11, Geiß, Leclerc, and Schröer provided an initial quantum seed in $A_q(\mathfrak{n}(w))$ whose quantum cluster variables are unipotent quantum minors. The unipotent quantum minors are elements in $A_q(\mathfrak{n})$, which are regarded as a $q$-analogue of a generalization of the minors of upper triangular matrices. In particular, they are elements in $\mathbf{B}^\mathrm{up}$. We define the determinantial module $\mathsf{M}(\mu ,\zeta )$ to be the simple module in $R \text{-gmod}$ corresponding to the unipotent quantum minor $\mathrm{D}(\mu ,\zeta )$ under the isomorphism $A_q( \mathfrak{n})_{\mathbb{Z}[q^{\pm 1}]} \simeq K(R \text{-gmod})$. Here $(\mu ,\zeta )$ is a pair of elements in the weight lattice of $\mathfrak{g}$ satisfying certain conditions.

Our main theorem of the second part is as follows.

Main Theorem 2 (Theorem 11.2.2).

Let $( \{ D(k,0) \}_{1 \le k \le r}, \widetilde{B}, L)$ be the initial quantum seed of $A_q(\mathfrak{n}(w))$ in 11 with respect to a reduced expression $\widetilde{w}=s_{i_r}\cdots s_{i_1}$ of $w$. Let $\mathsf{M}(k,0):=\mathsf{M}(s_{i_1}\cdots s_{i_k}\varpi _{i_k},\varpi _{i_k})$ be the determinantial module corresponding to the unipotent quantum minor $D(k,0)$. Then the pair

$$\begin{equation*} ( \{ \mathsf{M}(k,0) \}_{1 \le k \le r}, \widetilde{B}) \end{equation*}$$

is admissible in $\mathcal{C}_w$.

Combining these theorems, the category $\mathcal{C}_w$ gives a monoidal categorification of the quantum cluster algebra $A_q(\mathfrak{n}(w))$. If we take the base field of the symmetric KLR algebra to be of characteristic $0$, these theorems, along with Theorem 2.1.4 due to 3940, imply the quantization conjecture.

The most essential condition for an admissible pair is that there exists the first mutation $\mathsf{M}(k,0)'$ in the exact sequences 0.2 for each $k \in {J}_\mathrm{ex}$. To establish this, we investigate the properties of determinantial modules and those of their convolution products. Note that a unipotent quantum minor is the image of a global basis element of the quantum coordinate ring $A_q(\mathfrak{g})$ under a natural projection $A_q(\mathfrak{g}) \to A_q(\mathfrak{n})$. Since there exists a bicrystal embedding from the crystal basis $B(A_q(\mathfrak{g}))$ of $A_q(\mathfrak{g})$ to the crystal basis $B(\widetilde{U}_q(\mathfrak{g}))$ of the modified quantum groups $\widetilde{U}_q(\mathfrak{g})$, this investigation amounts to the study of the interplay among the crystal and global bases of $A_q(\mathfrak{g})$, $\widetilde{U}_q(\mathfrak{g})$, and $A_q(\mathfrak{n})$. Hence we start the second part of the paper with the studies of those algebras and their crystal/global bases along the line of the works in 171819.

Next, we recall the (unipotent) quantum minors and the T-system, an equation consisting of three terms in products of unipotent quantum minors studied in 311. A detailed study of the relation between $A_q(\mathfrak{g})$, $\widetilde{U}_q(\mathfrak{g})$, and $A_q(\mathfrak{n})$ and their global bases enables us to establish several equations involving unipotent quantum minors in the algebra $A_q(\mathfrak{n})$. The upshot is that those equations can be translated into exact sequences in the category $R\text{-gmod}$ involving convolution products of determinantial modules via the categorification of $U_q^-(\mathfrak{g})$. It enables us to show that the pair $( \{ \mathsf{M}(k,0) \}_{1 \le k \le r}, \widetilde{B})$ is admissible.

The paper is organized as follows. In Section 1, we briefly review basic materials on quantum group $U_q(\mathfrak{g})$ and KLR algebra $R$. In Section 2, we continue the study in 15 of the R-matrices between $R$-modules. In Section 3, we derive certain properties of $\widetilde{\Lambda }(M,N)$ and $\operatorname {\mathfrak{d}}(M,N)$. In Section 4, we prove the first conjecture of Leclerc in 29. In Section 5, we recall the definition of quantum cluster algebras. In Section 6, we give the definitions of a monoidal seed, a quantum monoidal seed, a monoidal categorification of a cluster algebra, and a monoidal categorification of a quantum cluster algebra. In Section 7, we prove Main Theorem 1. In Section 8, we review the algebras $A_q(\mathfrak{g})$, $\widetilde{U}_q(\mathfrak{g})$, and $A_q(\mathfrak{n})$, and study the relations among them. In Section 9, we study the properties of quantum minors including $T$-systems and generalized $T$-systems. In Section 10, we study the determinantial modules over KLR algebras. Finally, in Section 11, we establish Main Theorem 2.

1. Quantum groups and global bases

In this section, we briefly recall the quantum groups and the crystal and global bases theory for $U_q(\mathfrak{g})$. We refer to 161720 for materials in this subsection.

1.1. Quantum groups

Let $I$ be an index set. A Cartan datum is a quintuple $(A,\mathsf{P}, \Pi ,\mathsf{P}^{\vee },\Pi ^{\vee })$ consisting of

(i)

an integer-valued matrix $A=(a_{ij})_{i,j \in I}$, called the symmetrizable generalized Cartan matrix, which satisfies

(a)

$a_{ii} = 2$ $(i \in I)$,

(b)

$a_{ij} \le 0$ $(i \neq j)$,

(c)

there exists a diagonal matrix $D=\text{diag} (\mathsf{s}_i \mid i \in I)$ such that $DA$ is symmetric, and $\mathsf{s}_i$ are relatively prime positive integers,

(ii)

a free abelian group $\mathsf{P}$, called the weight lattice,

(iii)

$\Pi = \{ \alpha _i \in \mathsf{P}\mid \ i \in I \}$, called the set of simple roots,

(iv)

$\mathsf{P}^{\vee }\mathbin{:=}\operatorname {Hom}_\mathbb{Z}(\mathsf{P}, \mathbb{Z})$, called the co-weight lattice,

(v)

$\Pi ^{\vee }= \{ h_i \ | \ i \in I \}\subset \mathsf{P}^{\vee }$, called the set of simple coroots, satisfying the following properties:

(1)

$\langle h_i,\alpha _j \rangle = a_{ij}$ for all $i,j \in I$,

(2)

$\Pi$ is linearly independent over $\mathbb{Q}$,

(3)

for each $i \in I$, there exists $\varpi _i \in \mathsf{P}$ such that $\langle h_j, \varpi _i \rangle =\delta _{ij}$ for all $j \in I$.

We call $\varpi _i$ the fundamental weights.

The free abelian group $\mathsf{Q}\mathbin{:=}\mathop{\textstyle \bigoplus }\limits _{i \in I} \mathbb{Z}\alpha _i$ is called the root lattice. Set $\mathsf{Q}^{+}= \sum _{i \in I} \mathbb{Z}_{\ge 0} \alpha _i\subset \mathsf{Q}$ and $\mathsf{Q}^{-}{=} \sum _{i \in I} \mathbb{Z}_{\le 0} \alpha _i\subset \mathsf{Q}$. For $\beta {=}\sum _{i\in I}m_i\alpha _i\in \mathsf{Q}$, we set $|\beta |{=}\sum _{i\in I}|m_i|$.

Set $\mathfrak{h}=\mathbb{Q}\otimes _\mathbb{Z}\mathsf{P}^{\vee }$. Then there exists a symmetric bilinear form $(\quad , \quad )$ on $\mathfrak{h}^*$ satisfying

$$\begin{equation*} (\alpha _i , \alpha _j) =\mathsf{s}_i a_{ij} \quad (i,j \in I) \quad \text{and $\langle h_i,\lambda \rangle = \dfrac{2(\alpha _i,\lambda )}{(\alpha _i,\alpha _i)}$ for any $\lambda \in \mathfrak{h}^*$ and $i \in I$}. \end{equation*}$$

The Weyl group of $\mathfrak{g}$ is the group of linear transformations on $\mathfrak{h}^*$ generated by $s_i$ $(i\in I)$, where

$$\begin{eqnarray*} s_i(\lambda ) := \lambda - \langle h_i, \lambda \rangle \alpha _i \quad \text{for } \ \lambda \in \mathfrak{h}^*, \ i \in I. \end{eqnarray*}$$

Let $q$ be an indeterminate. For each $i \in I$, set $q_i = q^{\,\mathsf{s}_i}$.

Definition 1.1.1.

The quantum group associated with a Cartan datum $(A,\mathsf{P},\Pi , \mathsf{P}^{\vee }, \Pi ^{\vee })$ is the algebra $U_q(\mathfrak{g})$ over $\mathbb{Q}(q)$ generated by $e_i,f_i$ $(i \in I)$ and $q^{h}$ $(h \in \mathsf{P}^\vee )$ satisfying the following relations:

$$\begin{equation*} \begin{aligned} & q^0=1,\ q^{h} q^{h'}=q^{h+h'} \ \ \text{for} \ h,h' \in \mathsf{P}^\vee ,\\ & q^{h}e_i q^{-h}= q^{\langle h, \alpha _i\rangle } e_i, \ \ \ q^{h}f_i q^{-h} = q^{-\langle h, \alpha _i\rangle } f_i \ \ \text{for} \ h \in \mathsf{P}^\vee , i \in I, \\ & e_if_j - f_je_i = \delta _{ij} \dfrac{t_i -t^{-1}_i}{q_i- q^{-1}_i }, \ \ \text{ where } t_i=q^{\mathsf{s}_i h_i}, \\ & \sum ^{1-a_{ij}}_{r=0} (-1)^r \left[\begin{matrix} 1-a_{ij} \\ r\\ \end{matrix} \right]_i e^{1-a_{ij}-r}_i e_j e^{r}_i =0 \quad \text{ if } i \ne j, \\ & \sum ^{1-a_{ij}}_{r=0} (-1)^r \left[\begin{matrix} 1-a_{ij} \\ r\\ \end{matrix} \right]_i f^{1-a_{ij}-r}_if_j f^{r}_i=0 \quad \text{ if } i \ne j. \end{aligned} \end{equation*}$$

Here, we set $[n]_i =\dfrac{ q^n_{i} - q^{-n}_{i} }{ q_{i} - q^{-1}_{i} },\; [n]_i! = \prod ^{n}_{k=1} [k]_i$, and $\left[\begin{matrix} m \\ n\\ \end{matrix} \right]_i= \dfrac{ [m]_i! }{[m-n]_i! [n]_i! }\;$ for $i \in I$ and $m,n \in \mathbb{Z}_{\ge 0}$ such that $m\ge n$.

Let $U_q^{+}(\mathfrak{g})$ (resp. $U_q^{-}(\mathfrak{g})$) be the subalgebra of $U_q(\mathfrak{g})$ generated by $e_i$’s (resp. $f_i$’s), and let $U^0_q(\mathfrak{g})$ be the subalgebra of $U_q(\mathfrak{g})$ generated by $q^{h}$ $(h \in \mathsf{P}^{\vee })$. Then we have the triangular decomposition

$$\begin{equation*} U_q(\mathfrak{g}) \simeq U^{-}_q(\mathfrak{g}) \otimes U^{0}_q(\mathfrak{g}) \otimes U^{+}_q(\mathfrak{g}), \end{equation*}$$

and the weight space decomposition

$$\begin{equation*} U_q(\mathfrak{g}) = \bigoplus _{\beta \in \mathsf{Q}} U_q(\mathfrak{g})_{\beta }, \end{equation*}$$

where $U_q(\mathfrak{g})_{\beta }\mathbin{:=}\left\{ x \in U_q(\mathfrak{g}) \mid \text{$q^{h}x q^{-h} =q^{\langle h, \beta \rangle }x$ for any $h \in \mathsf{P}$} \right\}$.

There are $\mathbb{Q}(q)$-algebra antiautomorphisms $\varphi$ and $^*$ of $U_q(\mathfrak{g})$ given as follows:

$$\begin{align*} &\varphi (e_i)=f_i, \quad \varphi (f_i)=e_i, \quad \varphi (q^h)=q^h, \\ &e_i^*=e_i, \quad \quad \ f_i^*=f_i, \quad \quad \ (q^h)^*=q^{-h}. \end{align*}$$

There is also a $\mathbb{Q}$-algebra automorphism $\overline{\phantom {a}}$ of $U_q(\mathfrak{g})$ given by

$$\begin{align*} & \overline{e}_i=e_i, \quad \overline{f}_i=f_i, \quad \overline{q^h}=q^{-h}, \quad \overline{q}=q^{-1}. \end{align*}$$

We define the divided powers by

$$\begin{equation*} e_i^{(n)} = e_i^n / [n]_i!, \quad f_i^{(n)} = f_i^n / [n]_i! \ \ (n \in \mathbb{Z}_{\ge 0}). \end{equation*}$$

Let us denote by $U_q(\mathfrak{g})_\mathbb{Z} [q^{\pm 1}]$ the $\mathbb{Z} [q^{\pm 1}]$-subalgebra of $U_q(\mathfrak{g})$ generated by $e_i^{(n)}$, $f_i^{(n)}$, $q^h$, and $\displaystyle \prod _{k=1}^n \dfrac{\{q^{1-k}q^h\}}{[k]}$ ($i\in I, \ n \in \mathbb{Z}_{\ge 0},\ h \in \mathsf{P}^\vee$), where $\{x\}:=(x-x^{-1})/(q-q^{-1})$. Let us also denote by $U_q^{-}(\mathfrak{g})_\mathbb{Z} [q^{\pm 1}]$ the $\mathbb{Z} [q^{\pm 1}]$-subalgebra of $U^-_q(\mathfrak{g})$ generated by $f_i^{(n)}$ ($i\in I$, $n \in \mathbb{Z}_{\ge 0}$), and by $U_q^{+}(\mathfrak{g})_\mathbb{Z} [q^{\pm 1}]$ the $\mathbb{Z} [q^{\pm 1}]$-subalgebra of $U^+_q(\mathfrak{g})$ generated by $e_i^{(n)}$ ($i\in I$, $n \in \mathbb{Z}_{\ge 0}$).

1.2. Integrable representations

A $U_q(\mathfrak{g})$-module $M$ is called integrable if $M= \bigoplus _{\eta \in \mathsf{P}} M_\eta$ where $M_\eta \mathbin{:=}\{ m \in M \ | \ q^hm=q^{\langle h,\eta \rangle } m \}$, $\dim M_\eta < \infty$, and the actions of $e_i$ and $f_i$ on $M$ are locally nilpotent for all $i \in I$. We denote by $\mathcal{O}_{\mathrm{int}}(\mathfrak{g})$ the category of integrable left $U_q(\mathfrak{g})$-modules $M$ satisfying that there exist finitely many weights $\lambda _1$, …, $\lambda _m$ such that $\operatorname {wt}(M)\subset \cup _{j}(\lambda _j+\mathsf{Q}^-)$. The category $\mathcal{O}_{\mathrm{int}}(\mathfrak{g})$ is semisimple with its simple objects being isomorphic to the highest weight modules $V(\lambda )$ with highest weight vector $u_\lambda$ of highest weight $\lambda \in \mathsf{P}^+ :=\left\{\mu \in \mathsf{P} \mid \langle h_i, \mu \rangle \ge 0 \ \text{for all} \ i \in I \right\}$, the set of dominant integral weights.

For $M \in \mathcal{O}_{\mathrm{int}}(\mathfrak{g})$, let us denote by $\mathbf{D}_\varphi M$ the left $U_q(\mathfrak{g})$-module $\bigoplus _{\eta \in \mathsf{P}} \operatorname {Hom}_{\mathbb{Q}(q)}(M_\eta ,\mathbb{Q}(q))$ with the action of $U_q(\mathfrak{g})$ given by

$$\begin{equation*} (a\psi )(m)=\psi (\varphi (a)m)\quad \text{for $\psi \in \mathbf{D}_\varphi M$, $m\in M$, and $a\in U_q(\mathfrak{g})$.} \end{equation*}$$

Then $\mathbf{D}_\varphi M$ belongs to $\mathcal{O}_{\mathrm{int}}(\mathfrak{g})$.

For a left $U_q(\mathfrak{g})$-module $M$, we denote by $M^{\mspace {1mu}\mathrm{r}}$ the right $U_q(\mathfrak{g})$-module $\{m^{\mspace {1mu}\mathrm{r}}\ | \ m\in M \}$ with the right action of $U_q(\mathfrak{g})$ given by

$$\begin{equation*} \text{$(m^{\mspace {1mu}\mathrm{r}})\, x=(\varphi (x)m)^{\mspace {1mu}\mathrm{r}}$ for $m\in M$ and $x\in U_q(\mathfrak{g})$.} \end{equation*}$$

We denote by $\mathcal{O}_{\mathrm{int}}^{{\mspace {1mu}\mathrm{r}}} (\mathfrak{g})$ the category of right integrable $U_q(\mathfrak{g})$-modules $M^{\mspace {1mu}\mathrm{r}}$ such that $M \in \mathcal{O}_{\mathrm{int}}(\mathfrak{g})$.

There are two comultiplications $\Delta _+$ and $\Delta _-$ on $U_q(\mathfrak{g})$ defined as follows:

$$\begin{align} &\Delta _+(e_i)=e_i \mathop{\otimes }1 + t_i \mathop{\otimes }e_i,\quad \Delta _+(f_i)=f_i \mathop{\otimes }t_i^{-1} + 1 \mathop{\otimes }f_i,\quad \Delta _+(q^h)=q^h \mathop{\otimes }q^h, {\tag{1.1}}\\ &\Delta _-(e_i)=e_i \mathop{\otimes }t_i^{-1} + 1 \mathop{\otimes }e_i,\quad \Delta _-(f_i)=f_i \mathop{\otimes }1 + t_i \mathop{\otimes }f_i,\quad \Delta _-(q^h)=q^h \mathop{\otimes }q^h. {\tag{1.2}} \end{align}$$

For two $U_q(\mathfrak{g})$-modules $M_1$ and $M_2$, let us denote by $M_1 \otimes _{_+}\mspace {-1mu}M_2$ and $M_1 \otimes _{_-}\mspace {-1mu}M_2$ the vector space $M_1 \mathop{\otimes }_{\mathbb{Q}(q)}M_2$ endowed with $U_q(\mathfrak{g})$-module structure induced by the comultiplications $\Delta _+$ and $\Delta _-$, respectively. Then we have

$$\begin{equation*} \mathbf{D}_\varphi (M_1 \otimes _{\pm } M_2 ) \simeq (\mathbf{D}_\varphi M_1) \otimes _{\mp } (\mathbf{D}_\varphi M_2). \end{equation*}$$

For any $i \in I$, there exists a unique $\mathbb{Q}(q)$-linear endomorphism $e'_i$ of $U^-_q(\mathfrak{g})$ such that

$$\begin{eqnarray*} e'_i(f_j)=\delta _{i,j} \ (j \in I), & e'_i(xy) = (e'_i x) y + q_i^{\langle h_i, \beta \rangle } x (e'_iy) \ (x \in U^{-}_q(\mathfrak{g})_{\beta }, y \in U^-_q(\mathfrak{g})). \end{eqnarray*}$$

The quantum boson algebra $B_q(\mathfrak{g})$ is defined as the subalgebra of $\operatorname {End}_{\mathbb{Q}(q)}(U_q(\mathfrak{g}))$ generated by $f_i$ and $e'_i$ $(i \in I)$. Then $B_q(\mathfrak{g})$ has a $\mathbb{Q}(q)$-algebra anti-automorphism $\varphi$ which sends $e_i'$ to $f_i$ and $f_i$ to $e_i'$. As a $B_q(\mathfrak{g})$-module, $U^-_q(\mathfrak{g})$ is simple.

The simple $U_q(\mathfrak{g})$-module $V(\lambda )$ and the simple $B_q(\mathfrak{g})$-module $U_q^-(\mathfrak{g})$ have a unique non-degenerate symmetric bilinear form $( \ , \ )$ such that

$$\begin{align*} & (u_\lambda ,u_\lambda ) = 1 \text{ and } (xu,v)=(u,\varphi (x)v) \text{ for } u,v \in V(\lambda ) \text{ and } x \in U_q(\mathfrak{g}), \\ & \ ( {\mathbf{{1}}},{\mathbf{{1}}}) = 1 \text{ and } (xu,v)=(u,\varphi (x)v) \text{ for } u,v \in U^-_q(\mathfrak{g}) \text{ and } x \in B_q(\mathfrak{g}). \end{align*}$$

Note that $( \ , \ )$ induces the non-degenerate bilinear form

$$\begin{equation*} \langle \cdot , \cdot \rangle \colon V(\lambda )^{\mspace {1mu}\mathrm{r}}\times V(\lambda ) \to \mathbb{Q}(q) \end{equation*}$$

given by $\langle u^{\mspace {1mu}\mathrm{r}}, v \rangle = (u,v)$, by which $\mathbf{D}_\varphi V(\lambda )$ is canonically isomorphic to $V(\lambda )$.

1.3. Crystal bases and global bases

For a subring $A$ of $\mathbb{Q}(q)$, we say that $L$ is an $A$-lattice of a $\mathbb{Q}(q)$-vector space $V$ if $L$ is a free $A$-submodule of $V$ such that $V=\mathbb{Q}(q)\mathop{\otimes }_AL$.

Let us denote by $\mathbf{A}_0$ (resp. $\mathbf{A}_\infty$) the ring of rational functions in $\mathbb{Q}(q)$ which are regular at $q=0$ (resp. $q=\infty$). Set $\mathbf{A}\mathbin{:=}\mathbb{Q}[q^{\pm 1}]$.

Let $M$ be a $U_q(\mathfrak{g})$-module in $\mathcal{O}_{\mathrm{int}}(\mathfrak{g})$. Then, for each $i \in I$, any $u\in M$ can be uniquely written as

$$\begin{equation*} u=\sum _{n \ge 0}f_i^{(n)}u_n\quad \text{with $e_iu_n=0$.} \end{equation*}$$

We define the lower Kashiwara operators by

$$\begin{eqnarray*} &&\tilde{e}^{\mathrm{low}}_i(u)=\sum _{n\ge 1}f_i^{(n-1)}u_n \quad \text{and}\quad \tilde{f}^{\mathrm{low}}_i(u)=\sum _{n\ge 0}f_i^{(n+1)}u_n, \end{eqnarray*}$$

and the upper Kashiwara operators by

$$\begin{eqnarray*} && \tilde{e}^\mathrm{up}_i(u)=\tilde{e}_i^{{\mathrm{low}}}q_i^{-1} t_i^{-1} u\quad \text{and}\quad \tilde{f}^\mathrm{up}_i(u)=\tilde{f}_i^{{\mathrm{low}}}q_i^{-1} t_i u. \end{eqnarray*}$$

Similarly, for each $i \in I$, any element $x \in U_q^-(\mathfrak{g})$ can be written uniquely as

$$\begin{eqnarray*} x = \sum _{n \ge 0} f_i^{(n)} x_n\quad \text{ with $e'_i x_n =0$.} \end{eqnarray*}$$

We define the Kashiwara operators $\tilde{e}_i, \tilde{f}_i$ on $U_q^-(\mathfrak{g})$ by

$$\begin{eqnarray*} \tilde{e}_i x = \sum _{n \ge 1} f_i^{(n-1)}x_n, & \tilde{f}_i x = \displaystyle \sum _{n \ge 0} f_i^{(n+1)}x_n. \end{eqnarray*}$$

We say that an $\mathbf{A}_0$-lattice $L$ of $M$ is a lower (resp. upper) crystal lattice of $M$ if $L=\mathop{\textstyle \bigoplus }\limits _{\eta \in \mathsf{P}}L_\eta$, where $L_\eta = L \cap M_\eta$ and it is invariant by the lower (resp. upper) Kashiwara operators.

Lemma 1.3.1.

Let $L$ be a lower crystal lattice of $M\in \mathcal{O}_{\mathrm{int}}(\mathfrak{g})$. Then we have

(i)

$\mathop{\textstyle \bigoplus }\limits \nolimits _{\lambda \in \mathsf{P}}q^{-(\lambda ,\lambda )/2}L_\lambda$ is an upper crystal lattice of $M$.

(ii)

$L^\vee \mathbin{:=}\left\{\psi \in \mathbf{D}_\varphi M \mid \langle \psi ,L\rangle \in \mathbf{A}_0 \right\}$ is an upper crystal lattice of $\mathbf{D}_\varphi M$.

Proof.

(i) Let $\phi _M$ be the endomorphism of $M$ given by $\phi _M\vert _{M_\lambda }=q^{-(\lambda ,\lambda )/2}\operatorname {id}_{M_\lambda }$. Then we have $\tilde{e}_i^\mathrm{up}=\phi _M\circ \tilde{e}_i^{\mathrm{low}}\circ \phi _M^{-1}$ and $\tilde{f}_i^\mathrm{up}=\phi _M\circ \tilde{f}_i^{\mathrm{low}}\circ \phi _M^{-1}$.

Item (ii) follows from $(3.2.1)$, $(3.2.2)$ in 17. Note that the definition of upper Kashiwara operators are slightly different from the ones in 17, but similar properties hold.

■

Definition 1.3.2.

A lower (resp. upper) crystal basis of $M$ consists of a pair $(L,B)$ satisfying the following conditions:

(i)

$L$ is a lower (resp. upper) crystal lattice of $M$,

(ii)

$B= \sqcup _{\eta \in \mathsf{P}} B_\eta$ is a basis of the $\mathbb{Q}$-vector space $L/qL$, where $B_\eta =B \cap (L_\eta /qL_\eta )$,

(iii)

the induced maps $\tilde{e}_i$ and $\tilde{f}_i$ on $L/qL$ satisfy$$\begin{equation*} \tilde{e}_iB, \tilde{f}_iB \subset B \sqcup \{0\}, \text{ and } \tilde{f}_ib=b' \text{ if and only if } b=\tilde{e}_ib' \text{ for } b,b' \in B. \end{equation*}$$

Here $\tilde{e}_i$ and $\tilde{f}_i$ denote the lower (resp. upper) Kashiwara operators.

For $\lambda \in \mathsf{P}^+$, let $u_\lambda$ be the highest weight vector of $V(\lambda )$. Let $L^{\mathrm{low}}(\lambda )$ be the $\mathbf{A}_0$-submodule of $V(\lambda )$ generated by $\left\{\tilde{f}_{i_1} \cdots \tilde{f}_{i_l} u_\lambda \mid l \in \mathbb{Z}_{\ge 0}, \ i_1 , \ldots , i_l \in I \right\}$, and let $B(\lambda )$ be the subset of $L^{\mathrm{low}}(\lambda ) / qL^{\mathrm{low}}(\lambda )$ given by

$$\begin{eqnarray*} B^{\mathrm{low}}(\lambda ) = \left\{\tilde{f}_{i_1} \cdots \tilde{f}_{i_l} u_\lambda \mod q L(\lambda ) \mid l \in \mathbb{Z}_{\ge 0}, \ i_1, \ldots , i_l \in I \right\} \backslash \{0\}. \end{eqnarray*}$$

It is shown in 16 that $(L^{\mathrm{low}}(\lambda ),B^{\mathrm{low}}(\lambda ))$ is a lower crystal basis of $V(\lambda )$. Using the non-degenerate symmetric bilinear form $( \ , \ )$, $V(\lambda )$ has the upper crystal basis $(L^\mathrm{up}(\lambda ),B^\mathrm{up}(\lambda ))$ where

$$\begin{equation*} L^\mathrm{up}(\lambda ) \mathbin{:=}\{ u \in V(\lambda ) \ | \ (u,L^{{\mathrm{low}}} (\lambda )) \subset \mathbf{A}_0 \}, \end{equation*}$$

and $B^\mathrm{up}(\lambda )\subset L^\mathrm{up}(\lambda )/qL^\mathrm{up}(\lambda )$ is the dual basis of $B^{\mathrm{low}}(\lambda )$ with respect to the induced non-degenerate pairing between $L^\mathrm{up}(\lambda )/qL^\mathrm{up}(\lambda )$ and $L^{\mathrm{low}}(\lambda )/qL^{\mathrm{low}}(\lambda )$.

An (abstract) crystal is a set $B$ together with maps

$$\begin{equation*} \operatorname {wt}\colon B \to \mathsf{P}, \ \ \varepsilon _i,\varphi _i\colon B \to \mathbb{Z}\sqcup \{ \infty \} \text{ and } \tilde{e}_i,\tilde{f}_i\colon B \to B \sqcup \{ 0 \} \text{ for } i \in I, \end{equation*}$$

such that

(C1)

$\varphi _i(b)=\varepsilon _i(b)+\langle h_i,\operatorname {wt}(b) \rangle$ for any $i$,

(C2)

if $b \in B$ satisfies $\tilde{e}_i(b) \ne 0$, then$$\begin{equation*} \varepsilon _i(\tilde{e}_ib)=\varepsilon _i(b)-1, \ \varphi _i(\tilde{e}_ib)=\varphi _i(b)+1, \ \operatorname {wt}(\tilde{e}_i b)=\operatorname {wt}(b)+\alpha _i, \end{equation*}$$

(C3)

if $b \in B$ satisfies $\tilde{f}_i(b) \ne 0$, then$$\begin{equation*} \varepsilon _i(\tilde{f}_ib)=\varepsilon _i(b)+1, \ \varphi _i(\tilde{f}_ib)=\varphi _i(b)-1, \ \operatorname {wt}(\tilde{f}_i b)=\operatorname {wt}(b)-\alpha _i, \end{equation*}$$

(C4)

for $b,b' \in B$, $b'=\tilde{f}_i b$ if and only if $b=\tilde{e}_i b'$,

(C5)

if $\varphi _i(b)=-\infty$, then $\tilde{e}_ib=\tilde{f}_ib=0$.

Recall that, with the notions of morphism and tensor product rule of crystals, the category of crystals becomes a monoidal category 19. If $(L,B)$ is a crystal basis of $M$, then $B$ is an abstract crystal. Since $B^{\mathrm{low}}(\lambda ) \simeq B^\mathrm{up}(\lambda )$, we drop the superscripts for simplicity.

Let $V$ be a $\mathbb{Q}(q)$-vector space, and let $L_0$ be an $\mathbf{A}_0$-lattice of $V$, $L_\infty$ an $\mathbf{A}_\infty$-lattice of $V$, and $V_{\mathbf{A}}$ an $\mathbf{A}$-lattice of $V$. We say that the triple $(V_{\mathbf{A}},L_0,L_\infty )$ is balanced if the following canonical map is a $\mathbb{Q}$-linear isomorphism:

$$\begin{equation*} E\mathbin{:=}V_{\mathbf{A}} \cap L_0 \cap L_\infty \similarrightarrow L_0/qL_0. \end{equation*}$$

The inverse of the above isomorphism $G\colon L_0/qL_0\similarrightarrow E$ is called the globalizing map. If $(V_{\mathbf{A}},L_0,L_\infty )$ is balanced, then we have

$$\begin{eqnarray*} &&\text{$\mathbb{Q}(q)\mathop{\otimes }_\mathbb{Q}E\similarrightarrow V$, $\mathbf{A}\mathop{\otimes }_\mathbb{Q}E\similarrightarrow V_{\mathbf{A}}$, $\mathbf{A}_0\mathop{\otimes }_\mathbb{Q}E\similarrightarrow L_0$, and $\mathbf{A}_\infty \mathop{\otimes }_\mathbb{Q}E\similarrightarrow L_\infty $.} \end{eqnarray*}$$

Hence, if $B$ is a basis of $L_0/qL_0$, then $G(B)$ is a basis of $V$, $V_{\mathbf{A}}$, $L_0$, and $L_\infty$. We call $G(B)$ a global basis.

We define the two $\mathbf{A}$-lattices of $V(\lambda )$ by

$$\begin{eqnarray*} &&V^{\mathrm{low}}(\lambda )_\mathbf{A}\mathbin{:=}\bigl (\mathbb{Q}\mathop{\otimes }U_q^-(\mathfrak{g})_{\mathbb{Z}[q^{\pm 1}]}\bigr )u_\lambda \quad \text{and}\\ &&V^\mathrm{up}(\lambda )_\mathbf{A}\mathbin{:=}\left\{ u \in V(\lambda ) \mid \bigl (u, V^{\mathrm{low}}(\lambda )_\mathbf{A}\bigr )\subset \mathbf{A} \right\}. \end{eqnarray*}$$

Recall that there is a $\mathbb{Q}$-linear automorphism—on $V(\lambda )$ defined by

$$\begin{equation*} \overline{P u_{\lambda }}=\overline{P} u_\lambda ,\quad \text{for } \ P \in {U_q(\mathfrak{g})}. \end{equation*}$$

Then $\bigl (V^{\mathrm{low}}(\lambda )_\mathbf{A},\; L^{\mathrm{low}}(\lambda ),\; \overline{ L^{\mathrm{low}}(\lambda )}\bigr )$ and $\bigl (V^\mathrm{up}(\lambda )_\mathbf{A},\; L^\mathrm{up}(\lambda ),\; \overline{ L^\mathrm{up}(\lambda )}\bigr )$ are balanced. Let us denote by $G^{\mathrm{low}}_\lambda$ and $G^\mathrm{up}_\lambda$ the associated globalizing maps, respectively. (If there is no danger of confusion, we simply denote them $G^{\mathrm{low}}$ and $G^\mathrm{up}$, respectively.) Then the sets

$$\begin{equation*} {\mathbf{B}}^{\mathrm{low}}(\lambda ) \mathbin{:=}\{G^{\mathrm{low}}_\lambda (b) \ | \ b \in B^{\mathrm{low}}(\lambda ) \} \ \ \text{ and } \ \ {\mathbf{B}}^\mathrm{up}(\lambda ) \mathbin{:=}\{G^\mathrm{up}_\lambda (b) \ | \ b \in B^\mathrm{up}(\lambda ) \} \end{equation*}$$

form $\mathbb{Z} [q^{\pm 1}]$-bases of

$$\begin{eqnarray*} && V^{\mathrm{low}}(\lambda )_\mathbb{Z} [q^{\pm 1}]:=U_q(\mathfrak{g})_\mathbb{Z} [q^{\pm 1}]u_\lambda \quad \text{and} \\ &&V^\mathrm{up}(\lambda )_\mathbb{Z} [q^{\pm 1}]\mathbin{:=}\left\{ u \in V(\lambda ) \mid \bigl (u, V^{\mathrm{low}}(\lambda )_\mathbb{Z} [q^{\pm 1}]\bigr )\subset \mathbb{Z} [q^{\pm 1}] \right\}, \end{eqnarray*}$$

respectively. They are called the lower global basis and the upper global basis of $V(\lambda )$.

Set

$$\begin{eqnarray*} &&L(\infty ) := \sum _{l \in \mathbb{Z}_{\ge 0}, \, i_1, \ldots , i_l \in I} \mathbf{A}_0 \tilde{f}_{i_1} \cdots \tilde{f}_{i_l} \cdot {\mathbf{{1}}}\subset U_q^-(\mathfrak{g}) \quad \text{and} \\ &&B(\infty ) := \left\{\tilde{f}_{i_1} \cdots \tilde{f}_{i_l} \cdot {\mathbf{{1}}}\mod q L(\infty ) \mid l \in \mathbb{Z}_{\ge 0}, i_1, \ldots , i_l \in I \right\} {\subset }\, L(\infty ) / q L(\infty ). \end{eqnarray*}$$

Then $(L(\infty ),B(\infty ))$ is a lower crystal basis of the simple $B_q(\mathfrak{g})$-module $U_q^-(\mathfrak{g})$ and the triple $( \mathbb{Q}\mathop{\otimes }U_q^-(\mathfrak{g})_{\mathbb{Z}[q^{\pm 1}]}, L(\infty ),\overline{L(\infty )})$ is balanced. Let us denote the globalizing map by $G^{{\mathrm{low}}}$. Then the set

$$\begin{equation*} {\mathbf{B}}^{\mathrm{low}}(U^-_q(\mathfrak{g}))\mathbin{:=}\{ G^{\mathrm{low}}(b )\ | \ b \in B(\infty ) \} \end{equation*}$$

forms a $\mathbb{Z} [q^{\pm 1}]$-basis of $U^-_q(\mathfrak{g})_\mathbb{Z} [q^{\pm 1}]$ and is called the lower global basis of $U_q^-(\mathfrak{g})$.

Let us denote by

$$\begin{eqnarray} {\mathbf{B}}^\mathrm{up}(U^-_q(\mathfrak{g})) \mathbin{:=}\{G^\mathrm{up}(b) \ | \ b \in B(\infty ) \} {\tag{1.3}} \end{eqnarray}$$

the dual basis of ${\mathbf{B}}^{\mathrm{low}}(U^-_q(\mathfrak{g}))$ with respect to $( \ , \ )$. Then it is a $\mathbb{Z} [q^{\pm 1}]$-basis of

$$\begin{equation*} U^-_q(\mathfrak{g})_\mathbb{Z} [q^{\pm 1}]^\vee \mathbin{:=}\{ x \in U^-_q(\mathfrak{g}) \ | \ (x,U^-_q(\mathfrak{g})_\mathbb{Z} [q^{\pm 1}]) \subset \mathbb{Z} [q^{\pm 1}]\} \end{equation*}$$

and called the upper global basis of $U_q^-(\mathfrak{g})$. Note that $U^-_q(\mathfrak{g})_\mathbb{Z} [q^{\pm 1}]^\vee$ has a $\mathbb{Z} [q^{\pm 1}]$-algebra structure as a subalgebra of $U^-_q(\mathfrak{g})$ (see also Section 8.2).

2. KLR algebras and R-matrices

2.1. KLR algebras

We recall the definition of Khovanov–Lauda–Rouquier algebra or quiver Hecke algebra (hereafter, we abbreviate it as KLR algebra) associated with a given Cartan datum $(A, \mathsf{P}, \Pi , \mathsf{P}^{\vee }, \Pi ^{\vee })$.

Let $\mathbf{k}$ be a base field. For $i,j\in I$ such that $i\not =j$, set

$$\begin{equation*} S_{i,j}=\left\{(p,q)\in \mathbb{Z}_{\ge 0}^2 \mid (\alpha _i , \alpha _i)p+(\alpha _j , \alpha _j)q=-2(\alpha _i , \alpha _j) \right\}. \end{equation*}$$

Let us take a family of polynomials $(Q_{ij})_{i,j\in I}$ in $\mathbf{k}[u,v]$ which are of the form

$$\begin{gather} Q_{ij}(u,v) = \begin{cases} \hspace{5ex} 0 \ \ & \text{if $i=j$,} \\[7.5pt] \sum \limits _{(p,q)\in S_{i,j}} t_{i,j;p,q} u^p v^q\quad & \text{if $i \neq j$} \end{cases} {\tag{2.1}}\\ \text{with $t_{i,j;p,q}\in \mathbf{k}$ such that $Q_{i,j}(u,v)=Q_{j,i}(v,u)$ and $t_{i,j:-a_{ij},0} \in \mathbf{k}^{\times }$.} \end{gather}$$

We denote by $\mathfrak{S}_{n} = \langle s_1, \ldots , s_{n-1} \rangle$ the symmetric group on $n$ letters, where $s_i\mathbin{:=}(i, i+1)$ is the transposition of $i$ and $i+1$. Then $\mathfrak{S}_n$ acts on $I^n$ by place permutations.

For $n \in \mathbb{Z}_{\ge 0}$ and $\beta \in \mathsf{Q}^+$ such that $|\beta | = n$, we set

$$\begin{equation*} I^{\beta } = \left\{\nu = (\nu _1, \ldots , \nu _n) \in I^{n} \mid \alpha _{\nu _1} + \cdots + \alpha _{\nu _n} = \beta \right\}. \end{equation*}$$

Definition 2.1.1.

For $\beta \in \mathsf{Q}^+$ with $|\beta |=n$, the KLR algebra $R(\beta )$ at $\beta$ associated with a Cartan datum $(A,\mathsf{P}, \Pi ,\mathsf{P}^{\vee },\Pi ^{\vee })$ and a matrix $(Q_{ij})_{i,j \in I}$ is the algebra over $\mathbf{k}$ generated by the elements $\{ e(\nu ) \}_{\nu \in I^{\beta }}$, $\{x_k \}_{1 \le k \le n}$, $\{ \tau _m \}_{1 \le m \le n-1}$ satisfying the following defining relations:

$$\begin{equation*} \begin{aligned} & e(\nu ) e(\nu ') = \delta _{\nu , \nu '} e(\nu ), \ \ \sum _{\nu \in I^{\beta } } e(\nu ) = 1, \\ & x_{k} x_{m} = x_{m} x_{k}, \ \ x_{k} e(\nu ) = e(\nu ) x_{k}, \\ & \tau _{m} e(\nu ) = e(s_{m}(\nu )) \tau _{m}, \ \ \tau _{k} \tau _{m} = \tau _{m} \tau _{k} \ \ \text{if} \ |k-m|>1, \\ & \tau _{k}^2 e(\nu ) = Q_{\nu _{k}, \nu _{k+1}} (x_{k}, x_{k+1}) e(\nu ), \\ & (\tau _{k} x_{m} - x_{s_k(m)} \tau _{k}) e(\nu ) = \begin{cases} -e(\nu ) \ \ & \text{if} \ m=k, \nu _{k} = \nu _{k+1}, \\ e(\nu ) \ \ & \text{if} \ m=k+1, \nu _{k}=\nu _{k+1}, \\ 0 \ \ & \text{otherwise}, \end{cases} \\ & (\tau _{k+1} \tau _{k} \tau _{k+1}-\tau _{k} \tau _{k+1} \tau _{k}) e(\nu )\\ & =\begin{cases} \dfrac{Q_{\nu _{k}, \nu _{k+1}}(x_{k}, x_{k+1}) - Q_{\nu _{k}, \nu _{k+1}}(x_{k+2}, x_{k+1})}{x_{k} - x_{k+2}}e(\nu ) \ \ & \text{if} \ \nu _{k} = \nu _{k+2}, \\ 0 \ \ & \text{otherwise}. \end{cases} \end{aligned} \end{equation*}$$

The above relations are homogeneous provided that

$$\begin{equation*} \deg e(\nu ) =0, \quad \deg \, x_{k} e(\nu ) = (\alpha _{\nu _k} , \alpha _{\nu _k}), \quad \deg \, \tau _{l} e(\nu ) = - (\alpha _{\nu _l} , \alpha _{\nu _{l+1}}), \end{equation*}$$

and hence $R( \beta )$ is a $\mathbb{Z}$-graded algebra.

For a graded $R(\beta )$-module $M=\bigoplus _{k \in \mathbb{Z}} M_k$, we define $qM =\bigoplus _{k \in \mathbb{Z}} (qM)_k$, where

$$\begin{align*} (qM)_k = M_{k-1} & \ (k \in \mathbb{Z}). \end{align*}$$

We call $q$ the grading shift functor on the category of graded $R(\beta )$-modules.

If $M$ is an $R(\beta )$-module, then we set $\operatorname {wt}(M)=-\beta \in \mathsf{Q}^-$ and call it the weight of $M$.

We denote by $R(\beta ) \text{-Mod}$ the category of $R(\beta )$-modules, and by $R(\beta ) \text{-mod}$ the full subcategory of $R(\beta )\text{-Mod}$ consisting of modules $M$ such that $M$ are finite-dimensional over $\mathbf{k}$, and the actions of the $x_k$’s on $M$ are nilpotent.

Similarly, we denote by $R(\beta )\text{-gMod}$ and by $R(\beta )\text{-gmod}$ the category of graded $R(\beta )$-modules and the category of graded $R(\beta )$-modules which are finite-dimensional over $\mathbf{k}$, respectively. We set

$$\begin{equation*} R\text{-gmod}=\mathop{\textstyle \bigoplus }\limits _{\beta \in \mathsf{Q}^+}R(\beta )\text{-gmod}\quad \text{and}\quad R\text{-mod}=\mathop{\textstyle \bigoplus }\limits _{\beta \in \mathsf{Q}^+}R(\beta )\text{-mod}. \end{equation*}$$

For $\beta , \gamma \in \mathsf{Q}^+$ with $|\beta |=m$, $|\gamma |= n$, set

$$\begin{equation*} e(\beta ,\gamma )=\displaystyle \sum _{\substack{\nu \in I^{\beta +\gamma }, \\ (\nu _1, \ldots ,\nu _m) \in I^{\beta }, \\ (\nu _{m+1}, \ldots ,\nu _{m+n} ) \in I^{\gamma } }} e(\nu ) \in R(\beta +\gamma ). \end{equation*}$$

Then $e(\beta ,\gamma )$ is an idempotent. Let

$$\begin{eqnarray*} R( \beta )\mathop{\otimes }R( \gamma )\to e(\beta ,\gamma )R( \beta +\gamma )e(\beta ,\gamma ) \end{eqnarray*}$$

be the $\mathbf{k}$-algebra homomorphism given by $e(\mu )\mathop{\otimes }e(\nu )\mapsto e(\mu *\nu )$ ($\mu \in I^{\beta }$ and $\nu \in I^{\gamma }$) $x_k\mathop{\otimes }1\mapsto x_ke(\beta ,\gamma )$ ($1\le k\le m$), $1\mathop{\otimes }x_k\mapsto x_{m+k}e(\beta ,\gamma )$ ($1\le k\le n$), $\tau _k\mathop{\otimes }1\mapsto \tau _ke(\beta ,\gamma )$ ($1\le k<m$), and $1\mathop{\otimes }\tau _k\mapsto \tau _{m+k}e(\beta ,\gamma )$ ($1\le k<n$). Here $\mu *\nu$ is the concatenation of $\mu$ and $\nu$; i.e., $\mu *\nu =(\mu _1,\ldots ,\mu _m,\nu _1,\ldots ,\nu _n)$.

For an $R(\beta )$-module $M$ and an $R(\gamma )$-module $N$, we define the convolution product $M\circ N$ by

$$\begin{equation*} M\circ N=R(\beta + \gamma ) e(\beta ,\gamma ) \mathop{\otimes }_{R(\beta )\otimes R( \gamma )}(M\otimes N). \end{equation*}$$

For $M \in R( \beta ) \text{-mod}$, the dual space

$$\begin{equation*} M^* \mathbin{:=}\operatorname {Hom}_{\mathbf{k}}(M, \mathbf{k}) \end{equation*}$$

admits an $R(\beta )$-module structure via

$$\begin{align*} (r \cdot f)(u) \mathbin{:=}f(\psi (r) u) \quad (r \in R( \beta ), \ u \in M), \end{align*}$$

where $\psi$ denotes the $\mathbf{k}$-algebra anti-involution on $R(\beta )$ which fixes the generators $e(\nu )$, $x_m$, and $\tau _k$ for $\nu \in I^{\beta }, 1 \leq m \leq |\beta |$, and $1 \leq k<|\beta |$.

It is known that (see 28, Theorem 2.2 (2))

$$\begin{eqnarray*} &&(M_1 \circ M_2)^* \simeq q^{(\beta ,\gamma )} (M_2^* \circ M_1 ^*) \end{eqnarray*}$$

for any $M_1 \in R(\beta ) \text{-gmod}$ and $M_2 \in R(\gamma ) \text{-gmod}$.

A simple module $M$ in $R \text{-gmod}$ is called self-dual if $M^* \simeq M$. Every simple module is isomorphic to a grading shift of a self-dual simple module 21, Section 3.2. Note also that we have $\operatorname {End}_{R(\beta )}{M} \simeq \mathbf{k}$ for every simple module $M$ in $R(\beta ) \text{-gmod}$ 21, Corollary 3.19.

Let us denote by $K(R\text{-gmod})$ the Grothendieck group of $R\text{-gmod}$. Then, $K(R\text{-gmod})$ is an algebra over $\mathbb{Z} [q^{\pm 1}]$ with the multiplication induced by the convolution product and the $\mathbb{Z} [q^{\pm 1}]$-action induced by the grading shift functor $q$.

In 2138, it is shown that a KLR algebra categorifies the negative half of the corresponding quantum group. More precisely, we have the following theorem.

Theorem 2.1.2 (2138).

For a given Cartan datum $(A,\mathsf{P}, \Pi ,\mathsf{P}^{\vee },\Pi ^{\vee })$, we take a parameter matrix $(Q_{ij})_{i,j \in {J}}$ satisfying the conditions in 2.1, and let $U_q(\mathfrak{g})$ and $R(\beta )$ be the associated quantum group and the KLR algebras, respectively. Then there exists a $\mathbb{Z} [q^{\pm 1}]$-algebra isomorphism

$$\begin{align} U^-_q(\mathfrak{g})_\mathbb{Z} [q^{\pm 1}]^{\vee } \simeq K(R\text{-gmod}). {\tag{2.2}} \end{align}$$

KLR algebras also categorify the upper global bases.

Definition 2.1.3.

We say that a KLR algebra $R$ is symmetric if $Q_{i,j}(u,v)$ is a polynomial in $u-v$ for all $i,j\in I$.

In particular, the corresponding generalized Cartan matrix $A$ is symmetric. In symmetric case, we assume $(\alpha _i,\alpha _i)=2$ for $i \in I$.

Theorem 2.1.4 (3940).

Assume that the KLR algebra $R$ is symmetric and the base field $\mathbf{k}$ is of characteristic $0$. Then under the isomorphism 2.2 in Theorem 2.1.2, the upper global basis corresponds to the set of the isomorphism classes of self-dual simple $R$-modules.

2.2. R-matrices for KLR algebras

For $|\beta |=n$ and $1\le a<n$, we define $\varphi _a\in R( \beta )$ by

$$\begin{eqnarray*} &&\begin{array}{l} \varphi _a e(\nu )= \begin{cases} \bigl (\tau _ax_a-x_{a}\tau _a\bigr )e(\nu ) & \text{if $\nu _a=\nu _{a+1}$,} \\[10.0pt] \tau _ae(\nu )& \text{otherwise.} \end{cases} \end{array} \end{eqnarray*}$$

They are called the intertwiners. Since $\{\varphi _a\}_{1\le a<n}$ satisfies the braid relation, $\varphi _w\mathbin{:=}\varphi _{i_1}\cdots \varphi _{i_\ell }$ does not depend on the choice of reduced expression $w=s_{i_1}\cdots s_{i_\ell }$.

For $m,n\in \mathbb{Z}_{\ge 0}$, let us denote by $w[{m,n}]$ the element of $\mathfrak{S}_{m+n}$ defined by

$$\begin{eqnarray*} &&w[{m,n}](k)=\begin{cases} k+n&\text{if $1\le k\le m$,}\\ k-m&\text{if $m<k\le m+n$.}\end{cases} \end{eqnarray*}$$

Let $\beta ,\gamma \in \mathsf{Q}^+$ with $|\beta |=m$, $|\gamma |=n$, and let $M$ be an $R(\beta )$-module and $N$ an $R(\gamma )$-module. Then the map $M\mathop{\otimes }N\to N\circ M$ given by $u\mathop{\otimes }v\longmapsto \varphi _{w[n,m]}(v\mathop{\otimes }u)$ is $R( \beta )\mathop{\otimes }R(\gamma )$-linear, and hence it extends to an $R( \beta +\gamma )$-module homomorphism

$$\begin{eqnarray*} &&R_{M,N}\colon M\circ N\xrightarrow {\,\hspace{2ex}\,}N\circ M. \end{eqnarray*}$$

Assume that the KLR algebra $R(\beta )$ is symmetric. Let $z$ be an indeterminate which is homogeneous of degree $2$, and let $\psi _z$ be the graded algebra homomorphism

$$\begin{eqnarray*} &&\psi _z\colon R( \beta )\to \mathbf{k}[z]\mathop{\otimes }R( \beta ) \end{eqnarray*}$$

given by

$$\begin{equation*} \psi _z(x_k)=x_k+z,\quad \psi _z(\tau _k)=\tau _k, \quad \psi _z(e(\nu ))=e(\nu ). \end{equation*}$$

For an $R( \beta )$-module $M$, we denote by $M_z$ the $\bigl (\mathbf{k}[z]\mathop{\otimes }R( \beta )\bigr )$-module $\mathbf{k}[z]\mathop{\otimes }M$ with the action of $R( \beta )$ twisted by $\psi _z$. Namely,

$$\begin{eqnarray*} && \begin{array}{l} e(\nu )(a\mathop{\otimes }u)=a\mathop{\otimes }e(\nu )u,\\[5.0pt] x_k(a\mathop{\otimes }u)=(za)\mathop{\otimes }u+a\mathop{\otimes }(x_ku),\\[5.0pt] \tau _k(a\mathop{\otimes }u)=a\mathop{\otimes }(\tau _k u) \end{array} \end{eqnarray*}$$

for $\nu \in I^\beta$, $a\in \mathbf{k}[z]$, and $u\in M$. Note that the multiplication by $z$ on $\mathbf{k}[z]$ induces an $R(\beta )$-module endomorphism on $M_z$. For $u\in M$, we sometimes denote by $u_z$ the corresponding element $1\mathop{\otimes }u$ of the $R( \beta )$-module $M_z$.

For a non-zero $M\in R(\beta )\text{-mod}$ and a non-zero $N\in R(\gamma )\text{-mod}$,

(2.3)

let $s$ be the order of zero of $R_{M_z,N}\colon M_z\circ N\xrightarrow {\,\hspace{2ex}\,}N\circ M_z$; i.e., the largest non-negative integer such that the image of $R_{M_z,N}$ is contained in $z^s(N\circ M_z)$.

Note that such an $s$ exists because $R_{M_z,N}$ does not vanish 14, Proposition 1.4.4 (iii). We denote by $R^{\mathrm{ren}}_{M_z,N}$ the morphism $z^{-s}R_{M_z,N}$.

Definition 2.2.1.

Assume that $R(\beta )$ is symmetric. For a non-zero $M\in R(\beta )\text{-mod}$ and a non-zero $N\in R(\gamma )\text{-mod}$, let $s$ be an integer as in 2.3. We define

$$\begin{equation*} \mathbf{r}_{\!_{M,N}}\colon M\circ N\to N\circ M \end{equation*}$$

by

$$\begin{equation*} \mathbf{r}_{\!_{M,N}} = R^{\mathrm{ren}}_{M_z,N}\vert _{z=0}, \end{equation*}$$

and call it the renormalized R-matrix.

By the definition, the renormalized R-matrix $\mathbf{r}_{\!_{M,N}}$ never vanishes.

We define also

$$\begin{equation*} \mathbf{r}_{\!_{N,M}}\colon N\circ M\to M\circ N \end{equation*}$$

by

$$\begin{equation*} \mathbf{r}_{\!_{N,M}} = \bigl ((-z)^{-t}R_{N,M_z}\bigr )\vert _{z=0}, \end{equation*}$$

where $t$ is the order of zero of $R_{N,M_z}$.

If $R(\beta )$ and $R(\gamma )$ are symmetric, then $s$ coincides with the order of zero of $R_{M,N_z}$, and $\bigl (z^{-s}R_{M_z,N}\bigr )\vert _{z=0}=\bigl ((-z)^{-s}R_{M,N_z}\bigr )\vert _{z=0}$ (see 15, (1.11)).

By the construction, if the composition $(N_1 \circ \mathbf{r}_{\!_{M,N_2}}) \circ (\mathbf{r}_{\!_{M,N_1}} \circ N_2)$ for $M,N_1,N_2 \in R \text{-mod}$ does not vanish, then it is equal to $\mathbf{r}_{\!_{M,N_1 \circ N_2}}$.

Definition 2.2.2.

A simple $R(\beta )$-module $M$ is called real if $M \circ M$ is simple.

The following lemma was used significantly in 15.

Lemma 2.2.3 (15, Lemma 3.1).

Let $\beta _k\in \mathsf{Q}^+$ and $M_k\in R(\beta _k)\text{-mod}$ $(k=1,2,3)$. Let $X$ be an $R(\beta _1+\beta _2)$-submodule of $M_1\circ M_2$ and $Y$ an $R(\beta _2+\beta _3)$-submodule of $M_2\circ M_3$ such that $X\circ M_3\subset M_1\circ Y$ as submodules of $M_1\circ M_2\circ M_3$. Then there exists an $R(\beta _2)$-submodule $N$ of $M_2$ such that $X\subset M_1\circ N$ and $N\circ M_3\subset Y$.

One of the main results in 15 is the following theorem.

Theorem 2.2.4 (15, Theorem 3.2).

Let $\beta , \gamma \in \mathsf{Q}^+$ and assume that $R(\beta )$ is symmetric. Let $M$ be a real simple module in $R(\beta ) \text{-mod}$ and $N$ a simple module in $R(\gamma ) \text{-mod}$. Then

(i)

$M \circ N$ and $N \circ M$ have simple socles and simple heads.

(ii)

Moreover, $\operatorname {Im}(\mathbf{r}_{\!_{M,N}})$ is equal to the head of $M \circ N$ and socle of $N \circ M$. Similarly, $\operatorname {Im}(\mathbf{r}_{\!_{N,M}})$ is equal to the head of $N \circ M$ and socle of $M \circ N$.

We will use the following convention frequently.

Definition 2.2.5.

For simple $R$-modules $M$ and $N$, we denote by $M \mathbin{\text{$\nabla $}}N$ the head of $M \circ N$ and by $M \mathbin{\text{$\Delta $}}N$ the socle of $M \circ N$.

3. Simplicity of heads and socles of convolution products

In this section, we assume that $R(\beta )$ is symmetric for any $\beta \in \mathsf{Q}^+$; i.e., $Q_{ij}(u,v)$ is a function in $u-v$ for any $i,j\in I$.

We also work always in the category of graded modules. For the sake of simplicity, we simply say that $M$ is an $R$-module instead of saying that $M$ is a graded $R(\beta )$-module for $\beta \in \mathsf{Q}^+$. We also sometimes ignore grading shifts if there is no danger of confusion. Hence, for $R$-modules $M$ and $N$, we sometimes say that $f\colon M\to N$ is a homomorphism if $f\colon q^aM\to N$ is a morphism in $R\text{-gmod}$ for some $a\in \mathbb{Z}$. If we want to emphasize that $f\colon q^aM\to N$ is a morphism in $R\text{-gmod}$, we say so.

3.1. Homogeneous degrees of R-matrices

Definition 3.1.1.

For non-zero $M,N \in R \text{-gmod}$, we denote by $\Lambda (M,N)$ the homogeneous degree of the R-matrix $\mathbf{r}_{\!_{M,N}}$.

Hence

$$\begin{eqnarray*} &&R^{\mathrm{ren}}_{M_z,N}:M_z \circ N \to q^{-\Lambda (M,N)} N \circ M_z \quad \text{and}\\ &&\mathbf{r}_{\!_{M,N}}:M\circ N \to q^{-\Lambda (M,N)} N \circ M \end{eqnarray*}$$

are morphisms in $R\text{-gMod}$ and in $R \text{-gmod}$, respectively.

Lemma 3.1.2.

For non-zero $R$-modules $M$ and $N$, we have

$$\begin{eqnarray*} \Lambda (M,N) \equiv \bigl (\operatorname {wt}(M),\operatorname {wt}(N)\bigr )\mod 2. \end{eqnarray*}$$

Proof.

Set $\beta \mathbin{:=}-\operatorname {wt}(M)$ and $\gamma \mathbin{:=}-\operatorname {wt}(N)$. By 14, (1.3.3), the homogeneous degree of $R_{M_z,N}$ is $-(\beta ,\gamma )+2{({\beta ,\gamma })_{\mathrm{n}}}$, where ${({\mdsmblkcircle ,\mdsmblkcircle })_{\mathrm{n}}}$ is the symmetric bilinear form on $\mathsf{Q}$ given by ${({\alpha _i,\alpha _j})_{\mathrm{n}}}=\delta _{ij}$. Hence $R^{\mathrm{ren}}_{M_z,N}=z^{-s}R_{M_z,N}$ has degree $-(\beta ,\gamma )+2{({\beta ,\gamma })_{\mathrm{n}}}-2s$.

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Definition 3.1.3.

For non-zero $R$-modules $M$ and $N$, we set

$$\begin{eqnarray*} \widetilde{\Lambda }(M,N) \mathbin{:=}\frac{1}{2}\bigl (\Lambda (M,N) +(\operatorname {wt}(M), \operatorname {wt}(N))\bigr )\in \mathbb{Z}. \end{eqnarray*}$$

Lemma 3.1.4.

Let $M$ and $N$ be self-dual simple modules. If one of them is real, then

$$\begin{eqnarray*} q^{\widetilde{\Lambda }(M,N)} M \mathbin{\text{$\nabla $}}N \end{eqnarray*}$$

is a self-dual simple module.

Proof.

Set $\beta =\operatorname {wt}(M)$ and $\gamma =\operatorname {wt}(N)$. Set $M \mathbin{\text{$\nabla $}}N = q^c L$ for some self-dual simple module $L$ and some $c \in \mathbb{Z}$. Then we have

$$\begin{eqnarray*} M \circ N \twoheadrightarrow q^c L \rightarrowtail q^{-\Lambda (M,N)} N \circ M, \end{eqnarray*}$$

since $M \mathbin{\text{$\nabla $}}N=\operatorname {Im}\mathbf{r}_{\!_{M,N}}$. Taking dual, we obtain

$$\begin{eqnarray*} q^{\Lambda (M,N)+(\beta ,\gamma )} M \circ N \twoheadrightarrow q^{-c} L \rightarrowtail q^{(\beta ,\gamma )} N \circ M. \end{eqnarray*}$$

In particular, $q^{-c-\Lambda (M,N)-(\beta ,\gamma )} L$ is a simple quotient of $M\circ N$. Hence we have $c=-c-\Lambda (M,N)-(\beta ,\gamma )$, which implies $c= -\widetilde{\Lambda }(M,N)$.

■

Lemma 3.1.5.

(i)

Let $M_k$ be non-zero modules $( k=1,2,3)$, and let $\varphi _1: L \to M_1\circ M_2$ and $\varphi _2: M_2 \circ M_3 \to L'$ be non-zero homomorphisms. Assume further that $M_2$ is a simple module. Then the composition$$\begin{eqnarray*} L \circ M_3 \xrightarrow {\,\varphi _1 \circ M_3\,} M_1 \circ M_2 \circ M_3 \xrightarrow {\,M_1 \circ \varphi _2\,} M_1 \circ L' \end{eqnarray*}$$

does not vanish.

(ii)

Let $M$ be a simple module, and let $N_1, N_2$ be non-zero modules. Then the composition$$\begin{eqnarray*} M \circ N_1 \circ N_2 \xrightarrow {\,\mathbf{r}_{\!_{M,N_1}} \circ N_2\,} N_1 \circ M \circ N_2 \xrightarrow {\,N_1 \circ \; \mathbf{r}_{\!_{M,N_2}}\,} N_1 \circ N_2 \circ M \end{eqnarray*}$$

coincides with $\mathbf{r}_{\!_{M,N_1 \circ N_2}}$, and the composition$$\begin{eqnarray*} N_1 \circ N_2 \circ M \xrightarrow {\,N_1\circ \; \mathbf{r}_{\!_{N_2,M}}\,} N_1 \circ M \circ N_2 \xrightarrow {\,\mathbf{r}_{\!_{N_1,M}} \circ N_2\,} M \circ N_1 \circ N_2 \end{eqnarray*}$$

coincides with $\mathbf{r}_{\!_{N_1 \circ N_2,M}}$.

In particular, we have$$\begin{eqnarray*} \Lambda (M, N_1\circ N_2) = \Lambda (M,N_1)+\Lambda (M,N_2) \end{eqnarray*}$$

and$$\begin{eqnarray*} \Lambda (N_1\circ N_2,M) = \Lambda (N_1,M)+\Lambda (N_2,M). \end{eqnarray*}$$

Proof.

(i)

Assume that the composition vanishes. Then we have $\mathrm{Im}\varphi _1 \circ M_3 \subset M_1 \circ \operatorname {Ker}\varphi _2$. By Lemma 2.2.3, there is a submodule $N$ of $M_2$ such that $\mathrm{Im}\varphi _1 \subset M_1 \circ N$ and $N \circ M_3 \subset \operatorname {Ker}\varphi _2$. The first inclusion implies that $N\neq 0$ since $\varphi _1$ is non-zero, and the second implies $N\neq M_2$ since $\varphi _2$ is non-zero. It contradicts the simplicity of $M_2$.

(ii)

It is enough to show that the compositions $(N_1 \circ \mathbf{r}_{\!_{M,N_2}}) \circ (\mathbf{r}_{\!_{M,N_1}} \circ N_2)$ and $(\mathbf{r}_{\!_{N_1,M}} \circ N_2) \circ (N_1 \circ \mathbf{r}_{\!_{N_2,M}})$ do not vanish, but these immediately follow from (i).

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3.2. Properties of $\widetilde{\Lambda }(M,N)$ and $\operatorname {\mathfrak{d}}(M,N)$

Lemma 3.2.1.

Let $M$ and $N$ be simple $R$-modules. Then we have

(i)

$\Lambda (M,N)+\Lambda (N,M) \in 2 \mathbb{Z}_{\ge 0}.$

(ii)

If $\Lambda (M,N)+\Lambda (N,M) =2m$ for some $m \in \mathbb{Z}_{\ge 0}$, then$$\begin{eqnarray*} R^{\mathrm{ren}}_{M_z,N} \circ R^{\mathrm{ren}}_{N,M_z}=z^m \operatorname {id}_{N \circ M_z} \quad \text{and} \quad R^{\mathrm{ren}}_{N,M_z} \circ R^{\mathrm{ren}}_{M_z,N}=z^m \operatorname {id}_{M_z \circ N} \end{eqnarray*}$$

up to constant multiples.

Proof.

By 14, Proposition 1.6.2, the morphism

$$\begin{eqnarray*} R^{\mathrm{ren}}_{N,M_z} \circ R^{\mathrm{ren}}_{M_z,N} : M_z \circ N \to M_z \circ N \end{eqnarray*}$$

is equal to $f(z) \operatorname {id}_{M_z \circ N}$ for some $0 \neq f(z) \in \mathbf{k}[z]$. Since $R^{\mathrm{ren}}_{N,M_z} \circ R^{\mathrm{ren}}_{M_z,N}$ is homogeneous of degree $\Lambda (M,N)+\Lambda (N,M)$, we have $f(z)=c z^{\frac{1}{2}(\Lambda (M,N)+\Lambda (N,M))}$ for some $c \in \mathbf{k}^\times$.

■

Definition 3.2.2.

For non-zero modules $M$ and $N$, we set

$$\begin{eqnarray*} \operatorname {\mathfrak{d}}(M,N) = \frac{1}{2}\bigl (\Lambda (M,N)+\Lambda (N,M)\bigr ). \end{eqnarray*}$$

Note that if $M$ and $N$ are simple modules, then we have $\operatorname {\mathfrak{d}}(M,N) \in \mathbb{Z}_{\ge 0}$. Note also that if $M,N_1,N_2$ are simple modules, then we have $\operatorname {\mathfrak{d}}(M,N_1 \circ N_2) =\operatorname {\mathfrak{d}}(M,N_1)+\operatorname {\mathfrak{d}}(M,N_2)$ by Lemma 3.1.5 (ii).

Lemma 3.2.3 (15).

Let $M,N$ be simple modules and assume that one of them is real. Then the following conditions are equivalent:

(i)

$\operatorname {\mathfrak{d}}(M,N)=0$.

(ii)

$\mathbf{r}_{\!_{M,N}}$ and $\mathbf{r}_{\!_{N,M}}$ are inverse to each other up to a constant multiple.

(iii)

$M \circ N$ and $N \circ M$ are isomorphic up to a grading shift.

(iv)

$M \mathbin{\text{$\nabla $}}N$ and $N \mathbin{\text{$\nabla $}}M$ are isomorphic up to a grading shift.

(v)

$M \circ N$ is simple.

Proof.

By specializing the equations in Lemma 3.2.1 (ii) at $z=0$, we obtain that $\operatorname {\mathfrak{d}}(M,N)=0$ if and only if $\mathbf{r}_{\!_{M,N}} \circ \mathbf{r}_{\!_{N,M}} = \operatorname {id}_{N\circ M}$ and $\mathbf{r}_{\!_{N,M}} \circ \mathbf{r}_{\!_{M,N}} = \operatorname {id}_{M\circ N}$ up to non-zero constant multiples. Hence the conditions (i) and (ii) are equivalent.

The conditions (ii), (iii), (iv), and (v) are equivalent by 15, Theorem 3.2, Proposition 3.8, and Corollary 3.9.

■

Definition 3.2.4.

Let $M,N$ be simple modules.

(i)

We say that $M$ and $N$ commute if $\operatorname {\mathfrak{d}}(M,N)=0$.

(ii)

We say that $M$ and $N$ are simply linked if $\operatorname {\mathfrak{d}}(M,N)=1$.

Proposition 3.2.5.

Let $M_1,\ldots , M_r$ be a commuting family of real simple modules. Then the convolution product

$$\begin{eqnarray*} M_{1} \circ \cdots \circ M_{r} \end{eqnarray*}$$

is a real simple module.

Proof.

We shall first show the simplicity of the convolutions. By induction on $r$, we may assume that $M_{2} \circ \cdots \circ M_{r}$ is a simple module. Then we have

$$\begin{eqnarray*} \operatorname {\mathfrak{d}}(M_{1}, M_{2} \circ \cdots \circ M_{r}) =\sum _{s=2}^r \operatorname {\mathfrak{d}}(M_{1},M_{s})=0 \end{eqnarray*}$$

so that $M_{1} \circ \cdots \circ M_{r}$ is simple by Lemma 3.2.3.

Since $(M_{1} \circ \cdots \circ M_{r}) \circ (M_{1} \circ \cdots \circ M_{r})$ is also simple, $M_{1} \circ \cdots \circ M_{r}$ is real.

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Definition 3.2.6.

Let $M_1,\ldots ,M_m$ be real simple modules. Assume that they commute with each other. We set

$$\begin{eqnarray*} &&M_1\mathop{\textstyle \bigodot }\limits M_2\mathbin{:=}q^{\widetilde{\Lambda }(M_1,M_2)}M_1\circ M_2,\\ &&\mathop{\textstyle \bigodot }\limits _{1 \le k \le m}M_k \mathbin{:=}(\cdots (M_1\mathop{\textstyle \bigodot }\limits M_2)\cdots )\mathop{\textstyle \bigodot }\limits M_{m-1})\mathop{\textstyle \bigodot }\limits M_m\\ &&\hspace{20ex}\simeq q^{\sum _{1 \le i <j\le m} \widetilde{\Lambda }(M_i,M_j)}M_1 \circ \cdots \circ M_m. \end{eqnarray*}$$

It is invariant under the permutations of $M_1,\ldots ,M_m$.

Lemma 3.2.7.

Let $M_1,\ldots , M_m$ be real simple modules commuting with each other. Then for any $\sigma \in \mathfrak{S}_m$, we have

$$\begin{equation*} \mathop{\text{$\odot $}}_{1\le k\le m} M_k\simeq \mathop{\text{$\odot $}}_{1\le k\le m} M_{\sigma (k)}\quad \text{in $R\text{-gmod}$.} \end{equation*}$$

Moreover, if the $M_k$’s are self-dual, then so is $\mathop{\text{$\odot $}}_{1\le k\le m} M_k$.

Proof.

It follows from Lemma 3.1.4 and $q^{\widetilde{\Lambda }(M_i,M_j)}M_i\circ M_j\simeq q^{\widetilde{\Lambda }(M_j,M_i)}M_j\circ M_i$.

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Proposition 3.2.8.

Let $f:N_1 \to N_2$ be a morphism between non-zero $R$-modules $N_1, N_2$, and let $M$ be a non-zero $R$-module.

(i)

If $\Lambda (M,N_1)=\Lambda (M,N_2)$, then the following diagram is commutative:$$\begin{eqnarray*} \vcenter{\img[][123pt][56pt][{$\xymatrix@C=5em{ M\conv N_1 \ar[r]^{\rmat{M,N_1}} \ar[d]_{M \circ f} &N_1 \conv M \ar[d]^{f \circ M} \\ M\conv N_2 \ar[r]^{\rmat{M,N_2}}& N_2 \conv M. }$}]{Images/img7e840daa03f56782fa3cfd4794012148.svg}} \end{eqnarray*}$$

(ii)

If $\Lambda (M,N_1) < \Lambda (M,N_2)$, then the composition$$\begin{eqnarray*} M \circ N_1 \xrightarrow {\,M \circ f\,} M \circ N_2 \xrightarrow {\,\mathbf{r}_{\!_{M,N_2}}\,} N_2 \circ M \end{eqnarray*}$$

vanishes.

(iii)

If $\Lambda (M,N_1) > \Lambda (M,N_2)$, then the composition$$\begin{eqnarray*} M \circ N_1 \xrightarrow {\,\mathbf{r}_{\!_{M,N_1}}\,} N_1 \circ M \xrightarrow {\,f \circ M\,} N_2 \circ M \end{eqnarray*}$$

vanishes.

(iv)

If $f$ is surjective, then we have$$\begin{eqnarray*} \Lambda (M,N_1) \ge \Lambda (M,N_2) \quad \text{and} \quad \Lambda (N_1,M) \ge \Lambda (N_2,M). \end{eqnarray*}$$

If $f$ is injective, then we have$$\begin{eqnarray*} \Lambda (M,N_1) \le \Lambda (M,N_2) \quad \text{and}\quad \Lambda (N_1,M) \le \Lambda (N_2,M). \end{eqnarray*}$$

Proof.

Let $s_i$ be the order of zero of $R_{M_z,N_i}$ for $i=1,2$. Then we have $\Lambda (M,N_1) - \Lambda (M,N_2)=2(s_2-s_1)$.

Set $m\mathbin{:=}\min \{s_1,s_2\}$. Then the following diagram is commutative:

$$\begin{eqnarray*} \vcenter{\img[][184pt][56pt][{$\xymatrix@C=10em{ M_z\conv N_1 \ar[r]^{z^{-m}R_{M_z,N_1}} \ar[d]_{M_z \conv f} & N_1 \conv M_z \ar[d]^{f \conv M_z} \\ M_z\conv N_2 \ar[r]^{z^{-m}R_{M_z,N_2}}& N_2 \conv M_z. }$}]{Images/img46060ae1559137c3ef4df649a48878e2.svg}} \end{eqnarray*}$$

(i) If $s_1=s_2$, then by specializing $z=0$ in the above diagram, we obtain the commutativity of the diagram in (i).

(ii) If $s_1 > s_2$, then we have

$$\begin{eqnarray*} z^{-m}R_{M_z,N_1}=z^{s_1-m} \bigl (z^{-s_1}R_{M_z,N_1}\bigr ) \end{eqnarray*}$$

so that $z^{-m}R_{M_z,N_1}|_{z=0}$ vanishes. Hence we have

$$\begin{eqnarray*} \mathbf{r}_{\!_{M,N_2}} \circ (M\circ f) = z^{-m}R_{M_z,N_2}|_{z=0} \circ (M\circ f) =0, \end{eqnarray*}$$

as desired. In particular, $f$ is not surjective.

(iii) Similarly, if $s_1 < s_2$, then we have $(f \circ M) \circ \mathbf{r}_{\!_{M,N_1}} =0,$ and $f$ is not injective.

(iv) The statements for $\Lambda (M,N_1)$ and $\Lambda (M, N_2)$ follow from (ii) and (iii). The other statements can be shown in a similar way.

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Proposition 3.2.9.

Let $M$ and $N$ be simple modules. We assume that one of them is real. Then we have

$$\begin{equation*} \operatorname {Hom}_{R\text{-mod}}(M\circ N,N\circ M)=\mathbf{k}\,\mathbf{r}_{\!_{M,N}}. \end{equation*}$$

Proof.

Since the other case can be proved similarly, we assume that $M$ is real. Let $f\colon M\circ N\to N\circ M$ be a morphism. Note that we have $\mathbf{r}_{\!_{M,\,M\circ N}}=M\circ \mathbf{r}_{\!_{M,N}}$ and $\mathbf{r}_{\!_{M,\,N\circ M}}=\mathbf{r}_{\!_{M,N}}\circ M$ by Lemma 3.1.5 (ii) and by the fact that $\mathbf{r}_{\!_{M,M}}=\operatorname {id}_{M\circ M}$ up to a constant multiple. Thus, by Proposition 3.2.8, we have a commutative diagram (up to a constant multiple)

$$\begin{equation*} \vcenter{\img[][143pt][52pt][{$\xymatrix@C=10ex{ M\conv M\conv N\ar[r]^{M\circ\rmat{M,N}}\ar[d]_{M\circ f} &M\conv N\conv M\ar[d]_{f\circ M}\\ M\conv N\conv M\ar[r]^{\rmat{M,N}\circ M} &N\conv M\conv M. }$}]{Images/imgfaa05ed0403fb77c08101cc0f9a87f41.svg}} \end{equation*}$$

Hence we have

$$\begin{equation*} M\circ {\mathrm{Im}}(\mathbf{r}_{\!_{M,N}})\subset f^{-1}\bigl ({\mathrm{Im}}(\mathbf{r}_{\!_{M,N}})\bigr )\circ M. \end{equation*}$$

Hence there exists a submodule $K$ of $N$ such that ${\mathrm{Im}}(\mathbf{r}_{\!_{M,N}})\subset K\circ M$ and $M\circ K\subset f^{-1}\bigl ({\mathrm{Im}}(\mathbf{r}_{\!_{M,N}})\bigr )$ by Lemma 2.2.3. Since $K\not =0$, we have $K=N$. Hence $f(M\circ N)\subset {\mathrm{Im}}(\mathbf{r}_{\!_{M,N}})$, which means that $f$ factors as $M\circ N\to {\operatorname {soc}}(N\circ M)\rightarrowtail N\circ M$. It remains to remark that $\operatorname {Hom}_{R\text{-mod}}\bigl (M\circ N,{\operatorname {soc}}(N\circ M)\bigr )=\mathbf{k}\,\mathbf{r}_{\!_{M,N}}$.

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Proposition 3.2.10.

Let $L$, $M$, and $N$ be simple modules. Then we have

$$\begin{eqnarray} &&\begin{array}{l} \Lambda (L,S)\le \Lambda (L,M)+\Lambda (L,N),\ \Lambda (S,L)\le \Lambda (M,L)+\Lambda (N,L),\ \text{and}\\[5.0pt] \operatorname {\mathfrak{d}}(S,L)\le \operatorname {\mathfrak{d}}(M,L)+\operatorname {\mathfrak{d}}(N,L)\end{array} {\tag{3.1}} \end{eqnarray}$$

for any subquotient $S$ of $M\circ N$. Moreover, when $L$ is real, the following conditions are equivalent:

(i)

$L$ commutes with $M$ and $N$.

(ii)

Any simple subquotient $S$ of $M\circ N$ commutes with $L$ and satisfies $\Lambda (L,S)=\Lambda (L,M)+\Lambda (L,N)$.

(iii)

Any simple subquotient $S$ of $M\circ N$ commutes with $L$ and satisfies $\Lambda (S,L)=\Lambda (M,L)+\Lambda (N,L)$.

Proof.

The inequalities 3.1 are consequences of Proposition 3.2.8. Let us show the equivalence of (i)–(iii).

Let $M\circ N=K_0\supset K_1\supset \cdots \supset K_\ell \supset K_{\ell +1}=0$ be a Jordan–Hölder series of $M\circ N$. Then the renormalized R-matrix $R^{\mathrm{ren}}_{L_z,M\circ N}=(M\circ R^{\mathrm{ren}}_{L_z,N})\circ (R^{\mathrm{ren}}_{L_z,M}\circ N) \colon L_z\circ M\circ N\to M\circ N\circ L_z$ is homogeneous of degree $\Lambda (L,M)+\Lambda (L,N)$, and it sends $L_z\circ K_k$ to $K_k\circ L_z$ for any $k\in \mathbb{Z}$. Hence $f\mathbin{:=}\mathbf{r}_{\!_{L,M\circ N}}=R^{\mathrm{ren}}_{L_z,M\circ N}\vert _{z=0}$ sends $L\circ K_k$ to $K_k\circ L$.

First assume (i). Then $f$ is an isomorphism. Hence $f\vert _{L\circ K_k}\colon L\circ K_k\to K_k\circ L$ is injective. By comparing their dimension, $f\vert _{L\circ K_k}$ is an isomorphism, Hence $f\vert _{L\circ (K_k/K_{k+1})}$ is an isomorphism of homogeneous degree $\Lambda (L,M)+\Lambda (L,N)$. Hence we obtain (ii).

Conversely, assume (ii). Then, $R^{\mathrm{ren}}_{L_z,M\circ N}\vert _{L_z\circ (K_k/K_{k+1})}$ and $R^{\mathrm{ren}}_{L_z,K_k/K_{k+1}}$ have the same homogeneous degree, and hence they should coincide. It implies that $f\vert _{L\circ (K_k/K_{k+1})}=\mathbf{r}_{\!_{L,K_k/K_{k+1}}}$ is an isomorphism for any $k$. Therefore $f=(M\circ \mathbf{r}_{\!_{L,N}})\circ (\mathbf{r}_{\!_{L,M}}\circ N)$ is an isomorphism, which implies that $\mathbf{r}_{\!_{L,N}}$ and $\mathbf{r}_{\!_{L,M}}$ are isomorphisms. Thus we obtain (i).

Similarly, (i) and (iii) are equivalent.

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Lemma 3.2.11.

Let $L$, $M$, and $N$ be simple modules. We assume that $L$ is real and commutes with $M$. Then the diagram

$$\begin{equation*} \vcenter{\img[][155pt][56pt][{$\xymatrix@C=10ex {L\conv(M\conv N)\ar[r]^{\rmat{L,M\circ N}}\ar[d] &(M\conv N)\conv L\ar[d]\\ L\conv(M\hconv N)\ar[r]^{\rmat{L,M\shconv N}} &(M \hconv N)\conv L}$}]{Images/img55e2050d7bfffdb491f749c5b091a8ad.svg}} \end{equation*}$$

commutes.

Proof.

Otherwise the composition

$$\begin{equation*} \vcenter{\img[][303pt][18pt][{$\xymatrix@C=8ex{ L\conv M\conv N\ar[r]^-\sim_-{\rmat{L,M}\circ N} &M\conv L\conv N\ar[r]_-{M\circ\rmat{L,N}}&M\conv N\conv L \ar[r]& (M\hconv N)\conv L}$}]{Images/imgf9e469820839b07a3a90374c5209aaa6.svg}} \end{equation*}$$ vanishes by Proposition 3.2.8. Hence we have

$$\begin{equation*} M\circ \operatorname {Im}(\mathbf{r}_{\!_{L,N}})\subset \operatorname {Ker}(M\circ N\to M\mathbin{\text{$\nabla $}}N)\circ L. \end{equation*}$$

Hence, by Lemma 2.2.3, there exists a submodule $K$ of $N$ such that

$$\begin{equation*} \text{$\operatorname {Im}(\mathbf{r}_{\!_{L,N}})\subset K\circ L$ and $M\circ K\subset \operatorname {Ker}(M\circ N\to M\mathbin{\text{$\nabla $}}N)$.} \end{equation*}$$

The first inclusion implies $K\not =0$ and the second implies $K\not =N$, which contradicts the simplicity of $N$.

■

The following lemma can be proved similarly.

Lemma 3.2.12.

Let $L$, $M$, and $N$ be simple modules. We assume that $L$ is real and commutes with $N$. Then the diagram

$$\begin{equation*} \vcenter{\img[][154pt][56pt][{$\xymatrix@C=10ex {(M\conv N)\conv L\ar[r]^{\rmat{M\circ N,L}}\ar[d] &L\conv(M\conv N)\ar[d]\\ (M\hconv N)\conv L\ar[r]^{\rmat{M\shconv N,L}} &L\conv(M\hconv N)}$}]{Images/imgda1a17ccd08fb5201ac6778ca03a332e.svg}} \end{equation*}$$

commutes.

The following proposition follows from Lemma 3.2.11 and Lemma 3.2.12.

Proposition 3.2.13.

Let $L$, $M$, and $N$ be simple modules. Assume that $L$ is real. Then we have the following:

(i)

If $L$ and $M$ commute, then$$\begin{equation*} \Lambda (L,M\mathbin{\text{$\nabla $}}N)=\Lambda (L,M)+\Lambda (L,N). \end{equation*}$$

(ii)

If $L$ and $N$ commute, then$$\begin{equation*} \Lambda (M\mathbin{\text{$\nabla $}}N,L)=\Lambda (M,L)+\Lambda (N,L). \end{equation*}$$

Proposition 3.2.14.

Let $M$ be a real simple module, and let $N$ be a module with a simple socle. If the following diagram

$$\begin{equation*} \vcenter{\img[][155pt][55pt][{$\xymatrix@C=12ex {\wb{\soc(N)\conv M}\ar[r]^{\rmat{\soc(N),M}}\ar@{>->}[d] &\wb{M\conv\soc(N)}\ar@{>->}[d]\\ N\conv M\ar[r]^{\rmat{N,M}}&M\conv N }$}]{Images/img8a2cd948ebd3a371805a8060cb1c9217.svg}} \end{equation*}$$

commutes up to a non-zero constant multiple, then ${\operatorname {soc}}\bigl (M\circ {\operatorname {soc}}(N)\bigr )$ is equal to the socle of $M\circ N$. In particular, $M\circ N$ has a simple socle.

Proof.

Let $S$ be an arbitrary simple submodule of $M\circ N$. Then we have the following commutative diagram:

$$\begin{equation*} \vcenter{\img[][157pt][56pt][{$\xymatrix@C=12ex {\wb{S\conv M_z}\ar[r]^{ R_{S\circ M_z} } \ar@{>->}[d] &\wb{M_z\conv S}\ar@{>->}[d]\\ M\conv N\conv M_z\ar[r]^{R_{M\circ N,M_z}} &M\conv M\conv N. }$}]{Images/img2cbe97836a48b9ace48b9b477ef4c2cd.svg}} \end{equation*}$$

By multiplying $z^{-m}$, where $m$ is the order of zero of $R_{M\circ N, M}$, and specializing at $z=0$, we have a commutative diagram (up to a constant multiple)

$$\begin{equation*} \vcenter{\img[][153pt][47pt][{$\xymatrix@C=12ex {\wb{S\conv M}\ar[r] \ar@{>->}[d] &\wb{M\conv S}\ar@{>->}[d]\\ M\conv N\conv M\ar[r]^{M\circ\rmat{N,M}}&M\conv M\conv N. }$}]{Images/img644f2938e96b8f3815c99123f3704ea0.svg}} \end{equation*}$$

Here, we use the fact that $\mathbf{r}_{\!_{M\circ N, M}} = (\mathbf{r}_{\!_{M,M}} \circ N) \circ (M\circ \mathbf{r}_{\!_{N,M}})$ from Lemma 3.1.5 and the fact that $\mathbf{r}_{\!_{M,M}}$ is equal to $\operatorname {id}_{M\circ M}$ up to a non-zero constant multiple, because $M$ is a real simple module.

It follows that $S\circ M\subset M\circ (\mathbf{r}_{\!_{N,M}})^{-1}(S)$. Hence there exists a submodule $K$ of $N$ such that $S\subset M\circ K$ and $K\circ M\subset (\mathbf{r}_{\!_{N,M}})^{-1}(S)$ by Lemma 2.2.3. Hence $K\not =0$ and ${\operatorname {soc}}(N)\subset K$ by the assumption. Hence $\mathbf{r}_{\!_{N,M}}\bigl ({\operatorname {soc}}(N)\circ M\bigr )\subset \mathbf{r}_{\!_{N,M}}\bigl (K\circ M\bigr )\subset S$. Since $\mathbf{r}_{\!_{N,M}}\bigl ({\operatorname {soc}}(N)\circ M\bigr )$ is non-zero by the assumption, we have $\mathbf{r}_{\!_{N,M}}\bigl ({\operatorname {soc}}(N)\circ M\bigr )=S$. Thus we obtain the desired result.

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The following is a dual form of the preceding proposition.

Proposition 3.2.15.

Let $M$ be a real simple module. Let $N$ be a module with a simple head. If the following diagram

$$\begin{equation*} \vcenter{\img[][152pt][55pt][{$\xymatrix@C=12ex {{M\conv N}\ar[r]^{\rmat{M,N}}\ar@{->>}[d] &{N\conv M}\ar@{->>}[d]\\ M\conv\hd(N)\ar[r]^{\rmat{M,\hd(N)}}&\hd(N)\conv M }$}]{Images/imgefdfb4c6dabc4f75fc4e9a4b23722dac.svg}} \end{equation*}$$

commutes up to a non-zero constant multiple, then $M\mathbin{\text{$\nabla $}}{\operatorname {hd}}(N)$ is equal to the simple head of $M\circ N$.

Proposition 3.2.16.

Let $L$, $M$, and $N$ be simple modules. We assume that $L$ is real and one of $M$ and $N$ is real.

(i)

If $\Lambda (L,M\mathbin{\text{$\nabla $}}N)=\Lambda (L,M)+\Lambda (L,N)$, then $L\circ M\circ N$ has a simple head and $N\circ M\circ L$ has a simple socle.

(ii)

If $\Lambda (M\mathbin{\text{$\nabla $}}N,L)=\Lambda (M,L)+\Lambda (N,L)$, then $M\circ N\circ L$ has a simple head and $L\circ N\circ M$ has a simple socle.

(iii)

If $\operatorname {\mathfrak{d}}(L,M\mathbin{\text{$\nabla $}}N)=\operatorname {\mathfrak{d}}(L,M)+\operatorname {\mathfrak{d}}(L,N)$, then $L\circ M\circ N$ and $M\circ N\circ L$ have simple heads, and $N\circ M\circ L$ and $L\circ N\circ M$ have simple socles.

Proof.

(i) Denote $k=\Lambda (L,M\mathbin{\text{$\nabla $}}N)=\Lambda (L,M)+\Lambda (M,N)$ and $m=\Lambda ( M,N)$. Then the diagram

$$\begin{equation*} \vcenter{\img[][189pt][82pt][{$\xymatrix@C=12ex@R=3ex {{L\conv M\conv N}\ar[r]^{\rmat{L,M\circ N}}\ar@{->>}[d] & q^{-k}{M\conv N\conv L}\ar@{->>}[d] \\ \wb{L\conv(M\hconv N)}\ar[r]^{\rmat{L,M\shconv N}}\ar@{>->}[d] &\wb{q^{-k}(M\hconv N)\conv L}\ar@{>->}[d]\\ q^{-m}L\conv N\conv M\ar[r]^{\rmat{L,N\circ M}}&q^{-k-m} N\conv M\conv L }$}]{Images/img8aa4278d14bb90cb02b1dcc8f034c567.svg}} \end{equation*}$$

commutes. Hence Proposition 3.2.14 and Proposition 3.2.15 imply that $L\circ M\circ N$ has a simple head and $N\circ M\circ L$ has a simple socle. Item (ii) is proved similarly.

(iii) If $\operatorname {\mathfrak{d}}(L,M\mathbin{\text{$\nabla $}}N)=\operatorname {\mathfrak{d}}(L,M)+\operatorname {\mathfrak{d}}(L,N)$, then we have $\Lambda (L,M\mathbin{\text{$\nabla $}}N)=\Lambda (L,M)+\Lambda (L,N)$ and $\Lambda (M\mathbin{\text{$\nabla $}}N,L)=\Lambda (M,L)+\Lambda (N,L)$ by Proposition 3.2.8. Thus the statements in (iii) follow from (i) and (ii).

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Proposition 3.2.17.

Let $M$ and $N$ be simple modules. Assume that one of them is real and $\operatorname {\mathfrak{d}}(M,N)=1$. Then we have an exact sequence

$$\begin{eqnarray*} &&0\to M\mathbin{\text{$\Delta $}}N \to M\circ N\to M\mathbin{\text{$\nabla $}}N\to 0. \end{eqnarray*}$$

In particular, $M\circ N$ has length $2$.

Proof.

In the course of the proof, we ignore the grading.

Set $X=M_z\circ N$ and $Y=N\circ M_z$. By $R^{\mathrm{ren}}_{N,M_z}\colon Y\rightarrowtail X$ let us regard $Y$ as a submodule of $X$. By the condition, we have $R^{\mathrm{ren}}_{N,M_z}\circ R^{\mathrm{ren}}_{M_z,N}=z\operatorname {id}_X$ up to a constant multiple (see Lemma 3.2.1 (ii)), and hence we have

$$\begin{equation*} zX\subset Y\subset X. \end{equation*}$$

We have an exact sequence

$$\begin{equation*} 0\xrightarrow {\,\hspace{2ex}\,}\dfrac{Y}{zX}\xrightarrow {\,\hspace{2ex}\,}\dfrac{X}{zX}\xrightarrow {\,\hspace{2ex}\,}\dfrac{X}{Y}\xrightarrow {\,\hspace{2ex}\,}0. \end{equation*}$$

Since

$$\begin{equation*} M\circ N\simeq \dfrac{X}{zX}\twoheadrightarrow \dfrac{X}{Y}\rightarrowtail \dfrac{z^{-1}Y}{Y} \simeq N\circ M, \end{equation*}$$

we have $\dfrac{X}{Y}\simeq M\mathbin{\text{$\nabla $}}N$ by Proposition 3.2.9. Similarly,

$$\begin{equation*} N\circ M\simeq \dfrac{Y}{zY}\twoheadrightarrow \dfrac{Y}{zX}\rightarrowtail \dfrac{X}{zX} \simeq M\circ N \end{equation*}$$

implies that $\dfrac{Y}{zX}\simeq M\mathbin{\text{$\Delta $}}N$ by Proposition 3.2.9.

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Lemma 3.2.18.

Let $M$ and $N$ be simple modules. Assume that one of them is real. If there is an exact sequence

$$\begin{equation*} 0\to q^mX\xrightarrow {\,\hspace{2ex}\,}M\circ N\xrightarrow {\,\hspace{2ex}\,}q^n Y\xrightarrow {\,\hspace{2ex}\,}0 \end{equation*}$$

for self-dual simple modules $X$, $Y$ and integers $m$, $n$, then we have

$$\begin{equation*} \operatorname {\mathfrak{d}}(M,N)=m-n. \end{equation*}$$

Proof.

We may assume that $M$ and $N$ are self-dual without loss of generality. Then we have $n=-\widetilde{\Lambda }(M,N)$. Since

$$\begin{equation*} q^mX\simeq q^{\Lambda (N,M)}N\mathbin{\text{$\nabla $}}M \simeq q^{\Lambda (N,M)-\widetilde{\Lambda }(N,M)}\bigl (q^{\widetilde{\Lambda }(N,M)}N\mathbin{\text{$\nabla $}}M\bigr ), \end{equation*}$$

we have $m=\Lambda (N,M)-\widetilde{\Lambda }(N,M)$. Thus we obtain

$$\begin{equation*} m-n=\Lambda (N,M)-\widetilde{\Lambda }(N,M)+\widetilde{\Lambda }(M,N)=\operatorname {\mathfrak{d}}(M,N). \end{equation*}$$ ■

Lemma 3.2.19.

Let $M$ and $N$ be simple modules. Assume that one of them is real. If the equation

$$\begin{equation*} [M][N]=q^m[X]+q^n[Y] \end{equation*}$$

holds in $K(R\text{-gmod})$ for self-dual simple modules $X$, $Y$ and integers $m$, $n$ such that $m\ge n$, then we have

(i)

$\operatorname {\mathfrak{d}}(M,N)=m-n>0$,

(ii)

there exists an exact sequence$$\begin{equation*} 0\xrightarrow {\,\hspace{2ex}\,}q^mX\xrightarrow {\,\hspace{2ex}\,}M\circ N\xrightarrow {\,\hspace{2ex}\,}q^nY\xrightarrow {\,\hspace{2ex}\,}0, \end{equation*}$$

(iii)

$q^mX$ is the socle of $M\circ N$ and $q^nY$ is the head of $M\circ N$.

Proof.

First note that $\operatorname {\mathfrak{d}}(M,N)>0$ since $M\circ N$ is not simple. By the assumption, there exists either an exact sequence

$$\begin{equation*} 0\xrightarrow {\,\hspace{2ex}\,}q^mX\xrightarrow {\,\hspace{2ex}\,}M\circ N\xrightarrow {\,\hspace{2ex}\,}q^nY\xrightarrow {\,\hspace{2ex}\,}0, \end{equation*}$$

or

$$\begin{equation*} 0\xrightarrow {\,\hspace{2ex}\,}q^nY\xrightarrow {\,\hspace{2ex}\,}M\circ N\xrightarrow {\,\hspace{2ex}\,}q^mX\xrightarrow {\,\hspace{2ex}\,}0. \end{equation*}$$

The second sequence cannot exist by Lemma 3.2.18 because $\operatorname {\mathfrak{d}}(M,N)=n-m\le 0$. Hence the first sequence exists, and the assertion (iii) follows from Theorem 2.2.4.

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Proposition 3.2.20.

Let $X,Y,M$, and $N$ be simple $R$-modules. Assume that there is an exact sequence

$$\begin{eqnarray*} 0 \to X \to M \circ N \to Y \to 0, \end{eqnarray*}$$

$X \circ N$ and $Y \circ N$ are simple, and $X \circ N \not \simeq Y \circ N$ are ungraded modules. Then $N$ is a real simple module.

Proof.

Assume that $N$ is not real. Then $N \circ N$ is reducible, and we have $\mathbf{r}_{\!_{N,N}} \neq c \operatorname {id}_{N \circ N}$ for any $c \in \mathbf{k}$ by 15, Corollary 3.3. Note that $N \circ N$ is of length $2$, because $M\circ N \circ N$ is of length $2$.

Let $S$ be a simple submodule of $N \circ N$. Consider an exact sequence

$$\begin{equation*} 0\xrightarrow {\,\hspace{2ex}\,}X\circ N\xrightarrow {\,\hspace{2ex}\,}M\circ N\circ N\xrightarrow {\,\hspace{2ex}\,}Y\circ N\xrightarrow {\,\hspace{2ex}\,}0. \end{equation*}$$

Then we have

$$\begin{eqnarray} (X \circ N) \cap (M\circ S) =0. {\tag{3.2}} \end{eqnarray}$$

Indeed, if $(X\circ N) \subset (M \circ S)$, then there exists a submodule $Z$ of $N$ such that $X \subset M \circ Z$ and $Z \circ N \subset S$ by 15, Lemma 3.1. It contradicts the simplicity of $N$. Thus 3.2 holds.

Note that 3.2 implies

$$\begin{equation*} M\circ S\simeq Y\circ N \end{equation*}$$

since $Y \circ N$ is simple.

(a)

Assume first that $N \circ N$ is semisimple so that $N \circ N=S \oplus S'$ for some simple submodule $S'$ of $N \circ N$. Then $M\circ S\simeq Y\circ N\simeq M\circ S'$. Hence $M\circ S\simeq X\circ N\simeq M\circ S'$. Therefore we obtain $X \circ N \simeq Y \circ N,$ which is a contradiction.

(b)

Assume that $N \circ N$ is not semisimple so that $S$ is a unique non-zero proper submodule of $N \circ N$ and $(N \circ N) / S$ is a unique non-zero proper quotient of $N \circ N$. Without loss of generality, we may assume that $\mathbf{k}$ is algebraically closed 21, Corollary 3.19. Let $x \in \mathbf{k}$ be an eigenvalue of $\mathbf{r}_{\!_{N,N}}$. Since $\mathbf{r}_{\!_{N,N}} \not \in \mathbf{k}\operatorname {id}_{N \circ N}$, we have $0 \subsetneq \operatorname {Im}(\mathbf{r}_{\!_{N,N}}-x \operatorname {id}_{N\circ N}) \subsetneq N\circ N$. It follows that$$\begin{eqnarray*} S = \operatorname {Im}(\mathbf{r}_{\!_{N,N}}-x \operatorname {id}_{N\circ N}) \simeq (N \circ N) /S, \end{eqnarray*}$$

and hence we have an exact sequence$$\begin{eqnarray*} 0 \xrightarrow {\,\hspace{2ex}\,}M\circ S \xrightarrow {\,\hspace{2ex}\,}M \circ N \circ N \xrightarrow {\,\hspace{2ex}\,}M \circ \bigl ((N \circ N) /S\bigr )\xrightarrow {\,\hspace{2ex}\,}0. \end{eqnarray*}$$

Since $M \circ N \circ N$ is of length $2$, we have$$\begin{eqnarray*} X \circ N \simeq M \circ S \simeq M\circ \bigl ((N \circ N) /S\bigr )\simeq Y \circ N, \end{eqnarray*}$$

which is a contradiction.

■

Corollary 3.2.21.

Let $X,Y,N$ be simple $R$-modules, and let $M$ be a real simple $R$-module. If we have an exact sequence

$$\begin{eqnarray*} 0 \to X \to M \circ N \to Y \to 0 \end{eqnarray*}$$

and if $X \circ N$ and $Y \circ N$ are simple, then $N$ is a real simple module.

Proof.

Since $M$ is real and $M \circ N$ is not simple, $X$ is not isomorphic to $Y$ as an ungraded module by Lemma 3.2.3 (iv). It follows that $X \circ N$ is not isomorphic to $Y \circ N$, because $K(R\text{-gmod})$ is a domain so that $[X\circ N]=q^m [Y \circ N]$ for some $m \in \mathbb{Z}$ implies $[X]=q^m [Y]$. Now the assertion follows from Proposition 3.2.20.

■

Lemma 3.2.22.

Let $\{M_i\}_{1\le i\le n}$ and $\{N_i\}_{1\le i\le n}$ be a pair of commuting families of real simple modules. We assume that

(a)

$\{M_i\mathbin{\text{$\nabla $}}N_i\}_{1\le i\le n}$ is a commuting family of real simple modules,

(b)

$M_i\mathbin{\text{$\nabla $}}N_i$ commutes with $N_{j}$ for any $1\le i,j\le n$.

Then we have

$$\begin{equation*} (\circ _{1\le i\le n}M_i)\mathbin{\text{$\nabla $}}(\circ _{1\le j\le n}N_j) \simeq \circ _{1\le i\le n}(M_i\mathbin{\text{$\nabla $}}N_i)\quad \text{up to a grading shift.} \end{equation*}$$

Proof.

Since $\circ _{1\le i\le n}(M_i\mathbin{\text{$\nabla $}}N_i)$ is simple, it is enough to give an epimorphism $(\circ _{1\le i\le n}M_i)\circ (\circ _{1\le j\le n}N_j) \twoheadrightarrow \circ _{1\le i\le n}(M_i\mathbin{\text{$\nabla $}}N_i)$. We shall show it by induction on $n$. For $n>0$, we have

$$\begin{eqnarray*} &&(\circ _{1\le i\le n}M_i)\circ (\circ _{1\le j\le n}N_j)\simeq (\circ _{1\le i\le n-1}M_i)\circ M_n\circ N_n\circ (\circ _{1\le j\le n-1}N_j)\\ &&\twoheadrightarrow (\circ _{1\le i\le n-1}M_i)\circ (M_n\mathbin{\text{$\nabla $}}N_n) \circ (\circ _{1\le j\le n-1}N_j) \\ &&\simeq (\circ _{1\le i\le n-1}M_i)\circ (\circ _{1\le j\le n-1}N_j) \circ (M_n\mathbin{\text{$\nabla $}}N_n) \\ &&\twoheadrightarrow (\circ _{1\le i\le n-1}(M_i \mathbin{\text{$\nabla $}}N_i)) \circ (M_n\mathbin{\text{$\nabla $}}N_n), \end{eqnarray*}$$

as desired.

■

4. Leclerc’s conjecture

In this section, $R$ is assumed to be a symmetric KLR algebra over a base field $\mathbf{k}$.

4.1. Leclerc’s conjecture

The following theorem is a part of Leclerc’s conjecture stated in the Introduction.

Theorem 4.1.1.

Let $M$ and $N$ be simple modules. We assume that $M$ is real. Then we have the equalities in the Grothendieck group $K(R\text{-gmod})$ as follows:

(i)

$[M\circ N]=[M\mathbin{\text{$\nabla $}}N]+\sum _{k}[S_k]$

with simple modules $S_k$ such that $\Lambda (M,S_k)<\Lambda (M,M\mathbin{\text{$\nabla $}}N)=\Lambda (M,N)$,

(ii)

$[M\circ N]=[ M\mathbin{\text{$\Delta $}}N]+\sum _{k}[S_k]$

with simple modules $S_k$ such that $\Lambda (S_k,M)<\Lambda ( M\mathbin{\text{$\Delta $}}N,M)=\Lambda (N,M)$,

(iii)

$[N\circ M]=[N\mathbin{\text{$\nabla $}}M]+\sum _{k}[S_k]$

with simple modules $S_k$ such that $\Lambda (S_k,M)<\Lambda (N\mathbin{\text{$\nabla $}}M,M)=\Lambda (N,M)$,

(iv)

$[N\circ M]=[ N\mathbin{\text{$\Delta $}}M]+\sum _{k}[S_k]$

with simple modules $S_k$ such that $\Lambda (M,S_k)<\Lambda (M, N \mathbin{\text{$\Delta $}}M )=\Lambda (M,N)$.

In particular, $M\mathbin{\text{$\nabla $}}N$ as well as $M \mathbin{\text{$\Delta $}}N$ appears only once in the Jordan–Hölder series of $M\circ N$ in $R\text{-mod}$.

The following result is an immediate consequence of this theorem.

Corollary 4.1.2.

Let $M$ and $N$ be simple modules. We assume that one of them is real. Assume that $M$ and $N$ do not commute, Then we have the equality in the Grothendieck group $K(R\text{-gmod})$

$$\begin{equation*} [M\circ N]=[M\mathbin{\text{$\nabla $}}N]+[ M\mathbin{\text{$\Delta $}}N ]+\sum _{k}[S_k] \end{equation*}$$

with simple modules $S_k$. Moreover we have the following:

(i)

If $M$ is real, then we have $\Lambda (M, M\mathbin{\text{$\Delta $}}N)<\Lambda (M,N)$, $\Lambda (M\mathbin{\text{$\nabla $}}N,M)<\Lambda (N,M)$ and $\Lambda (M, S_k)<\Lambda (M,N)$, $\Lambda (S_k,M)<\Lambda (N,M)$.

(ii)

If $N$ is real, then we have $\Lambda (N, M\mathbin{\text{$\nabla $}}N)<\Lambda (N,M)$, $\Lambda ( M \mathbin{\text{$\Delta $}}N, N)<\Lambda ( M,N)$ and $\Lambda (N, S_k)<\Lambda (N,M)$, $\Lambda (S_k,N)<\Lambda (M,N)$.

Proof of Theorem 4.1.1.

We shall prove only (i). The other statements are proved similarly.

$$\begin{equation*} M\circ N=K_0\supset K_1\supset \cdots \supset K_\ell \supset K_{\ell +1}=0. \end{equation*}$$ Then we have $K_0/K_1\simeq M\mathbin{\text{$\nabla $}}N$. Let us consider the renormalized R-matrix $R^{\mathrm{ren}}_{M_z,M\circ N}=(M\circ R^{\mathrm{ren}}_{M_z,N})\circ (R^{\mathrm{ren}}_{M_z,M}\circ N)$

$$\begin{equation*} \vcenter{\img[][252pt][19pt][{$\xymatrix@C=10ex{ M_z\conv M\conv N\ar[r]^{\Rm_{M_z,M}\circ N} &M\conv M_z\conv N\ar[r]^{M\circ\Rm_{M_z,N}}&M\conv N\conv M_z.}$}]{Images/imge68665bc953f3dd9ffdfbd0818d6a3ed.svg}} \end{equation*}$$

Then $R^{\mathrm{ren}}_{M_z,M\circ N}$ sends $M_z\circ K_k$ to $K_k\circ M_z$ for any $k$. Hence evaluating the above diagram at $z=0$, we obtain

$$\begin{equation*} \vcenter{\img[][142pt][57pt][{$\xymatrix@C=10ex{ M\conv M\conv N\ar[r]^{M\circ\rmat{M,N}} &M\conv N\conv M\\ \wb{M\conv K_1}\ar[r]\ar@{^{(}->}[u] &\wb{K_1\conv M\,.}\ar@{^{(}->}[u]}$}]{Images/img7bdc7e206789c0f4f157425ab5d7ea9f.svg}} \end{equation*}$$

Since $\operatorname {Im}(\mathbf{r}_{\!_{M,N}}\colon M\circ N\to N\circ M)\simeq (M\circ N)/K_1$, we have $\mathbf{r}_{\!_{M,N}}(K_1)=0$. Hence, $R^{\mathrm{ren}}_{M_z,M\circ N}$ sends $M_z\circ K_1$ to $(K_1\circ M_z)\cap z\bigl ((M\circ N)\circ M_z\bigr )=z(K_1\circ M_z)$. Thus $z^{-1}R^{\mathrm{ren}}_{M_z,M\circ N}\vert _{M_z\circ K_1}$ is well defined. Then it sends $M_z\circ K_k$ to $K_k\circ M_z$ for $k\ge 1$. Thus we obtain an R-matrix

$$\begin{equation*} z^{-1}R^{\mathrm{ren}}_{M_z, M\circ N}\vert _{M_z\circ (K_k/K_{k+1})}\colon M_z\circ (K_k/K_{k+1})\to (K_k/K_{k+1})\circ M_z \quad \text{for $1\le k\le \ell $.} \end{equation*}$$

Hence we have

$$\begin{equation*} R^{\mathrm{ren}}_{M_z, K_k/K_{k+1}}=z^{-s_k}z^{-1}R^{\mathrm{ren}}_{M_z, M\circ N}\vert _{M_z\circ (K_k/K_{k+1})} \end{equation*}$$

for some $s_k\in \mathbb{Z}_{\ge 0}$. Since the homogeneous degree of $R^{\mathrm{ren}}_{M_z,M\circ N}$ is $\Lambda (M,M\circ N)=\Lambda (M,N)$, we obtain

$$\begin{equation*} \Lambda (M,K_k/K_{k+1})=\Lambda (M,N)-2(1+s_k)<\Lambda (M,N). \end{equation*}$$ ■

Recall that the isomorphism classes of self-dual simple modules in $R \text{-gmod}$ are parametrized by the crystal basis $B(\infty )$ 28. The following theorem is an application of the above theorem.

Theorem 4.1.3.

Let $\phi$ be an element of the Grothendieck group $K(R\text{-gmod})$ given by

$$\begin{eqnarray*} \phi = \sum _{b \in B(\infty )} a_b [L_b], \end{eqnarray*}$$

where $L_b$ is the self-dual simple module corresponding to $b \in B(\infty )$ and $a_b \in \mathbb{Z}[q^{\pm 1}]$. Let $A$ be a real simple module in $R\text{-gmod}$. Assume that we have an equality

$$\begin{eqnarray*} \phi [A] = q^l [A] \phi \end{eqnarray*}$$

in $K(R\text{-gmod})$ for some $l \in \mathbb{Z}$. Then $A$ commutes with $L_b$ and

$$\begin{eqnarray*} l= \Lambda (A,L_b) \end{eqnarray*}$$

for every $b \in B(\infty )$ such that $a_b \neq 0$.

Proof.

Note that we have

$$\begin{eqnarray*} \phi [A] = \sum _b a_b [L_b \circ A] = \sum _b a_b ([L_b \mathbin{\text{$\nabla $}}A] + \sum _k [S_{b,k}]) \quad \text{and} \\ q^l [A] \phi = q^l \sum _b a_b [A \circ L_b ] = q^l \sum _b a_b (q^{\Lambda (L_b,A)}[L_b \mathbin{\text{$\nabla $}}A] + \sum _k [S^{b,k}]), \end{eqnarray*}$$

for some simple modules $S_{b,k}$ and $S^{b,k}$ satisfying

$$\begin{eqnarray*} \Lambda (S_{b,k},A) < \Lambda (L_b,A) \quad \text{and} \quad \Lambda (S^{b,k},A) < \Lambda (L_b,A) \end{eqnarray*}$$

by Theorem 4.1.1.

We may assume that $\left\{ b\in B(\infty ) \mid a_b \neq 0 \right\}\not =\emptyset$. Set

$$\begin{eqnarray*} t\mathbin{:=}\displaystyle \max \left\{ \Lambda (L_b,A) \mid a_b \neq 0 \right\}. \end{eqnarray*}$$

By taking the classes of self-dual simple modules $S$ with $\Lambda (S,A)=t$ in the expansions of $\phi [A]$ and $q^l [A] \phi$, we obtain

$$\begin{eqnarray*} \sum _{\Lambda (L_b,A)=t} a_b [L_b \mathbin{\text{$\nabla $}}A ] =\sum _{\Lambda (L_b,A)=t} q^l a_b q^{\Lambda (L_b,A)} [L_b \mathbin{\text{$\nabla $}}A ]. \end{eqnarray*}$$

In particular, we have $t=-l$.

Set

$$\begin{eqnarray*} t'\mathbin{:=}\displaystyle \max \left\{ \Lambda (A,L_b) \mid a_b \neq 0 \right\}. \end{eqnarray*}$$

Then, by a similar argument we have $t'=l$.

It follows that

$$\begin{eqnarray*} 0= t+t' \ge \Lambda (L_b,A) + \Lambda (A,L_b)\ge 0 \end{eqnarray*}$$ for every $b$ such that $a_b \neq 0$. Hence $A$ and $L_b$ commute.

Since

$$\begin{eqnarray*} \sum a_b\nobreakspace q^{\Lambda (A,L_b)} [A\circ L_b] =\sum a_b [L_b \circ A] =\phi [A]=q^l [A] \phi =q^l \sum a_b [A\circ L_b], \end{eqnarray*}$$

we have

$$\begin{eqnarray*} l=\Lambda (A,L_b) \end{eqnarray*}$$

for any $b$ such that $a_b \neq 0$, as desired.

■

Corollary 4.1.4.

Let $M$ and $N$ be simple modules. Assume that one of them is real. If $[M]$ and $[N]$ q-commute (i.e., $[M][N]=q^n[N][M]$ for some $n\in \mathbb{Z}$), then $M$ and $N$ commute. In particular, $M\circ N$ is simple.

The following corollary is an immediate consequence of the corollary above and Theorem 2.1.4.

Corollary 4.1.5.

Assume that the generalized Cartan matrix $A$ is symmetric and that $b_1, b_2\in B(\infty )$ satisfy the following conditions:

(i)

one of $G^\mathrm{up}(b_1)^2$ and $G^\mathrm{up}(b_2)^2$ is a member of the upper global basis up to a power of $q$,

(ii)

$G^\mathrm{up}(b_1)$ and $G^\mathrm{up}(b_2)$ q-commute.

Then their product $G^\mathrm{up}(b_1)G^\mathrm{up}(b_2)$ is a member of the upper global basis of $U^-_q(\mathfrak{g})$ up to a power of $q$.

4.2. Geometric results

The result of this subsection (Theorem 4.2.1) was explained to us by Peter McNamara. It will be used in the proof of the crucial result Theorem 10.3.1. In this subsection, we assume further that the base field $\mathbf{k}$ is of characteristic $0$.

Theorem 4.2.1 (34, Lemma 7.5).

Assume that the base field $\mathbf{k}$ is of characteristic $0$. Assume that $M\in R\text{-gmod}$ has a head $q^cH$ with a self-dual simple module $H$ and $c\in \mathbb{Z}$. Then we have the equality in the Grothendieck group $K(R\text{-gmod})$

$$\begin{equation*} [M]=q^c[H]+\sum _{k}q^{c_k}[S_k] \end{equation*}$$

with self-dual simple modules $S_k$ and $c_k>c$.

By duality, we obtain the following corollary.

Corollary 4.2.2.

Assume that the base field $\mathbf{k}$ is a field of characteristic $0$. Assume that $M\in R\text{-gmod}$ has a socle $q^cS$ with a self-dual simple module $S$ and $c\in \mathbb{Z}$. Then we have the equality in $K(R\text{-gmod})$

$$\begin{equation*} [M]=q^c[S]+\sum _{k}q^{c_k}[S_k] \end{equation*}$$

with self-dual simple modules $S_k$ and $c_k<c$.

Applying this theorem to convolution products, we obtain the following corollary.

Corollary 4.2.3.

Assume that the base field $\mathbf{k}$ is of characteristic $0$. Let $M$ and $N$ be simple modules. We assume that one of them is real. Then we have the equalities in $K(R\text{-gmod})$ as follows:

(i)

$[M\circ N]=[M\mathbin{\text{$\nabla $}}N]+\sum _{k}q^{c_k}[S_k]$with self-dual simple modules $S_k$ and$$\begin{equation*} c_k>-\widetilde{\Lambda }(M,N)=\bigl (-\Lambda (M,N)-(\operatorname {wt}(M),\operatorname {wt}(N)\bigr )/2. \end{equation*}$$

(ii)

$[M\circ N]=[ M \mathbin{\text{$\Delta $}}N ]+\sum _{k}q^{c_k}[S_k]$with self-dual simple modules $S_k$ and $c_k<\bigl (\Lambda (N,M)-(\operatorname {wt}(N),\operatorname {wt}(M))\bigr )/2$.

Note that $q^{\widetilde{\Lambda }(M,N)}M\mathbin{\text{$\nabla $}}N$ is self-dual by Lemma 3.1.4.

Theorem 4.1.1 and Theorem 4.2.1 solve affirmatively Conjecture 1 of Leclerc 29 in the symmetric generalized Cartan matrix case, as stated in the Introduction. More precisely, let $R$ be a symmetric KLR algebra over a base field $\mathbf{k}$ of characteristic $0$, and let $M$ and $N$ be simple modules over $R$. Assume further that $M$ is real. Then by Theorem 4.1.1 $M \mathbin{\text{$\nabla $}}N$ and $M \mathbin{\text{$\Delta $}}N$ appear exactly once in a Jordan–Hölder series of $M \circ N$. Write $M \mathbin{\text{$\nabla $}}N = q^m H$ and $M \mathbin{\text{$\Delta $}}N=q^s S$ for some self-dual simple modules $H$, $S$, and $m,s \in \mathbb{Z}$. By Theorem 4.2.1, we have

$$\begin{eqnarray*} [M \circ N] = q^m [H] + q^s [S] + \sum _k q^{c_k} [S_k], \end{eqnarray*}$$

where $S_k$ are self-dual simple modules, and $m < c_k < s$ for all $k$. Collecting the terms, we obtain

$$\begin{eqnarray*} [M\circ N]=q^m [H] + q^s [S] + \sum _{L \not \simeq H, S} \gamma _{M,N}^L (q) [L], \end{eqnarray*}$$

with

$$\begin{equation*} \gamma ^L_{M,N}(q) \in q^{m+1}\mathbb{Z}[q] \cap q^{s-1}\mathbb{Z}[q^{-1}], \end{equation*}$$

which proves Leclerc’s first conjecture via Theorem 2.1.4.

We obtain the following result which is a generalization of Lemma 3.2.18 in the characteristic-zero case.

Corollary 4.2.4.

Assume that the base field $\mathbf{k}$ is of characteristic $0$. Let $M$ and $N$ be simple modules. We assume that one of them is real. Write

$$\begin{equation*} [M\circ N]=\sum _{k=1}^nq^{c_k}[S_k] \end{equation*}$$

with self-dual simple modules $S_k$ and $c_k\in \mathbb{Z}$. Then we have

$$\begin{equation*} \max \left\{c_k \mid 1\le k\le n \right\}-\min \left\{c_k \mid 1\le k\le n \right\}=\operatorname {\mathfrak{d}}(M,N). \end{equation*}$$

4.3. Proof of Theorem 4.2.1

Recall that the graded algebra $R(\beta )$ ($\beta \in \mathsf{Q}^+$) is geometrically realized as follows 40. There exist a reductive group $G$ and a $G$-equivariant projective morphism $f\colon X\to Y$ from a smooth algebraic $G$-variety $X$ to an affine $G$-variety $Y$ defined over the complex number field $\mathbb{C}$ such that

$$\begin{equation*} R(\beta )\simeq \operatorname {\widetilde{E}nd}_{{\operatorname {D^b_{G}}}(\mathbf{k}_Y)}(\operatorname {R}f_* (\mathbf{k}_X [\dim X])) \quad \text{as a graded $\mathbf{k}$-algebra.} \end{equation*}$$

Here, ${\operatorname {D^b_{G}}}(\mathbf{k}_Y)$ denotes the $G$-equivariant derived category of the $G$-variety $Y$ with coefficient $\mathbf{k}$, and $\operatorname {\widetilde{E}nd}_{{\operatorname {D^b_{G}}}(\mathbf{k}_Y)}(K)=\operatorname {\widetilde{H}om}_{{\operatorname {D^b_{G}}}(\mathbf{k}_Y)}(K, K)$ with

$$\begin{equation*} \operatorname {\widetilde{H}om}_{{\operatorname {D^b_{G}}}(\mathbf{k}_Y)}(K, K')\mathbin{:=}\mathop{\textstyle \bigoplus }\limits _{n\in \mathbb{Z}} \operatorname {Hom}_{{\operatorname {D^b_{G}}}(\mathbf{k}_Y)}(K, K'[n]). \end{equation*}$$

We denote by $\mathbf{k}_X[\dim X]$ the direct sum of the constant sheaves on each connected component of $X$, all of which are shifted by their dimensions. By the decomposition theorem 1, we have a decomposition

$$\begin{equation*} \operatorname {R}f_* (\mathbf{k}_X [\dim X])\simeq \mathop{\textstyle \bigoplus }\limits _{a\in J}E_a\mathop{\otimes }\mathcal{F}_a, \end{equation*}$$

where $\{\mathcal{F}_a\}_{a\in J}$ is a finite family of simple perverse sheaves on $Y$ and $E_a$ is a non-zero finite-dimensional graded $\mathbf{k}$-vector space such that

$$\begin{eqnarray} H^k(E_a)\simeq H^{-k}(E_a)\quad \text{for any $k\in \mathbb{Z}$.} {\tag{4.1}} \end{eqnarray}$$

The last fact 4.1 follows from the hard Lefschetz theorem 1.

Set $A_{a,b}=\operatorname {\widetilde{H}om}_{{\operatorname {D^b_{G}}}(\mathbf{k}_Y)}(\mathcal{F}_b, \mathcal{F}_a)$. Then we have the multiplication morphisms

$$\begin{equation*} A_{a,b}\mathop{\otimes }A_{b,c}\to A_{a,c} \end{equation*}$$

so that

$$\begin{equation*} A\mathbin{:=}\mathop{\textstyle \bigoplus }\limits _{a,b\in J}A_{a,b} \end{equation*}$$

has a structure of $\mathbb{Z}$-graded algebra such that

$$\begin{equation*} A_{\le 0}\mathbin{:=}\mathop{\textstyle \bigoplus }\limits _{n\le 0}A_n=A_0\simeq \mathbf{k}^J. \end{equation*}$$

Hence the family of the isomorphism classes of simple objects (up to a grading shift) in $A\text{-gmod}$ is $\{\mathbf{k}_a\}_{a\in J}$. Here, $\mathbf{k}_a$ is the module obtained by the algebra homomorphism $A\to A_{\le 0}\simeq \mathbf{k}^J\to \mathbf{k}$, where the last arrow is the $a$th projection. Hence we have

$$\begin{equation*} K(A\text{-gmod})\simeq \mathop{\textstyle \bigoplus }\limits _{a\in J}\mathbb{Z}[q^{\pm 1}][\mathbf{k}_a]. \end{equation*}$$

On the other hand, we have

$$\begin{equation*} R(\beta )\simeq \mathop{\textstyle \bigoplus }\limits _{a,b\in J}E_a\mathop{\otimes }A_{a,b}\mathop{\otimes }E_b^*. \end{equation*}$$

Set

$$\begin{equation*} L\mathbin{:=}\mathop{\textstyle \bigoplus }\limits _{a,b\in J}E_a\mathop{\otimes }A_{a,b}. \end{equation*}$$

Then, $L$ is endowed with a natural structure of $(\mathop{\textstyle \bigoplus }\limits _{a,b\in J}E_a\mathop{\otimes }A_{a,b}\mathop{\otimes }E_b^*, A)$-bimodule. It is well known that the functor $M\mapsto L\otimes _AM$ gives a graded Morita-equivalence

$$\begin{equation*} \Phi \colon A\text{-gmod}\similarrightarrow R(\beta )\text{-gmod}. \end{equation*}$$

Note that $\Phi (\mathbf{k}_a)\simeq E_a$ and $\{E_a\}_{a\in J}$ is the set of isomorphism classes of self-dual simple graded $R(\beta )$-modules by 4.1.

By the above observation, in order to prove the theorem, it is enough to show the corresponding statement for the graded ring $A$, which is obvious.

5. Quantum cluster algebras

In this section we recall the definition of skew-symmetric quantum cluster algebras following 3 and 11, Section 8.

5.1. Quantum seeds

Fix a finite index set ${J}={{J}}_{\mathrm{ex}}\sqcup {{J}}_{\mathrm{f\mspace {.01mu}r}}$ with the decomposition into the set ${{J}}_{\mathrm{ex}}$ of exchangeable indices and the set ${{J}}_{\mathrm{f\mspace {.01mu}r}}$ of frozen indices. Let $L=(\lambda _{ij})_{i,j\in {J}}$ be a skew-symmetric integer-valued ${J}\times {J}$-matrix.

Definition 5.1.1.

We define $\mathscr{P}(L)$ as the $\mathbb{Z}[q^{\pm 1/2}]$-algebra generated by a family of elements $\{X_i\}_{i\in {J}}$ with the defining relations

$$\begin{eqnarray*} &&X_iX_j=q^{\lambda _{ij}}X_j X_i \quad (i,j\in {J}). \end{eqnarray*}$$

We denote by $\mathscr{F}(L)$ the skew field of fractions of $\mathscr{P}(L)$.

For ${\mathbf{a}}=(a_i)_{i\in {J}}\in \mathbb{Z}^{J}$, we define the element $X^{\mathbf{a}}$ of $\mathscr{F}(L)$ as

$$\begin{eqnarray*} &&X^{\mathbf{a}}\mathbin{:=}q^{1/2 \sum _{i > j} a_ia_j\lambda _{ij}} \overset{\xrightarrow {\,\ \,}}{\prod }_{i\in {J}} X_i^{a_i}. \end{eqnarray*}$$

Here we take a total order $<$ on the set ${J}$ and $\overset{\xrightarrow {\,\ \,}}{\prod }_{i\in {J}} X_i^{a_i}=X_{i_1}^{a_{i_1}}\cdots X_{i_r}^{a_{i_r}}$ where ${J}=\{i_1,\ldots ,i_r\}$ with $i_1<\cdots <i_r$. Note that $X^{\mathbf{a}}$ does not depend on the choice of a total order of ${J}$.

We have

$$\begin{eqnarray} &&X^{\mathbf{a}}X^{\mathbf{b}}=q^{1/2\sum _{i,j\in {J}} a_ib_j\lambda _{ij}}X^{\mathbf{a+\mathbf{b}}}. {\tag{5.1}} \end{eqnarray}$$

If $\mathbf{a}\in \mathbb{Z}_{\ge 0}^{J}$, then $X^{\mathbf{a}}$ belongs to $\mathscr{P}(L)$.

It is well known that $\{X^{\mathbf{a}}\}_{\mathbf{a}\in \mathbb{Z}_{\ge 0}^{J}}$ is a basis of $\mathscr{P}(L)$ as a $\mathbb{Z}[q^{\pm 1/2}]$-module.

Let $A$ be a $\mathbb{Z}[q^{\pm 1/2}]$-algebra. We say that a family $\{x_i\}_{i\in {J}}$ of elements of $A$ is $L$-commuting if it satisfies $x_ix_j=q^{\lambda _{ij}}x_jx_i$ for any $i,j\in {J}$. In such a case we can define $x^\mathbf{a}$ for any $\mathbf{a}\in \mathbb{Z}_{\ge 0}^{J}$. We say that an $L$-commuting family $\{x_i\}_{i\in {J}}$ is algebraically independent if the algebra map $\mathscr{P}(L)\to A$ given by $X_i\mapsto x_i$ is injective.

Let $\widetilde{B} = (b_{ij})_{(i,j)\in {J}\times {{J}}_{\mathrm{ex}}}$ be an integer-valued ${J}\times {{J}}_{\mathrm{ex}}$-matrix. We assume that the principal part $B\mathbin{:=}(b_{ij})_{i,j\in {{J}}_{\mathrm{ex}}}$ of $\widetilde{B}$ is skew-symmetric.

To the matrix $\widetilde{B}$ we can associate the quiver $Q_{\widetilde{B}}$ without loops, $2$-cycles, and arrows between frozen vertices such that its vertices are labeled by $J$ and the arrows are given by

$$\begin{equation} b_{ij} = (\text{the number of arrows from $i$ to $j$}) - (\text{the number of arrows from $j$ to $i$}). {\tag{5.2}} \end{equation}$$

Here we extend the ${J}\times {{J}}_{\mathrm{ex}}$-matrix $\widetilde{B}$ to the skew-symmetric ${J}\times {J}$-matrix $\widetilde{B}'=(b_{ij})_{i,j\in {J}}$ by setting $b_{ij}=0$ for $i,j\in {{J}}_{\mathrm{f\mspace {.01mu}r}}$.

Conversely, whenever we have a quiver with vertices labeled by $J$ and without loops, $2$-cycles, and arrows between frozen vertices, we can associate a ${J}\times {J}_\mathrm{ex}$-matrix $\widetilde{B}$ by 5.2.

We say that the pair $(L, \widetilde{B})$ is compatible if there exists a positive integer $d$ such that

$$\begin{eqnarray} &&\sum _{k\in {J}} \lambda _{ik} b_{kj} =\delta _{ij} d \quad (i\in {J},\;j\in {{J}}_{\mathrm{ex}}). {\tag{5.3}} \end{eqnarray}$$

Let $(L, \widetilde{B})$ be a compatible pair and $A$ a $\mathbb{Z}[q^{\pm 1/2}]$-algebra. We say that $\mathscr{S}= (\{x_i\}_{i\in {J}},L, \widetilde{B})$ is a quantum seed in $A$ if $\{x_i\}_{i\in {J}}$ is an algebraically independent $L$-commuting family of elements of $A$.

The set $\{x_i\}_{i\in {J}}$ is called the cluster of $\mathscr{S}$ and its elements the cluster variables. The cluster variables $x_i$ ($i\in {{J}}_{\mathrm{f\mspace {.01mu}r}}$) are called the frozen variables. The elements $x^{\mathbf{a}}$ (${\mathbf{a}}\in \mathbb{Z}_{\ge 0}^{J}$) are called the quantum cluster monomials.

5.2. Mutation

For $k\in {{J}}_{\mathrm{ex}}$, we define a ${J}\times {J}$-matrix $E=(e_{ij})_{i,j\in {J}}$ and a ${{J}}_{\mathrm{ex}}\times {{J}}_{\mathrm{ex}}$-matrix $F=(f_{ij})_{i,j\in {{J}}_{\mathrm{ex}}}$ as follows:

$$\begin{eqnarray*} e_{ij}= \begin{cases} \delta _{ij} & \text{if} \ j \neq k, \\ -1 & \text{if} \ i= j = k, \\ \max (0, -b_{ik}) & \text{if} \ i \neq j = k, \end{cases} \hspace{10ex} f_{ij}= \begin{cases} \delta _{ij} & \text{if} \ i \neq k, \\ -1 & \text{if} \ i= j = k, \\ \max (0, b_{kj}) & \text{if} \ i = k \neq j. \end{cases} \end{eqnarray*}$$

The mutation $\mu _k(L,\widetilde{B})\mathbin{:=}(\mu _k(L),\mu _k(\widetilde{B}))$ of a compatible pair $(L,\widetilde{B})$ in direction $k$ is given by

$$\begin{eqnarray*} \mu _k(L)\mathbin{:=}(E^T) \, L \, E, \qquad \mu _k(\widetilde{B})\mathbin{:=}E \, \widetilde{B} \, F. \end{eqnarray*}$$

Then the pair $(\mu _k(L),\mu _k(\widetilde{B}))$ is also compatible with the same integer $d$ as in the case of $(L,\widetilde{B})$ 3.

Note that for each $k\in {{J}}_{\mathrm{ex}}$, we have

$$\begin{eqnarray} \mu _k(\widetilde{B})_{ij} = \begin{cases} -b_{ij} & \text{if} \ i=k \ \text{or} \ j=k, \\ b_{ij} + (-1)^{\delta (b_{ik} < 0)} \max (b_{ik} b_{kj}, 0) & \text{otherwise,} \end{cases} {\tag{5.4}} \end{eqnarray}$$

and

$$\begin{eqnarray*} \mu _k(L)_{ij} = \begin{cases} 0 & \text{if} \ i=j \\ -\lambda _{kj}+\displaystyle \sum _{t\in {J}} \max (0, -b_{tk}) \lambda _{tj} & \text{if} \ i=k, \ j\neq k, \\ -\lambda _{ik}+\displaystyle \sum _{t\in {J}} \max (0, -b_{tk}) \lambda _{it} & \text{if} \ i \neq k, \ j= k, \\ \lambda _{ij} & \text{otherwise.} \end{cases} \end{eqnarray*}$$

Note also that we have

$$\begin{equation*} \displaystyle \sum _{t\in {J}} \max (0, -b_{tk}) \lambda _{it} =\displaystyle \sum _{t\in {J}} \max (0, b_{tk}) \lambda _{it} \end{equation*}$$

for $i\in {J}$ with $i\neq k$, since $(L,\widetilde{B})$ is compatible.

We define

$$\begin{eqnarray} &&a_i'= \begin{cases} -1 & \text{if} \ i=k, \\ \max (0,b_{ik}) & \text{if} \ i\neq k, \end{cases} \qquad a_i''= \begin{cases} -1 & \text{if} \ i=k, \\ \max (0,-b_{ik}) & \text{if} \ i\neq k, \end{cases} {\tag{5.5}} \end{eqnarray}$$

and set ${\mathbf{a}}'\mathbin{:=}(a_i')_{i\in {J}}$ and ${\mathbf{a}}''\mathbin{:=}(a_i'')_{i\in {J}}$.

Let $A$ be a $\mathbb{Z}[q^{\pm 1/2}]$-algebra contained in a skew field $K$. Let $\mathscr{S}=(\{x_i\}_{i\in {J}},L, \widetilde{B})$ be a quantum seed in $A$. Define the elements $\mu _k(x)_i$ of $K$ by

$$\begin{eqnarray} \mu _k(x)_i\mathbin{:=}\begin{cases} x^{{\mathbf{a}}'} + x^{{\mathbf{a}}''}, & \text{if} \ i=k, \\ x_i & \text{if} \ i\neq k. \end{cases} {\tag{5.6}} \end{eqnarray}$$

Then $\{\mu _k(x)_i\}$ is an algebraically independent $\mu _k(L)$-commuting family in $K$. We call

$$\begin{eqnarray*} \mu _k(\mathscr{S})\mathbin{:=}\bigl (\{\mu _k(x)_i\}_{i\in {J}},\mu _k(L),\mu _k(\widetilde{B})\bigr ) \end{eqnarray*}$$

the mutation of $\mathscr{S}$ in direction $k$. It becomes a new quantum seed in $K$.

Definition 5.2.1.

Let $\mathscr{S}=(\{x_i\}_{i\in {J}},L, \widetilde{B})$ be a quantum seed in $A$. The quantum cluster algebra $\mathscr{A}_{q^{1/2}}(\mathscr{S})$ associated to the quantum seed $\mathscr{S}$ is the $\mathbb{Z}[q^{\pm 1/2}]$-subalgebra of the skew field $K$ generated by all the quantum cluster variables in the quantum seeds obtained from $\mathscr{S}$ by any sequence of mutations.

We call $\mathscr{S}$ the initial quantum seed of the quantum cluster algebra $\mathscr{A}_{q^{1/2}}(\mathscr{S})$.

6. Monoidal categorification of cluster algebras

Throughout this section, fix ${J}={{J}}_{\mathrm{ex}}\sqcup {{J}}_{\mathrm{f\mspace {.01mu}r}}$ and a base field $\mathbf{k}$.

Let $\mathcal{C}$ be a $\mathbf{k}$-linear abelian monoidal category. For the definition of monoidal category, see, for example, 14, Appendix A.1. Note that in 14, it was called the tensor category. A $\mathbf{k}$-linear abelian monoidal category is a $\mathbf{k}$-linear monoidal category such that it is abelian and the tensor functor $\;\mathop{\otimes }\;$ is $\mathbf{k}$-bilinear and exact.

We assume further the following conditions on $\mathcal{C}$:

$$\begin{equation} \begin{cases} \text{(i)} & \text{Any object of $\mathcal{C}$ has a finite length,}\\ \text{(ii)} & \text{$\mathbf{k}\similarrightarrow \operatorname {Hom}_{\mathcal{C}}(S,S)$ for any simple object $S$ of $\mathcal{C}$.} \end{cases} {\tag{6.1}} \end{equation}$$

A simple object $M$ in $\mathcal{C}$ is called real if $M \otimes M$ is simple.

6.1. Ungraded cases

Definition 6.1.1.

Let $\mathscr{S}= (\{ M_i\}_{i\in {J}},\widetilde{B})$ be a pair of a family $\{ M_i\}_{i\in {J}}$ of simple objects in $\mathcal{C}$ and an integer-valued ${J}\times {{J}}_{\mathrm{ex}}$-matrix $\widetilde{B} = (b_{ij})_{(i,j)\in {J}\times {{J}}_{\mathrm{ex}}}$ whose principal part is skew-symmetric. We call $\mathscr{S}$ a monoidal seed in $\mathcal{C}$ if

(i)

$M_i \otimes M_j \simeq M_j \otimes M_i$ for any $i,j\in {J}$,

(ii)

$\mathop{\textstyle \bigotimes }\limits _{i\in {J}} M_i^{\otimes a_i}$ is simple for any $(a_i)_{i\in {J}}\in \mathbb{Z}_{\ge 0}^{{J}}$.

Definition 6.1.2.

For $k\in {{J}}_{\mathrm{ex}}$, we say that a monoidal seed $\mathscr{S}= (\{ M_i\}_{i\in {J}},\widetilde{B})$ admits a mutation in direction $k$ if there exists a simple object $M_k' \in \mathcal{C}$ such that

(i)

there exist exact sequences in $\mathcal{C}$,$$\begin{eqnarray*} &&0 \to \mathop{\textstyle \bigotimes }\limits _{b_{ik} >0} M_i^{\otimes b_{ik}} \to M_k \otimes M_k' \to \mathop{\textstyle \bigotimes }\limits _{b_{ik} <0} M_i^{\otimes (-b_{ik})} \to 0,\\ &&0 \to \mathop{\textstyle \bigotimes }\limits _{b_{ik} <0} M_i^{\otimes (-b_{ik})} \to M_k' \otimes M_k \to \mathop{\textstyle \bigotimes }\limits _{b_{ik} >0} M_i^{\otimes b_{ik}} \to 0; \end{eqnarray*}$$

(ii)

the pair $\mu _k(\mathscr{S})\mathbin{:=}(\{M_i\}_{i\neq k}\cup \{M_k'\},\mu _k(\widetilde{B}))$ is a monoidal seed in $\mathcal{C}$.

Recall that a cluster algebra $A$ with an initial seed $(\{x_i\}_{i\in {J}},\widetilde{B})$ is the $\mathbb{Z}$-subalgebra of $\mathbb{Q}(x_i\vert i\in {J})$ generated by all the cluster variables in the seeds obtained from $(\{x_i\}_{i\in {J}},\widetilde{B})$ by any sequence of mutations. Here, the mutation $x_k'$ of a cluster variable $x_k$ $(k\in {{J}}_{\mathrm{ex}})$ is given by

$$\begin{eqnarray*} x_k' = \dfrac{\prod _{b_{ik} \ge 0} x_i^{b_{ik}}+\prod _{b_{ik} \le 0} x_i^{-b_{ik}}}{x_k}, \end{eqnarray*}$$

and the mutation of $\widetilde{B}$ is given in 5.4.

Definition 6.1.3.

A $\mathbf{k}$-linear abelian monoidal category $\mathcal{C}$ satisfying 6.1 is called a monoidal categorification of a cluster algebra $A$ if

(i)

the Grothendieck ring $K(\mathcal{C})$ is isomorphic to $A$,

(ii)

there exists a monoidal seed $\mathscr{S}= ( \{M_i\}_{i\in {J}},\widetilde{B})$ in $\mathcal{C}$ such that $[\mathscr{S}]\mathbin{:=}( \{[M_i]\}_{i\in {J}},\widetilde{B})$ is the initial seed of $A$ and $\mathscr{S}$ admits successive mutations in all directions.

Note that if $\mathcal{C}$ is a monoidal categorification of $A$, then every seed in $A$ is of the form $( \{[M_i]\}_{i\in {J}},\widetilde{B})$ for some monoidal seed $( \{M_i\}_{i\in {J}},\widetilde{B})$ in $\mathcal{C}$. In particular, all the cluster monomials in $A$ are the classes of real simple objects in $\mathcal{C}$.

6.2. Graded cases

Let $\mathsf{Q}$ be a free abelian group equipped with a symmetric bilinear form

$$\begin{eqnarray*} ( \ , \ ) : \mathsf{Q}\times \mathsf{Q}\to \mathbb{Z}\quad \text{such that} \ (\beta ,\beta ) \in 2\mathbb{Z}\ \text{for all} \ \beta \in \mathsf{Q}. \end{eqnarray*}$$

We consider a $\mathbf{k}$-linear abelian monoidal category $\mathcal{C}$ satisfying 6.1 and the following conditions:

(6.2)

(i)

We have a direct sum decomposition $\mathcal{C}= \mathop{\textstyle \bigoplus }\limits _{\beta \in \mathsf{Q}} \mathcal{C}_\beta$ such that the tensor product $\otimes$ sends $\mathcal{C}_\beta \times \mathcal{C}_\gamma$ to $\mathcal{C}_{\beta + \gamma }$ for every $\beta , \gamma \in \mathsf{Q}$.

(ii)

There exists an object $Q \in \mathcal{C}_0$ satisfying

(a)

there is an isomorphism$$\begin{equation*} R_{Q}(X) : Q \otimes X \similarrightarrow X \otimes Q \end{equation*}$$

functorial in $X \in \mathcal{C}$ such that$$\begin{equation*} \vcenter{\img[][220pt][37pt][{$\xymatrix@C=7ex{ Q\tensor X\tensor Y\ar[r]_-{R_Q(X)}\ar@/^3ex/[rr]^{R_Q(X\tensor Y)} &X\tensor Q\tensor Y\ar[r]_-{R_Q(Y)} &X\tensor Y\tensor Q}$}]{Images/img5cdbfbb55cbbdda96652369490dc9573.svg}} \end{equation*}$$

commutes for any $X,Y\in \mathcal{C}$;

(b)

the functor $X \mapsto Q \otimes X$ is an equivalence of categories.

(iii)

for any $M$, $N\in \mathcal{C}$, we have $\operatorname {Hom}_\mathcal{C}(M,Q^{\mathop{\otimes }n}\mathop{\otimes }N)=0$ except finitely many integers $n$.

We denote by $q$ the auto-equivalence $Q \otimes \mdsmblkcircle$ of $\mathcal{C}$, and call it the grading shift functor.

In such a case the Grothendieck group $K(\mathcal{C})$ is a $\mathsf{Q}$-graded $\mathbb{Z}[q^{\pm 1}]$-algebra: $K(\mathcal{C})=\mathop{\textstyle \bigoplus }\limits _{\beta \in \mathsf{Q}}K(\mathcal{C})_\beta$ where $K(\mathcal{C})_\beta =K(\mathcal{C}_\beta )$. Moreover, we have

$$\begin{equation*} K(\mathcal{C})=\mathop{\textstyle \bigoplus }\limits _{S}\mathbb{Z}[q^{\pm 1}][S], \end{equation*}$$

where $S$ ranges over equivalence classes of simple modules. Here, two simple modules $S$ and $S'$ are equivalent if $q^nS\simeq S'$ for some $n\in \mathbb{Z}$.

For $M\in \mathcal{C}_{\beta }$, we sometimes write $\beta =\operatorname {wt}(M)$ and call it the weight of $M$. Similarly, for $x\in \mathbb{Q}(q^{1/2})\mathop{\otimes }_{\mathbb{Z}[q^{\pm 1}]}K(\mathcal{C}_\beta )$, we write $\beta =\operatorname {wt}(x)$ and call it the weight of $x$.

Definition 6.2.1.

We call a quadruple $\mathscr{S}= (\{M_i\}_{i\in {J}}, L,\widetilde{B}, D)$ a quantum monoidal seed in $\mathcal{C}$ if it satisfies the following conditions:

(i)

$\widetilde{B} = (b_{ij})_{i\in {J},\,j\in {{J}}_{\mathrm{ex}}}$ is an integer-valued ${J}\times {{J}}_{\mathrm{ex}}$-matrix whose principal part is skew-symmetric,

(ii)

$L=(\lambda _{ij})_{i,j\in {J}}$ is an integer-valued skew-symmetric ${J}\times {J}$-matrix,

(iii)

$D=\{d_i\}_{i\in {J}}$ is a family of elements in $\mathsf{Q}$,

(iv)

$\{M_i\}_{i\in {J}}$ is a family of simple objects such that $M_i \in \mathcal{C}_{d_i}$ for any $i\in {J}$,

(v)

$M_i \otimes M_j \simeq q^{\lambda _{ij}} M_j \otimes M_i$ for all $i, j \in {J}$,

(vi)

$M_{i_1} \otimes \cdots \otimes M_{i_t}$ is simple for any sequence $(i_1,\ldots ,i_t)$ in ${J}$,

(vii)

The pair $(L,\widetilde{B})$ is compatible in the sense of 5.3 with $d=2$,

(viii)

$\lambda _{ij} - (d_i,d_j) \in 2\mathbb{Z}$ for all $i,j \in {J}$,

(ix)

$\displaystyle \sum _{i\in {J}}b_{ik}d_i =0$ for all $k\in {{J}}_{\mathrm{ex}}$.

Let $\mathscr{S}=(\{M_i\}_{i\in {J}}, L,\widetilde{B}, D)$ be a quantum monoidal seed. For any $X\in \mathcal{C}_\beta$ and $Y\in \mathcal{C}_\gamma$ such that $X\otimes Y\simeq q^{c}Y \otimes X$ and $c+(\beta ,\gamma )\in 2\mathbb{Z}$, we set

$$\begin{eqnarray*} &&\widetilde{\Lambda }(X,Y)=\frac{1}{2}\bigl (-c+(\beta ,\gamma )\bigr )\in \mathbb{Z} \end{eqnarray*}$$

and

$$\begin{eqnarray*} X\mathop{\text{$\odot $}}Y\mathbin{:=}q^{\widetilde{\Lambda }(X,Y)}X\mathop{\otimes }Y\simeq q^{\widetilde{\Lambda }(Y,X)}Y\mathop{\otimes }X. \end{eqnarray*}$$

Then $X\mathop{\text{$\odot $}}Y\simeq Y\mathop{\text{$\odot $}}X$. For any sequence $(i_1,\ldots , i_\ell )$ in ${J}$, we define

$$\begin{equation*} \mathop{\textstyle \bigodot }\limits _{s=1}^{\ell } M_{i_s}\mathbin{:=}(\cdots ((M_{i_1}\mathop{\text{$\odot $}}M_{i_2})\mathop{\text{$\odot $}}M_{i_3})\cdots )\mathop{\text{$\odot $}}M_{i_\ell }. \end{equation*}$$

Then we have

$$\begin{eqnarray*} \mathop{\textstyle \bigodot }\limits _{s=1}^{\ell } M_{i_s} = q^{\frac{1}{2}\sum _{1\le u<v \le \ell }(-\lambda _{i_u i_v} +(d_{i_u},d_{i_v}) ) } M_{i_1} \otimes \cdots \otimes M_{i_\ell }. \end{eqnarray*}$$

For any $w\in \mathfrak{S}_\ell$, we have

$$\begin{eqnarray*} &&\mathop{\textstyle \bigodot }\limits _{s=1}^{\ell } M_{ i_{w(s)}}\simeq \mathop{\textstyle \bigodot }\limits _{s=1}^{\ell } M_{i_s}. \end{eqnarray*}$$

Hence for any subset $A$ of ${J}$ and any set of non-negative integers $\{m_a\}_{a\in A}$, we can define $\mathop{\textstyle \bigodot }\limits _{a\in A}M_a^{\text{$\odot $}m_a}$.

For $(a_i)_{i\in {J}}\in \mathbb{Z}_{\ge 0}^{J}$ and $(b_i)_{i\in {J}}\in \mathbb{Z}_{\ge 0}^{J}$, we have

$$\begin{equation*} \bigl (\mathop{\textstyle \bigodot }\limits _{i\in {J}}M_i^{\text{$\odot $}a_i}\bigr )\mathop{\text{$\odot $}}\bigl (\mathop{\textstyle \bigodot }\limits _{i\in {J}}M_i^{\text{$\odot $}b_i}\bigr )\simeq \mathop{\textstyle \bigodot }\limits _{i\in {J}}M_i^{\text{$\odot $}(a_i+b_i)}. \end{equation*}$$

Let $\mathscr{S}=(\{M_i\}_{i\in {J}}, L,\widetilde{B}, D)$ be a quantum monoidal seed. When the $L$-commuting family $\{[M_i]\}_{i\in {J}}$ of elements of $\mathbb{Z}[q^{\pm 1/2}]\mathop{\otimes }_{\mathbb{Z}[q^{\pm 1}]}K(\mathcal{C})$ is algebraically independent, we shall define a quantum seed $[\mathscr{S}]$ in $\mathbb{Z}[q^{\pm 1/2}]\mathop{\otimes }_{\mathbb{Z}[q^{\pm 1}]}K(\mathcal{C})$ by

$$\begin{eqnarray*} &&[\mathscr{S}]=(\{q^{-(d_i,d_i)/4}[M_i]\}_{i\in {J}}, L,\widetilde{B}). \end{eqnarray*}$$

Set

$$\begin{eqnarray*} &&X_i:=q^{-(d_i,d_i)/4}[M_i]. \end{eqnarray*}$$

Then for any $\mathbf{a}=(a_i)_{i\in {J}}\in \mathbb{Z}_{\ge 0}^{J}$, we have

$$\begin{equation*} X^{\mathbf{a}}=q^{-(\mu ,\mu )/4}[\mathop{\textstyle \bigodot }\limits _{i\in {J}}M_i^{\text{$\odot $}a_i}], \end{equation*}$$

where $\mu =\operatorname {wt}(\mathop{\textstyle \bigodot }\limits _{i\in {J}}M_i^{\text{$\odot $}a_i})=\operatorname {wt}(X^{\mathbf{a}})=\sum _{i\in {J}}a_id_i$.

For a given $k\in {{J}}_{\mathrm{ex}}$, we define the mutation $\mu _k(D) \in \mathsf{Q}^{J}$ of $D$ in direction $k$ with respect to $\widetilde{B}$ by

$$\begin{eqnarray*} \mu _k(D)_i =d_i \ (i \neq k), \quad \mu _k(D)_k=-d_k+\sum _{b_{ik} >0} b_{ik} d_i. \end{eqnarray*}$$

Note that

$$\begin{eqnarray*} \mu _k(\mu _k(D))=D. \end{eqnarray*}$$

Note also that $(\mu _k(L),\mu _k(\widetilde{B}),\mu _k(D))$ satisfies conditions (viii) and (ix) in Definition 6.2.1 for any $k\in {{J}}_{\mathrm{ex}}$.

We have the following lemma.

Lemma 6.2.2.

Set $X'_k=\mu _k(X)_k$, the mutation of $X_k$ as in 5.6. Set $\zeta =\operatorname {wt}(X'_k)=-d_k+\sum _{b_{ik}>0}b_{ik}d_i$. Then we have

$$\begin{align*} q^{m_k}[M_k] q^{(\zeta ,\zeta )/4}X'_k&= q [\mathop{\textstyle \bigodot }\limits _{b_{ik} >0} M_i^{\text{$\odot $}b_{ik}}]+ [\mathop{\textstyle \bigodot }\limits _{b_{ik} <0} M_i^{\text{$\odot $}(-b_{ik})}],\\[12.5pt] q^{m'_k} q^{(\zeta ,\zeta )/4}X'_k[M_k]&= [\mathop{\textstyle \bigodot }\limits _{b_{ik} >0} M_i^{\text{$\odot $}b_{ik}}]+ q[\mathop{\textstyle \bigodot }\limits _{b_{ik} <0} M_i^{\text{$\odot $}(-b_{ik})}], \end{align*}$$

where

$$\begin{equation} \left\{\begin{array}{l} m_k=\dfrac{1}{2}(d_k,\zeta ) +\dfrac{1}{2} \displaystyle \sum _{b_{ik} < 0}\lambda _{ki} b_{ik}, \\[5.0pt] m'_k=\dfrac{1}{2}(d_k,\zeta ) +\dfrac{1}{2} \displaystyle \sum _{b_{ik} > 0} \lambda _{ki}b_{ik}. \end{array}\right. {\tag{6.3}} \end{equation}$$

Proof.

By 5.1, we have

$$\begin{eqnarray*} X_k X^{\mathbf{a}} = q^{\frac{1}{2} \sum _{i \in {J}} a_i \lambda _{ki}} X^{{{\mathbf{e}}_k} +\mathbf{a}} \quad \text{for ${\mathbf{a}}=(a_i)_{i\in {J}} \in \mathbb{Z}^{J}$ and $({\mathbf{e}}_k)_i = \delta _{ik}$ ($i \in {J}$).} \end{eqnarray*}$$

Let $\mathbf{a}'$ and ${\mathbf{a}}''$ be as in 5.5. Because

$$\begin{eqnarray*} \sum _{i \in {J}} a_i' \lambda _{ki} - \sum _{i \in {J}} a_i'' \lambda _{ki} = \sum _{b_{ik} >0} b_{ik} \lambda _{ki} - \sum _{b_{ik} <0} (-b_{ik}) \lambda _{ki} = \sum _{i \in {J}} b_{ik} \lambda _{ki} = 2, \end{eqnarray*}$$

we have

$$\begin{eqnarray*} X_k X'_k = X_k(X^{{\mathbf{a}}'} + X^{{\mathbf{a}}''}) =q^{\frac{1}{2} \sum _i a_i'' \lambda _{ki} } (qX^{{\mathbf{e_k}}+{\mathbf{a}}'} + X^{{\mathbf{e_k}}+{\mathbf{a}}''}). \end{eqnarray*}$$

Note that $\operatorname {wt}(X^{{\mathbf{e_k}}+{\mathbf{a}}'}) =\operatorname {wt}(X^{{\mathbf{e_k}}+{\mathbf{a}}''})= d_k +\zeta .$ It follows that

$$\begin{eqnarray*} m_k&=&-\dfrac{1}{4}((d_k,d_k)+(\zeta ,\zeta ))-\dfrac{1}{2} \sum _{i \in J} a_i'' \lambda _{ki} + \dfrac{1}{4}(\zeta +d_k,\zeta +d_k) \\ &=&\dfrac{1}{2} (d_k, \zeta ) +\dfrac{1}{2} \sum _{b_{ik} <0} b_{ik} \lambda _{ki}. \end{eqnarray*}$$

One can calculate $m_k'$ in a similar way.

■

Definition 6.2.3.

We say that a quantum monoidal seed $\mathscr{S}=(\{M_i\}_{i\in {J}}, L,\widetilde{B}, D)$ admits a mutation in direction $k\in {{J}}_{\mathrm{ex}}$ if there exists a simple object $M_k' \in \mathcal{C}_{\mu _k(D)_k}$ such that

(i)

there exist exact sequences in $\mathcal{C}$,$$\begin{eqnarray} &&0 \to q \mathop{\textstyle \bigodot }\limits _{b_{ik} >0} M_i^{\text{$\odot $}b_{ik}} \to q^{m_k} M_k \otimes M_k' \to \mathop{\textstyle \bigodot }\limits _{b_{ik} <0} M_i^{\text{$\odot $}(-b_{ik})} \to 0, {\tag{6.4}}\\ &&0 \to q \mathop{\textstyle \bigodot }\limits _{b_{ik} <0} M_i^{\text{$\odot $}(-b_{ik})} \to q^{m_k'} M_k' \otimes M_k \to \mathop{\textstyle \bigodot }\limits _{b_{ik} >0} M_i^{\text{$\odot $}b_{ik}} \to 0, {\tag{6.5}} \end{eqnarray}$$

where $m_k$ and $m'_k$ are as in 6.3.

(ii)

$\mu _k(\mathscr{S})\mathbin{:=}\bigl (\{M_i\}_{i\neq k}\sqcup \{M_k'\},\mu _k(L), \mu _k(\widetilde{B}), \mu _k(D)\bigr )$ is a quantum monoidal seed in $\mathcal{C}$.

We call $\mu _k(\mathscr{S})$ the mutation of $\mathscr{S}$ in direction $k$.

By Lemma 6.2.2, the following lemma is obvious.

Lemma 6.2.4.

Let $\mathscr{S}=(\{M_i\}_{i\in {J}}, L,\widetilde{B}, D)$ be a quantum monoidal seed which admits a mutation in direction $k\in {{J}}_{\mathrm{ex}}$. Then we have

$$\begin{equation*} [\mu _k(\mathscr{S})]=\mu _k([\mathscr{S}]). \end{equation*}$$

Definition 6.2.5.

Assume that a $\mathbf{k}$-linear abelian monoidal category $\mathcal{C}$ satisfies 6.1 and 6.2. The category $\mathcal{C}$ is called a monoidal categorification of a quantum cluster algebra $A$ over $\mathbb{Z}[q^{\pm 1/2}]$ if

(i)

the Grothendieck ring $\mathbb{Z}[q^{\pm 1/2}]\mathop{\otimes }_{\mathbb{Z}[q^{\pm 1}]} K(\mathcal{C})$ is isomorphic to $A$,

(ii)

there exists a quantum monoidal seed $\mathscr{S}=(\{M_i\}_{i\in {J}}, L,\widetilde{B}, D)$ in $\mathcal{C}$ such that $[\mathscr{S}]\mathbin{:=}(\{q^{-(d_i,d_i)/4}[M_i]\}_{i\in {J}}, L, \widetilde{B})$ is a quantum seed of $A$,

(iii)

$\mathscr{S}$ admits successive mutations in all the directions.

Note that if $\mathcal{C}$ is a monoidal categorification of a quantum cluster algebra $A$, then any quantum seed in $A$ obtained by a sequence of mutations from the initial quantum seed is of the form $(\{q^{-(d_i,d_i)/4}[M_i]\}_{i\in {J}}, L, \widetilde{B})$ for some quantum monoidal seed $(\{M_i\}_{i\in {J}}, L,\widetilde{B}, D)$. In particular, all the quantum cluster monomials in $A$ are the classes of real simple objects in $\mathcal{C}$ up to a power of $q^{1/2}$.

7. Monoidal categorification via modules over KLR algebras

7.1. Admissible pair

In this section, we assume that $R$ is a symmetric KLR algebra over a base field $\mathbf{k}$.

From now on, we focus on the case when $\mathcal{C}$ is a full subcategory of $R \text{-gmod}$ stable under taking convolution products, subquotients, extensions, and grading shift. In particular, we have

$$\begin{eqnarray*} \mathcal{C}= \mathop{\textstyle \bigoplus }\limits _{\beta \in \mathsf{Q}^-} \mathcal{C}_\beta , \quad \text{where} \ \mathcal{C}_\beta \mathbin{:=}\mathcal{C}\cap R(-\beta ) \text{-gmod}, \end{eqnarray*}$$

and we have the grading shift functor $q$ on $\mathcal{C}$. Hence we have

$$\begin{equation*} K(\mathcal{C}_\beta )\subset U^-_q(\mathfrak{g})_\beta , \end{equation*}$$

and $K(\mathcal{C})$ has a $\mathbb{Z}[q^{\pm 1}]$-basis consisting of the isomorphism classes of self-dual simple modules.

Definition 7.1.1.

A pair $(\{M_i\}_{i\in {J}}, \widetilde{B})$ is called admissible if

(i)

$\{M_i\}_{i\in {J}}$ is a family of real simple self-dual objects of $\mathcal{C}$ which commute with each other,

(ii)

$\widetilde{B}$ is an integer-valued ${J}\times {{J}}_{\mathrm{ex}}$-matrix with a skew-symmetric principal part,

(iii)

for each $k\in {{J}}_{\mathrm{ex}}$, there exists a self-dual simple object $M'_k$ of $\mathcal{C}$ such that there is an exact sequence in $\mathcal{C}$$$\begin{eqnarray} &&0 \to q \mathop{\textstyle \bigodot }\limits _{b_{ik} >0} M_i^{\text{$\odot $}b_{ik}} \to q^{\widetilde{\Lambda }(M_k,M_k')} M_k \circ M_k' \to \mathop{\textstyle \bigodot }\limits _{b_{ik} <0} M_i^{\text{$\odot $}(-b_{ik})} \to 0, {\tag{7.1}} \end{eqnarray}$$

and $M_k'$ commutes with $M_i$ for any $i \neq k$.

Note that $M'_k$ is uniquely determined by $k$ and $(\{M_i\}_{i\in {J}}, \widetilde{B})$. Indeed, it follows from $q^{\widetilde{\Lambda }(M_k,M_k')}M_k\mathbin{\text{$\nabla $}}M'_k\simeq \mathop{\textstyle \bigodot }\limits _{b_{ik} <0} M_i^{\text{$\odot $}(-b_{ik})}$ and 15, Corollary 3.7. Note also that if there is an epimorphism $q^mM_k\circ M'_k\twoheadrightarrow \mathop{\textstyle \bigodot }\limits _{b_{ik} <0} M_i^{\text{$\odot $}(-b_{ik})}$ for some $m\in \mathbb{Z}$, then $m$ should coincide with $\widetilde{\Lambda }(M_k,M'_k)$ by Lemma 3.1.4 and Lemma 3.2.7.

For an admissible pair $(\{M_i\}_{i\in {J}}, \widetilde{B})$, let $\Lambda =(\Lambda _{ij})_{i,j\in {J}}$ be the skew-symmetric matrix given by $\Lambda _{ij}=\Lambda (M_i,M_j)$. and let $D=\{d_i\}_{i\in {J}}$ be the family of elements of $\mathsf{Q}^-$ given by $d_i=\operatorname {wt}(M_i)$.

Now we can simplify the conditions in Definition 6.2.1 and Definition 6.2.3 as follows.

Proposition 7.1.2.

Let $(\{M_i\}_{i\in {J}},\widetilde{B})$ be an admissible pair in $\mathcal{C}$, and let $M'_k$ $(k\in {{J}}_{\mathrm{ex}})$ be as in Definition 7.1.1. Then we have the following properties:

(a)

The quadruple $\mathscr{S}\mathbin{:=}(\{M_i\}_{i\in {J}}, -\Lambda ,\widetilde{B},D)$ is a quantum monoidal seed in $\mathcal{C}$.

(b)

The self-dual simple object $M_k'$ is real for every $k\in {{J}}_{\mathrm{ex}}$.

(c)

The quantum monoidal seed $\mathscr{S}$ admits a mutation in each direction $k\in {{J}}_{\mathrm{ex}}$.

(d)

$M_k$ and $M'_k$ are simply linked for any $k\in {{J}}_{\mathrm{ex}}$ (i.e., $\operatorname {\mathfrak{d}}(M_k,M'_k)=1$).

(e)

For any $j\in {J}$ and $k\in {{J}}_{\mathrm{ex}}$, we have$$\begin{eqnarray} && \begin{array}{l}\Lambda (M_j,M'_k)=-\Lambda (M_j,M_k)-\sum _{b_{ik}<0}\Lambda (M_j,M_i)b_{ik},\\[5.0pt] \Lambda (M'_k,M_j)=-\Lambda (M_k,M_j)+\sum _{b_{ik}>0}\Lambda (M_i,M_j)b_{ik}. \end{array} {\tag{7.2}} \end{eqnarray}$$

Proof.

Item d follows from the exact sequence 7.1 and Lemma 3.2.18.

Item b follows from the exact sequence 7.1 by applying Corollary 3.2.21 to the case

$$\begin{equation*} M=M_k, \ N=M'_k, \ X= q \mathop{\textstyle \bigodot }\limits _{b_{ik} > 0} M_i^{\text{$\odot $}b_{ik}}, \ \text{and} \ Y= \mathop{\textstyle \bigodot }\limits _{b_{ik} < 0} M_i^{\text{$\odot $}(-b_{ik})}. \end{equation*}$$

Item e follows from

$$\begin{align*} \Lambda (M_j,M_k)+\Lambda (M_j,M'_k)&=\Lambda (M_j,M_k\mathbin{\text{$\nabla $}}M'_k) =\Lambda \bigl (M_j,\mathop{\textstyle \bigodot }\limits _{b_{ik} <0} M_i^{\text{$\odot $}(-b_{ik})}\bigr )\\ &=\sum _{b_{ik}<0}\Lambda (M_j,M_i)(-b_{ik}) \end{align*}$$

and

$$\begin{align*} \Lambda (M_k,M_j)+\Lambda (M'_k,M_j)&=\Lambda (M'_k\mathbin{\text{$\nabla $}}M_k,M_j) =\Lambda \bigl (\mathop{\textstyle \bigodot }\limits _{b_{ik} >0} M_i^{\text{$\odot $}b_{ik}}, M_j\bigr )\\ &=\sum _{b_{ik}>0}\Lambda (M_i,M_j)b_{ik}. \end{align*}$$

Let us show a. The conditions (i)–(v) in Definition 6.2.1 are satisfied by the construction. The condition (vi) follows from Proposition 3.2.5 and the fact that $M_i$ is real simple for every $i\in {J}$. The condition (viii) is nothing but Lemma 3.1.2. The condition (ix) follows easily from the fact that the weights of the first and the last terms in the exact sequence 7.1 coincide.

Let us show the condition (vii) in Definition 6.2.1. By 7.2 and d of this proposition, we have

$$\begin{align*} 2\delta _{jk}=2\operatorname {\mathfrak{d}}(M_j,M'_k) &=-2\operatorname {\mathfrak{d}}(M_j,M_k)-\sum _{b_{ik}<0}\Lambda (M_j,M_i)b_{ik} +\sum _{b_{ik}>0}\Lambda (M_i,M_j)b_{ik}\\ &=-\!\!\!\sum _{b_{ik}<0}\Lambda (M_j,M_i)b_{ik}-\!\!\!\sum _{b_{ik}>0}\Lambda (M_j,M_i)b_{ik} =-\!\!\sum _{i\in {J}}\Lambda (M_j,M_i)b_{ik} \end{align*}$$

for $k\in {{J}}_{\mathrm{ex}}$ and $j\in {J}$. Thus we have shown that $\mathscr{S}$ is a quantum monoidal seed in $\mathcal{C}$.

Let us show c. Let $k\in {{J}}_{\mathrm{ex}}$. The exact sequence 6.4 follows from 7.1 and the equality

$$\begin{eqnarray} \widetilde{\Lambda }(M_k,M'_k)=\dfrac{1}{2}\bigl ((\operatorname {wt}(M_k,M'_k) -\sum _{b_{ik}<0}\Lambda (M_k,M_i)b_{ik}\bigr )=m_k, \tag{7.3} \end{eqnarray}$$

which is an immediate consequence of 7.2.

Similarly, taking the dual of the exact sequence 7.1, we obtain an exact sequence

$$\begin{eqnarray*} &&0 \,{\to } \mathop{\textstyle \bigodot }\limits _{b_{ik} <0} M_i^{\text{$\odot $}(-b_{ik})} {\to } q^{-\widetilde{\Lambda }(M_k,M_k')+(\operatorname {wt}M_k, \operatorname {wt}M_k')} M_k' \circ M_k \,{\to }\, q^{-1}\mathop{\textstyle \bigodot }\limits _{b_{ik} >0} M_i^{\text{$\odot $}b_{ik}} \,{\to }\, 0, \end{eqnarray*}$$

which gives the exact sequence 6.5.

It remains to prove that $\mu _k(\mathscr{S})\mathbin{:=}(\{M_i\}_{i\neq k}\cup \{M_k'\},\mu _k(-\Lambda ), \mu _k(\widetilde{B}), \mu _k(D))$ is a quantum monoidal seed in $\mathcal{C}$ for any $k\in {{J}}_{\mathrm{ex}}$.

We see easily that $\mu _k(\mathscr{S})$ satisfies the conditions (i)–(iv) and (vii)–(ix) in Definition 6.2.1.

For the condition (v), it is enough to show that for $i,j\in {J}$ we have

$$\begin{eqnarray*} \mu _k(-\Lambda )_{ij} = -\Lambda (\mu _k(M)_i,\mu _k(M)_j), \end{eqnarray*}$$

where $\mu _k(M)_i = M_i$ for $i\neq k$ and $\mu _k(M)_k=M_k'$. In the case $i \neq k$ and $j \neq k$, we have

$$\begin{eqnarray*} \mu _{k}(-\Lambda )_{ij}=-\Lambda (M_i,M_j)=-\Lambda (\mu _k(M_i),\mu _k(M_j)). \end{eqnarray*}$$

The other cases follow from 7.2.

The condition (vi) in Definition 6.2.1 for $\mu _k(\mathscr{S})$ follows from Proposition 3.2.5 and the fact that $\{\mu _k(M)_i\}_{i\in {J}}$ is a commuting family of real simple modules.

■

Now we are ready to give one of our main theorems.

Theorem 7.1.3.

Let $(\{M_i\}_{i\in {J}},\widetilde{B})$ be an admissible pair in $\mathcal{C}$ and set

$$\begin{equation*} \mathscr{S}=(\{M_i\}_{i\in {J}}, -\Lambda ,\widetilde{B},D) \end{equation*}$$

as in Proposition 7.1.2. We set $[\mathscr{S}]\mathbin{:=}(\{q^{-\frac{1}{4}(\operatorname {wt}(M_i), \operatorname {wt}(M_i))}[M_i]\}_{i\in {J}}, -\Lambda ,\widetilde{B},D)$. We assume further that

$$\begin{equation} \begin{gathered} \text{The $\mathbb{Q}(q^{1/2})$-algebra $\mathbb{Q}(q^{1/2})\mathop{\otimes }\limits _{\mathbb{Z}[q^{\pm 1}]}K(\mathcal{C})$ is isomorphic to}\\ \mathbb{Q}(q^{1/2})\mathop{\otimes }\limits _{\mathbb{Z}[q^{\pm 1}]}\mathscr{A}_{q^{1/2}}([\mathscr{S}]). \end{gathered} {\tag{7.4}} \end{equation}$$

Then, for each $x\in {{J}}_{\mathrm{ex}}$, the pair $\bigl (\{\mu _x(M)_i\}_{i\in {J}},\mu _x(\widetilde{B})\bigr )$ is admissible in $\mathcal{C}$.

Proof.

In Proposition 7.1.2 (b), we have already shown that the condition (i) in Definition 7.1.1 holds for $(\{\mu _x(M)_i\}_{i\in {J}},\mu _x(\widetilde{B}))$. The condition (ii) is clear from the definition. Let us show (iii). Set $N_i\mathbin{:=}\mu _x(M)_i$ and $b_{ij}' \mathbin{:=}\mu _x(\widetilde{B})_{ij}$ for $i\in {J}$ and $j\in {{J}}_{\mathrm{ex}}$. It is enough to show that, for any $y\in {{J}}_{\mathrm{ex}}$, there exists a self-dual simple module $M_y'' \in \mathcal{C}$ such that there is a short exact sequence

$$\begin{equation} \vcenter{\img[][314pt][29pt][{$\xymatrix{ 0 \ar[r] & q \sodot_{b'_{iy} > 0} N_i^{\snconv b'_{iy}} \ar[r] &q^{\tLa(N_y,M_y'')} N_y \conv M_y'' \ar[r] & \sodot_{b'_{iy} <0} N_i^{\snconv(-b'_{iy})} \ar[r] & 0 }$}]{Images/img2c884be1464718be2d88efe64d2c875f.svg}} {\tag{7.5}} \end{equation}$$

and

$$\begin{equation*} \operatorname {\mathfrak{d}}(N_i,M_y'')=0 \quad \text{for $i \neq y$.} \end{equation*}$$

If $x=y$, then $b'_{iy}=-b_{ix}$, and hence $M''_y=M_x$ satisfies the desired condition.

Assume that $x\not =y$ and $b_{xy}=0$. Then $b'_{iy} = b_{iy}$ for any $i$ and $N_i=M_i$ for any $i\not =x$. Hence $M_y''=\mu _y(M)_y$ satisfies the desired condition.

We will show the assertion in the case $b_{xy} > 0$. We omit the proof of the case $b_{xy} <0$ because it can be shown in a similar way.

Recall that we have

$$\begin{eqnarray} b_{iy}'=\begin{cases} b_{iy}+b_{ix}b_{xy} & \text{if } b_{ix} >0, \\ b_{iy} & \text{if } b_{ix} \le 0 \end{cases} {\tag{7.6}} \end{eqnarray}$$

for $i \in {J}$ different from $x$ and $y$.

Set

$$\begin{eqnarray*} &&M_x'\mathbin{:=}\mu _x(M)_x, \hspace{3ex} M_y'\mathbin{:=}\mu _y(M)_y, \\ &&C\mathbin{:=}\mathop{\textstyle \bigodot }\limits _{b_{ix} >0} M_i^{\text{$\odot $}b_{ix}}, \quad S\mathbin{:=}\mathop{\textstyle \bigodot }\limits _{b_{ix} <0, \ i\neq y} M_i^{\text{$\odot $}-b_{ix}}, \\ &&P\mathbin{:=}\mathop{\textstyle \bigodot }\limits _{b_{iy} >0, i\neq x} M_i^{\text{$\odot $}b_{iy}}, \quad Q:=\mathop{\textstyle \bigodot }\limits _{b_{iy}' <0, \ i\neq x} M_i^{\text{$\odot $}-b_{iy}'}, \\ &&A\mathbin{:=}\mathop{\textstyle \bigodot }\limits _{b_{iy}' \le 0, \ b_{ix} >0} M_i^{\text{$\odot $}b_{ix}b_{xy}} \mathop{\text{$\odot $}}\mathop{\textstyle \bigodot }\limits _{\substack{b_{iy} <0,\; b_{iy}' >0,\;b_{ix} >0}} M_i^{\text{$\odot $}-b_{iy}} \\ &&\hspace{3ex}\simeq \mathop{\textstyle \bigodot }\limits _{\substack{b_{iy} <0,\; b_{ix} >0}} M_i^{\text{$\odot $}\min (b_{ix}b_{xy},-b_{iy})},\\ &&B\mathbin{:=}\mathop{\textstyle \bigodot }\limits _{ b_{iy} \ge 0, \ b_{ix} >0} M_i^{\text{$\odot $}b_{ix}b_{xy}} \mathop{\text{$\odot $}}\mathop{\textstyle \bigodot }\limits _{\substack{b_{iy}' >0, \; b_{iy} <0,\;b_{ix} >0}} M_i^{\text{$\odot $}b_{iy}'}. \end{eqnarray*}$$

Then using 7.6 repeatedly, we have

$$\begin{align*} Q \mathop{\text{$\odot $}}A \simeq \mathop{\textstyle \bigodot }\limits _{b_{iy} < 0} M_i^{\text{$\odot $}-b_{iy}}, \quad A \mathop{\text{$\odot $}}B \simeq C^{\text{$\odot $}b_{xy}}, \quad \text{and} \quad B\mathop{\text{$\odot $}}P \simeq \mathop{\text{$\odot $}}\mathop{\textstyle \bigodot }\limits _{ b_{iy}' > 0} M_i^{\text{$\odot $}b_{iy}'}. \end{align*}$$

Set

$$\begin{eqnarray*} &&L\mathbin{:=}(M_x')^{\text{$\odot $}b_{xy}}, \quad V\mathbin{:=}M_x^{\text{$\odot $}b_{xy}}, \end{eqnarray*}$$

and set

$$\begin{eqnarray*} X \mathbin{:=}\mathop{\textstyle \bigodot }\limits _{b_{iy} >0} M_i^{\text{$\odot $}b_{iy}}\simeq M_x^{\text{$\odot $}b_{xy}} \mathop{\text{$\odot $}}P =V \mathop{\text{$\odot $}}P, \quad Y\mathbin{:=}\mathop{\textstyle \bigodot }\limits _{b_{iy} <0} M_i^{\text{$\odot $}-b_{iy}}\simeq Q \mathop{\text{$\odot $}}A. \end{eqnarray*}$$

Then 7.5 is read as

$$\begin{eqnarray} \vcenter{\img[][272pt][15pt][{$\xymatrix{ 0 \ar[r] & q (B \nconv P)\ar[r] &q^{\tL(M_y,M''_y)}M_y \conv M''_y \ar[r] & L \nconv Q\ar[r] & 0. }$}]{Images/imgd514b2be1f441a7d4b93ef84fc4cb9e1.svg}} {\tag{7.7}} \end{eqnarray}$$

Note that we have

$$\begin{eqnarray} &&0 \to q C \to q^{\widetilde{\Lambda }(M_x,M_x')}M_x \circ M_x' \to M_y^{\text{$\odot $}b_{xy}} \mathop{\text{$\odot $}}S \to 0, \tag{7.8}\\ &&0 \to q X \to q^{\widetilde{\Lambda }(M_y,M_y')}M_y \circ M_y' \to Y \to 0. {\tag{7.9}} \end{eqnarray}$$

Taking the convolution products of $L=(M_x')^{\text{$\odot $}b_{xy}}$ and 7.9, we obtain

$$\begin{eqnarray*} && \vcenter{\img[][278pt][37pt][{$\xymatrix@R=.5ex{0 \ar[r]& qL \conv X \ar[r]& q^{\tLa(M_y,M_y')} L \conv(M_y \conv M_y')\ar[r]& L \conv Y \ar[r]& 0, \\ 0\ar[r]& qX \conv L \ar[r]& q^{\tLa(M_y,M_y')} (M_y \conv M_y') \conv L \ar[r]& Y\conv L \ar[r]& 0.}$}]{Images/img54c85a358b650614865f4830c1f78327.svg}} \end{eqnarray*}$$

Since $L$ commutes with $M_y$, we have

$$\begin{eqnarray*} &&\Lambda (L, Y)=\Lambda (L, M_y \mathbin{\text{$\nabla $}}M_y') \\ &&=\Lambda (L, M_y) +\Lambda (L, M_y') = \Lambda (L,M_y \circ M_y' ). \end{eqnarray*}$$

On the other hand, we have

$$\begin{eqnarray*} &&\Lambda (M_x',X) - \Lambda (M_x',Y) \\ &&=\Lambda (M_x', \mathop{\textstyle \bigodot }\limits _{b_{iy} >0} M_i^{\text{$\odot $}b_{iy}})-\Lambda (M_x',\mathop{\textstyle \bigodot }\limits _{b_{iy} <0} M_i^{\text{$\odot $}-b_{iy}}) \\ &&=\sum _{b_{iy} >0} \Lambda (M_x',M_i)b_{iy} -\sum _{b_{iy} <0} \Lambda (M_x',M_i)(-b_{iy}) \\ &&=\sum _{i\in {J}} \Lambda (M_x', M_i)b_{iy} =\sum _{i \neq x} \Lambda (M_x',M_i)b_{iy} + \Lambda (M_x',M_x)b_{xy} \\ &&=\sum _{i \neq x} \Lambda (M_x', M_i)(b_{iy}' -\delta (b_{ix} >0)b_{ix} b_{xy}) + \Lambda (M_x',M_x)b_{xy} \\ &&=\sum _{i \neq x} \Lambda (M_x', M_i) b_{iy}' -\sum _{b_{ix} >0} \Lambda (M_x', M_i)b_{ix}b_{xy} + \Lambda (M_x',M_x)b_{xy} \\ &&\mathop{=}\limits _{\mathrm{(}a)}0-\Lambda (M_x', \mathop{\textstyle \bigodot }\limits _{b_{ix} >0} M_i^{\text{$\odot $}b_{ix}})b_{xy} + \Lambda (M_x',M_x)b_{xy} \\ &&=\bigl (-\Lambda (M_x',\mathop{\textstyle \bigodot }\limits _{b_{ix} >0} M_i^{\text{$\odot $}b_{ix}} ) + \Lambda (M_x',M_x) \bigr )b_{xy} \\ &&=(-\Lambda (M_x', M_x' \mathbin{\text{$\nabla $}}M_x) + \Lambda (M_x',M_x) )b_{xy} \\ &&=(-\Lambda (M_x', M_x')-\Lambda (M_x', M_x) + \Lambda (M_x',M_x) )b_{xy}=0. \end{eqnarray*}$$

Note that we used the compatibility of the pair $\bigl (\bigl (-\Lambda (\mu _x(M_i),\mu _x(M_j))\bigr )_{i,j\in {J}},\, \mu _x(\widetilde{B})\!\bigr )$ when we derive the equality (a).

Since $L=(M'_x)^{\text{$\odot $}b_{xy}}$, the equality $\Lambda (M'_x,X)=\Lambda (M'_x,Y)$ implies

$$\begin{eqnarray*} && \Lambda (L,X) =\Lambda (L,Y)=\Lambda (L, M_y \circ M_y'). \end{eqnarray*}$$

Hence the following diagram is commutative by Proposition 3.2.8 (i):

$$\begin{eqnarray*} &&\begin{array}{c} \vcenter{\img[][314pt][59pt][{$\xymatrix{ 0 \ar[r] & q L \conv X \ar[r] \ar[d]^{\rmat{L,X}} & q^{\tLa(M_y,M_y')} L \conv(M_y \conv M_y') \ar[d]^{\rmat{L,M_y \circ M_y'}} \ar[r] & L \conv Y \ar[r] \ar[d]_{\rmat{L,Y}} ^{\bwr}&0\\ 0 \ar[r] & q^{d+1} X \conv L \ar[r] & q^{d+\tLa(M_y,M_y')} (M_y \conv M_y') \conv L \ar[r] & q^{d}\, Y \conv L \ar[r] & 0, }$}]{Images/img3eda5251586201044ed7ef571d12eaff.svg}}\end{array} \end{eqnarray*}$$

where $d=-\Lambda (L,X)=-\Lambda (L,M_y \circ M_y')=-\Lambda (L,Y)$. Note that since $L=(M_x')^{\text{$\odot $}b_{xy}}$ commutes with $Q$ and $A$, $\mathbf{r}_{\!_{L,Y}}$ is an isomorphism. Hence we have

$$\begin{equation*} \operatorname {Im}(\mathbf{r}_{\!_{L,Y}}) \simeq L \circ Y. \end{equation*}$$

Therefore we obtain an exact sequence

$$\begin{eqnarray} \vcenter{\img[][228pt][16pt][{$\xymatrix{0\ar[r]&\Im(\rmat{L,X})\ar[r]&\Im(\rmat{L, M_y\circ M'_y})\ar[r] &L\circ Y\ar[r]&0.}$}]{Images/img6d6930fb0f445156bd314f0c2b39f6a4.svg}} {\tag{7.10}} \end{eqnarray}$$

On the other hand, $\mathbf{r}_{\!_{L, M_y \circ M_y'}}$ decomposes (up to a grading shift) by Lemma 3.1.5 as follows:

$$\begin{eqnarray*} \vcenter{\img[][284pt][56pt][{$\xymatrix@C=6em{ L \conv M_y \conv M_y' \ar[r]^\sim_{\rmat{L,M_y} \circ M_y'} \ar@/^2pc/ [rr]^{\rmat{L, M_y \circ M_y'}}& M_y \conv L \conv M_y' \ar[r]_{M_y \circ\rmat{L,M_y'}} & M_y \conv M_y' \conv L. }$}]{Images/imgdee2c0267ec46a80fc987ec06c091085.svg}} \end{eqnarray*}$$

Since $L=(M_x')^{\text{$\odot $}b_{xy}}$ commutes with $M_y$, the homomorphisms $\mathbf{r}_{\!_{L,M_y}} \circ M_y'$ is an isomorphism, and hence we have

$$\begin{eqnarray*} \operatorname {Im}(\mathbf{r}_{\!_{L, M_y \circ M_y'}}) \simeq M_y \circ (L\mathbin{\text{$\nabla $}}M'_y) \quad \text{up to a grading shift.} \end{eqnarray*}$$

Similarly, $\mathbf{r}_{\!_{L, X}}$ decomposes (up to a grading shift) as follows:

$$\begin{eqnarray*} \vcenter{\img[][244pt][43pt][{$\xymatrix@C=6em{ L \conv V \conv P \ar[r]_{\rmat{L,V} \circ P} \ar@/^1.5pc/ [rr]^{\rmat{L,X}}& V \conv L \conv P \ar[r]_{ V\circ\rmat{L,P}}^\sim& V \conv P \conv L. }$}]{Images/imgdae2de98788cd577db1a4f425c4c033f.svg}} \end{eqnarray*}$$

Since $L$ commutes with $P$, the homomorphism $V \circ \mathbf{r}_{\!_{L,P}}$ is an isomorphism, and hence we have

$$\begin{eqnarray*} \operatorname {Im}(\mathbf{r}_{\!_{L,X}}) \simeq (L \mathbin{\text{$\nabla $}}V) \circ P \simeq \bigl ((M_x')^{\circ b_{xy}} \mathbin{\text{$\nabla $}}M_x^{\circ b_{xy}}\bigr )\circ P \quad \text{up to a grading shift.} \end{eqnarray*}$$

On the other hand, Lemma 3.2.22 implies that

$$\begin{eqnarray*} (M_x')^{\circ b_{xy}} \mathbin{\text{$\nabla $}}M_x^{\circ b_{xy}} \simeq (M_x' \mathbin{\text{$\nabla $}}M_x)^{\circ b_{xy}} \simeq C^{\circ b_{xy}} \simeq B \mathop{\text{$\odot $}}A, \end{eqnarray*}$$

and hence we obtain

$$\begin{eqnarray*} \operatorname {Im}(\mathbf{r}_{\!_{L,X}}) \simeq (B \mathop{\text{$\odot $}}P) \mathop{\text{$\odot $}}A\quad \text{up to a grading shift.} \end{eqnarray*}$$

Thus the exact sequence 7.10 becomes the exact sequence in $\mathcal{C}$,

$$\begin{eqnarray} && \vcenter{\img[][316pt][11pt][{$\xymatrix{ 0 \ar[r] & q^m (B \nconv P) \nconv A \ar[r] &q^nM_y \conv(L\hconv M_y') \ar[r] & (L \nconv Q) \nconv A \ar[r] & 0 }$}]{Images/imgd545889b41819a356569e63bab8e50c9.svg}} {\tag{7.11}} \end{eqnarray}$$

for some $m,n\in \mathbb{Z}$. Since $(L \mathop{\text{$\odot $}}Q) \mathop{\text{$\odot $}}A$ is self-dual, $n=\widetilde{\Lambda }(M_y,L\mathbin{\text{$\nabla $}}M_y')$. On the other hand, by Proposition 3.2.13 (i) and Proposition 7.1.2 (d), we have

$$\begin{eqnarray*} \operatorname {\mathfrak{d}}(M_y,L \mathbin{\text{$\nabla $}}M_y') \le \operatorname {\mathfrak{d}}(M_y, L) + \operatorname {\mathfrak{d}}(M_y, M_y') =1. \end{eqnarray*}$$

By the exact sequence 7.11, $M_y \circ (L\mathbin{\text{$\nabla $}}M_y')$ is not simple, and we conclude

$$\begin{equation*} \operatorname {\mathfrak{d}}(M_y,L \mathbin{\text{$\nabla $}}M_y')=1. \end{equation*}$$

Then Lemma 3.2.18 implies that $m=1$. Thus we obtain an exact sequence in $\mathcal{C}$,

$$\begin{eqnarray} && \vcenter{\img[][312pt][15pt][{$\xymatrix@C=3ex{ 0 \ar[r] & q (B \nconv P) \nconv A \ar[r] &q^{\tL(M_y,L\shconv M_y')}M_y \conv(L\hconv M_y') \ar[r] & (L \nconv Q) \nconv A \ar[r] & 0. }$}]{Images/img244e66a4a532928561bef6bbabddae9f.svg}} {\tag{7.12}} \end{eqnarray}$$

Now we shall rewrite 7.12 by using $\mdsmblkcircle \circ A$ instead of $\mdsmblkcircle \mathop{\text{$\odot $}}A$. We have

$$\begin{eqnarray*} &&\widetilde{\Lambda }(B,A)+\widetilde{\Lambda }(A,A) =b_{xy}\widetilde{\Lambda }(C,A) =b_{xy}\widetilde{\Lambda }(M_x' \mathbin{\text{$\nabla $}}M_x,A) \\ &&\hspace{10ex}=b_{xy}\widetilde{\Lambda }(M_x',A)+b_{xy}\widetilde{\Lambda }(M_x,A)=\widetilde{\Lambda }(L,A)+b_{xy}\widetilde{\Lambda }(M_x,A). \end{eqnarray*}$$

On the other hand, the exact sequence 7.9 gives

$$\begin{eqnarray*} &&b_{xy}\widetilde{\Lambda }(M_x,A)+\widetilde{\Lambda }(P,A)=\widetilde{\Lambda }(X,A)=\widetilde{\Lambda }(M'_y\mathbin{\text{$\nabla $}}M_y,A)\\ &&\hspace{5ex}=\widetilde{\Lambda }(M'_y,A)+\widetilde{\Lambda }(M_y,A)=\widetilde{\Lambda }(M_y\mathbin{\text{$\nabla $}}M'_y,A)= \widetilde{\Lambda }(Y,A)\\ &&\hspace{5ex}=\widetilde{\Lambda }(Q,A)+\widetilde{\Lambda }(A,A). \end{eqnarray*}$$

It follows that

$$\begin{eqnarray*} &&\widetilde{\Lambda }(B \circ P,A)=\widetilde{\Lambda }(B,A) + \widetilde{\Lambda }(P,A) \\ &&\hspace{5ex}= \bigl (\widetilde{\Lambda }(L,A)+b_{xy}\widetilde{\Lambda }(M_x,A)-\widetilde{\Lambda }(A,A)\bigr )\\ &&\hspace{10ex}+\bigl (\widetilde{\Lambda }(Q,A)+\widetilde{\Lambda }(A,A)-b_{xy} \widetilde{\Lambda }(M_x,A)\bigr )\\ &&\hspace{5ex}=\widetilde{\Lambda }(L,A) + \widetilde{\Lambda }(Q,A)=\widetilde{\Lambda }(L \circ Q, A). \end{eqnarray*}$$

Hence we have

$$\begin{eqnarray*} && \vcenter{\img[][302pt][11pt][{$\xymatrix{ 0 \ar[r] & q (B \nconv P) \conv A \ar[r] &q^{c} M_y \conv(L\hconv M_y') \ar[r] & (L \nconv Q) \conv A \ar[r] & 0, }$}]{Images/imgbd78a88b84b10a7d916e3656c5fbc960.svg}} \end{eqnarray*}$$

where $c=\widetilde{\Lambda }(M_y, L \mathbin{\text{$\nabla $}}M_y')-\widetilde{\Lambda }(B \mathop{\text{$\odot $}}P,A)$ by Lemma 3.1.4.

Thus we obtain the identity in $K(R \text{-gmod})$,

$$\begin{eqnarray*} q^c [M_y] [L \mathbin{\text{$\nabla $}}M_y'] = \bigl (q [B \mathop{\text{$\odot $}}P] + [L \mathop{\text{$\odot $}}Q]\bigr )[A]. \end{eqnarray*}$$

On the other hand, the hypothesis 7.4 implies that there exists $\phi \in \mathbb{Q}(q^{1/2}) \otimes _{\mathbb{Z}[q^{\pm 1}]} K(\mathcal{C})$ corresponding to $\mu _y\mu _x([M])$ so that it satisfies

$$\begin{eqnarray} [M_y] \phi = q [B \mathop{\text{$\odot $}}P] + [L \mathop{\text{$\odot $}}Q] {\tag{7.13}} \end{eqnarray}$$

and

$$\begin{eqnarray} \phi [\mu _x(M)_i] = q^{\lambda '_{yi}} [\mu _x(M)_i] \phi \quad \text{for} \ i \neq y, {\tag{7.14}} \end{eqnarray}$$

where $\mu _y\mu _x(-\Lambda )=(\lambda '_{ij})_{i,j\in {J}}$.

Hence, in $\mathbb{Q}(q^{1/2}) \otimes _{\mathbb{Z}[q^{\pm 1}]} K(\mathcal{C})$, we have

$$\begin{eqnarray*} [M_y] \phi [A] = \bigl (q [B \mathop{\text{$\odot $}}P] + [L \mathop{\text{$\odot $}}Q]\bigr )[A ]= q^c [M_y][L \mathbin{\text{$\nabla $}}M_y']. \end{eqnarray*}$$

Since $\mathbb{Q}(q^{1/2}) \otimes _{\mathbb{Z}[q^{\pm 1}]} K(\mathcal{C})$ is a domain, we conclude that

$$\begin{eqnarray*} \phi [A] = q^c [L \mathbin{\text{$\nabla $}}M_y']. \end{eqnarray*}$$

On the other hand, 7.14 implies

$$\begin{eqnarray*} \phi [A] = q^{l} [A] \phi \quad \text{for some $l\in \mathbb{Z}$.} \end{eqnarray*}$$

Hence, Theorem 4.1.3 implies that, when we write

$$\begin{eqnarray*} \phi =\sum _{b \in B(\infty )} a_b [L_b]\quad \text{for some $a_b \in \mathbb{Q}(q^{1/2})$,} \end{eqnarray*}$$

we have

$$\begin{equation*} L_b\circ A \simeq q^{l} A\circ L_b \quad \text{whenever } \ a_b\not =0. \end{equation*}$$

In particular, each module $L_b\circ A$ with $a_b\not =0$ is simple because $A$ is a real simple module. Thus we obtain

$$\begin{eqnarray*} q^c [ L \mathbin{\text{$\nabla $}}M_y' ] =\phi [A] = \sum _{b \in B(\infty )} a_b [L_b \circ A]. \end{eqnarray*}$$

Since $L \mathbin{\text{$\nabla $}}M_y'$ is simple, there exists $b_0$ such that $L_{b_0} \circ A$ is isomorphic to $L \mathbin{\text{$\nabla $}}M_y'$ up to a grading shift, and $a_b=0$ for $b \neq b_0$. Set $M_y'':=L_{b_0}$. Then we conclude that $\phi [A] = q^m [M_y'' \circ A] =q^m[M_y''][A]$ so that

$$\begin{equation*} \phi =q^m[M''_y] \quad \text{for some } \ m \in \mathbb{Z}. \end{equation*}$$

We emphasize that $M_y''$ is a self-dual simple module in $R \text{-gmod}$ which satisfies that $M_y'' \circ A \simeq L \mathbin{\text{$\nabla $}}M_y$ up to a grading shift.

Now 7.13 implies

$$\begin{equation*} q^m[M_y\circ M''_y] = q [B \mathop{\text{$\odot $}}P] + [L \mathop{\text{$\odot $}}Q]. \end{equation*}$$

Hence there exists an exact sequence

$$\begin{equation*} 0\xrightarrow {\,\hspace{2ex}\,}W\xrightarrow {\,\hspace{2ex}\,}q^m\;M_y\circ M''_y\xrightarrow {\,\hspace{2ex}\,}Z\xrightarrow {\,\hspace{2ex}\,}0, \end{equation*}$$

where $W=q B \mathop{\text{$\odot $}}P$ and $Z=L \mathop{\text{$\odot $}}Q$ or $W=L \mathop{\text{$\odot $}}Q$ and $Z=q B \mathop{\text{$\odot $}}P$. By Lemma 3.2.18, the second case does not occur, and we have an exact sequence

$$\begin{equation*} 0\xrightarrow {\,\hspace{2ex}\,}q B \mathop{\text{$\odot $}}P\xrightarrow {\,\hspace{2ex}\,}q^m\;M_y\circ M''_y\xrightarrow {\,\hspace{2ex}\,}L \mathop{\text{$\odot $}}Q\xrightarrow {\,\hspace{2ex}\,}0. \end{equation*}$$

Since $M_y$, $M_y''$, and $L \mathop{\text{$\odot $}}Q$ are self-dual, we have $m=\widetilde{\Lambda }(M_y,M_y'')$, and we obtain the desired short exact sequence 7.7.

Since $\phi$ commutes with $[\mu _x (M)_i]$ up to a power of $q$ in $K(\mathcal{C})$, and $\mu _x(M)_i$ is real simple, $M_y''$ commutes with $\mu _x (M)_i$ for $i\neq y$, by Corollary 4.1.4.

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Corollary 7.1.4.

Let $(\{M_i\}_{i\in {J}},\widetilde{B})$ be an admissible pair in $\mathcal{C}$. Under the assumption 7.4, $\mathcal{C}$ is a monoidal categorification of the quantum cluster algebra $\mathscr{A}_{q^{1/2}}([\mathscr{S}])$. Furthermore, the following statements hold:

(i)

The quantum monoidal seed $\mathscr{S}=(\{M_i\}_{i\in {J}}, -\Lambda ,\widetilde{B},D)$ admits successive mutations in all directions.

(ii)

Any cluster monomial in $\mathbb{Z}[q^{\pm 1/2}] \otimes _{\mathbb{Z}[q^\pm 1]} K(\mathcal{C})$ is the isomorphism class of a real simple object in $\mathcal{C}$ up to a power of $q^{1/2}$.

(iii)

Any cluster monomial in $\mathbb{Z}[q^{\pm 1/2}]\otimes _{\mathbb{Z}[q^\pm 1]} K(\mathcal{C})$ is a Laurent polynomial of the initial cluster variables with a coefficient in $\mathbb{Z}_{\ge 0}[q^{\pm 1/2}]$.

Proof.

Items (i) and (ii) are straightforward.

Let us show (iii). Let $x$ be a cluster monomial. By the Laurent phenomenon 3, we can write

$$\begin{equation*} xX^{\mathbf{c}}=\sum _{\mathbf{a}\in \mathbb{Z}_{\ge 0}^{J}}c_\mathbf{a}X^\mathbf{a}, \end{equation*}$$

where $X=(X_i)_{i\in {J}}$ is the initial cluster, $\mathbf{c}\in \mathbb{Z}_{\ge 0}^{J}$, and $c_\mathbf{a}\in \mathbb{Q}(q^{\pm 1/2})$. Since $x$ and $X^{\mathbf{c}}$ are the isomorphism classes of simple modules up to a power of $q^{1/2}$, their product $xX^{\mathbf{c}}$ can be written as a linear combination of the isomorphism classes of simple modules with coefficients in $\mathbb{Z}_{\ge 0}[q^{\pm 1/2}]$. Since every $X^\mathbf{a}$ is the isomorphism class of a simple module up to a power of $q^{1/2}$, we have $c_\mathbf{a}\in \mathbb{Z}_{\ge 0}[q^{\pm 1/2}]$.

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8. Quantum coordinate rings and modified quantized enveloping algebras

8.1. Quantum coordinate ring

Let $U_q(\mathfrak{g})^*$ be $\operatorname {Hom}_{\mathbb{Q}(q)}(U_q(\mathfrak{g}),\mathbb{Q}(q))$. Then the comultiplication $\Delta _+$ (see 1.1) induces the multiplication $\mu$ on $U_q(\mathfrak{g})^*$ as follows:

$$\begin{align*} \mu \colon & U_q(\mathfrak{g})^* \mathop{\otimes }U_q(\mathfrak{g})^* \to (U_q(\mathfrak{g})\mathop{\otimes }U_q(\mathfrak{g}))^*\xrightarrow {\,\;(\Delta _+)^*\;\,} U_q(\mathfrak{g})^* . \end{align*}$$

Later on, it will be convenient to use Sweedler’s notation $\Delta _+(x)=x_{(1)} \mathop{\otimes }x_{(2)}$. With this notation,

$$\begin{equation*} \text{$\bigl (fg\bigr )(x)=f(x_{(1)})\,g(x_{(2)})$ for $f,g \in U_q(\mathfrak{g})^*$ and $x\in U_q(\mathfrak{g})$.} \end{equation*}$$

The $U_q(\mathfrak{g})$-bimodule structure on $U_q(\mathfrak{g})$ induces a $U_q(\mathfrak{g})$-bimodule structure on $U_q(\mathfrak{g})^*$. Namely,

$$\begin{align*} (x \cdot f) (v) =f(vx) \quad \text{ and } \quad (f \cdot x) (v) =f(xv) \quad \text{ for } f \in U_q(\mathfrak{g})^* \text{ and } x,v \in U_q(\mathfrak{g}). \end{align*}$$

Then the multiplication $\mu$ is a morphism of a $U_q(\mathfrak{g})$-bimodule, where $U_q(\mathfrak{g})^* \mathop{\otimes }U_q(\mathfrak{g})^*$ has the structure of a $U_q(\mathfrak{g})$-bimodule via $\Delta _+$. That is, for $f,g \in U_q(\mathfrak{g})^*$ and $x,y \in U_q(\mathfrak{g})$, we have

$$\begin{equation*} x(fg)y=(x_{(1)}fy_{(1)})(x_{(2)}gy_{(2)}), \end{equation*}$$

where $\Delta _+(x)=x_{(1)}\mathop{\otimes }x_{(2)}$ and $\Delta _+(y)=y_{(1)}\mathop{\otimes }y_{(2)}$.

Definition 8.1.1.

We define the quantum coordinate ring $A_q(\mathfrak{g})$ as follows:

$$\begin{equation*} A_q(\mathfrak{g}) = \{ u \in U_q(\mathfrak{g})^* \ | \ \text{ $U_q(\mathfrak{g})u$ belongs to $\mathcal{O}_{\mathrm{int}}(\mathfrak{g})$ and $u U_q(\mathfrak{g})$ belongs to $\mathcal{O}_{\mathrm{int}}^{\mspace {1mu}\mathrm{r}}(\mathfrak{g})$}\}. \end{equation*}$$

Then, $A_q(\mathfrak{g})$ is a subring of $U_q(\mathfrak{g})^*$ because (i) $\mu$ is $U_q(\mathfrak{g})$-bilinear, and (ii) $\mathcal{O}_{\mathrm{int}}(\mathfrak{g})$ and $\mathcal{O}_{\mathrm{int}}^{\mspace {1mu}\mathrm{r}}(\mathfrak{g})$ are closed under the tensor product.

We have the weight decomposition $A_q(\mathfrak{g}) = \mathop{\textstyle \bigoplus }\limits _{\eta ,\zeta \in \mathsf{P}} A_q(\mathfrak{g})_{\eta ,\zeta }$, where

$$\begin{equation*} A_q(\mathfrak{g})_{\eta ,\zeta } \mathbin{:=}\{ \psi \in A_q(\mathfrak{g}) \ | \ q^{h_{\operatorname {l}}} \cdot \psi \cdot q^{h_{\mspace {1mu}\mathrm{r}}} = q^{\langle h_{\operatorname {l}},\eta \rangle +\langle h_{\mspace {1mu}\mathrm{r}},\zeta \rangle } \psi \text{ for } h_{\operatorname {l}},h_{\mspace {1mu}\mathrm{r}}\in \mathsf{P}^\vee \}, \end{equation*}$$

For $\psi \in A_q(\mathfrak{g})_{\eta ,\zeta }$, we write

$$\begin{equation*} \operatorname {wt}_{\operatorname {l}}(\psi )=\eta \quad \text{ and } \quad \operatorname {wt}_{\mspace {1mu}\mathrm{r}}(\psi )=\zeta . \end{equation*}$$

For any $V\in \mathcal{O}_{\mathrm{int}}(\mathfrak{g})$, we have the $U_q(\mathfrak{g})$-bilinear homomorphism

$$\begin{equation*} \Phi _V\colon V\mathop{\otimes }(\mathbf{D}_\varphi V)^{\mspace {1mu}\mathrm{r}}\to A_q(\mathfrak{g}) \end{equation*}$$

given by

$$\begin{equation*} \Phi _V(v\mathop{\otimes }\psi ^{\mspace {1mu}\mathrm{r}})(a)=\langle \psi ^{\mspace {1mu}\mathrm{r}}, av\rangle =\langle \psi ^{\mspace {1mu}\mathrm{r}}a, v\rangle \quad \text{for $v\in V$, $\psi \in \mathbf{D}_\varphi V$ and $a\in U_q(\mathfrak{g})$.} \end{equation*}$$

Proposition 8.1.2 (17, Proposition 7.2.2).

We have an isomorphism $\Phi$ of $U_q(\mathfrak{g})$-bimodules

$$\begin{equation} \Phi \colon \mathop{\textstyle \bigoplus }\limits _{\lambda \in \mathsf{P}^+} V(\lambda ) \mathop{\otimes }_{\mathbb{Q}(q)} V(\lambda )^{\mspace {1mu}\mathrm{r}}\overset{\sim }{\longrightarrow } A_q(\mathfrak{g}) {\tag{8.1}} \end{equation}$$

given by $\Phi |_{V(\lambda ) \mathop{\otimes }_{\mathbb{Q}(q)} V(\lambda )^{\mspace {1mu}\mathrm{r}}} = \Phi _\lambda \mathbin{:=}\Phi _{V(\lambda )}$. Namely,

$$\begin{equation*} \Phi (u \mathop{\otimes }v^{\mspace {1mu}\mathrm{r}})(x)= \langle v^{\mspace {1mu}\mathrm{r}},xu \rangle = \langle v^{\mspace {1mu}\mathrm{r}}x,u \rangle =(v,xu) \text{ for any } v,u \in V(\lambda ) \text{ and } x \in U_q(\mathfrak{g}). \end{equation*}$$

We introduce the crystal basis $\big (L^\mathrm{up}(A_q(\mathfrak{g})), B(A_q(\mathfrak{g})) \big )$ of $A_q(\mathfrak{g})$ as the images by $\Phi$ of

$$\begin{equation*} \mathop{\textstyle \bigoplus }\limits _{\lambda \in \mathsf{P}^+} L^\mathrm{up}(\lambda ) \otimes L^\mathrm{up}(\lambda )^{\mspace {1mu}\mathrm{r}}\text{ and } \bigsqcup _{\lambda \in \mathsf{P}^+} B(\lambda ) \otimes B(\lambda )^{\mspace {1mu}\mathrm{r}}. \end{equation*}$$

Hence it is a crystal base with respect to the left action of $U_q(\mathfrak{g})$ and also the right action of $U_q(\mathfrak{g})$. We sometimes write by $e_i^*$ and $f_i^*$ the operators of $A_q(\mathfrak{g})$ obtained by the right actions of $e_i$ and $f_i$.

We define the $\mathbb{Z} [q^{\pm 1}]$-form of $A_q(\mathfrak{g})$ by

$$\begin{equation*} A_q(\mathfrak{g})_\mathbb{Z} [q^{\pm 1}]\mathbin{:=}\left\{\psi \in A_q(\mathfrak{g}) \mid \langle \psi ,\, U_q(\mathfrak{g})_\mathbb{Z} [q^{\pm 1}]\rangle \subset \mathbb{Z} [q^{\pm 1}] \right\}. \end{equation*}$$

We define the bar-involution $-$ of $A_q(\mathfrak{g})$ by

$$\begin{equation*} \overline{\psi }(x) = \overline{\psi (\overline{x})} \quad \text{ for } \psi \in A_q(\mathfrak{g}), \ x \in U_q(\mathfrak{g}). \end{equation*}$$

Note that the bar-involution is not a ring homomorphism but it satisfies

$$\begin{eqnarray*} &&\overline{\psi \;\theta \;}=q^{(\operatorname {wt}_\mathrm{l}(\psi ),\operatorname {wt}_\mathrm{l}(\theta )) -(\operatorname {wt}_{\mspace {1mu}\mathrm{r}}(\psi ),\operatorname {wt}_{\mspace {1mu}\mathrm{r}}(\theta ))}\;\overline{\theta }\;\overline{\psi }\quad \text{for any $\psi $, $\theta \in A_q(\mathfrak{g})$.} \end{eqnarray*}$$

Since we do not use this formula and it is proved similarly to Proposition 8.1.4 below, we omit its proof.

The triple $\big (\mathbb{Q}\mathop{\otimes }A_q(\mathfrak{g})_\mathbb{Z} [q^{\pm 1}],\,L^\mathrm{up}(A_q(\mathfrak{g})),\,\overline{L^\mathrm{up}(A_q(\mathfrak{g}))}\big )$ is balanced 17, Theorem 1, and hence there exists an upper global basis of $A_q(\mathfrak{g})$,

$$\begin{equation*} {\mathbf{B}}^\mathrm{up}(A_q(\mathfrak{g}))\mathbin{:=}\{ G^\mathrm{up}(b) \ | \ b \in B^\mathrm{up}(A_q(\mathfrak{g})) \}. \end{equation*}$$

For $\lambda \in \mathsf{P}^+$ and $\mu \in W\lambda$, we denote by $u_{\mu }$ the unique member of the upper global basis of $V(\lambda )$ with weight $\mu$. It is also a member of the lower global basis.

Proposition 8.1.3.

Let $\lambda \in \mathsf{P}^+$, $w\in W$, and $b\in B(\lambda )$. Then, $\Phi (G^\mathrm{up}(b)\mathop{\otimes }u_{w\lambda }^{\mspace {1mu}\mathrm{r}})$ is a member of the upper global basis of $A_q(\mathfrak{g})$.

Proof.

The element $\psi \mathbin{:=}\Phi (G^\mathrm{up}(b)\mathop{\otimes }u_{w\lambda }^{\mspace {1mu}\mathrm{r}})$ is bar-invariant and a member of crystal basis modulo $q L^\mathrm{up}(A_q(\mathfrak{g}))$. For any $P\in U_q(\mathfrak{g})_\mathbb{Z} [q^{\pm 1}]$,

$$\begin{equation*} \langle \psi , P\rangle =\big ( u_{w\lambda }, PG^\mathrm{up}(b)\big ) \end{equation*}$$

belongs to $\mathbb{Z} [q^{\pm 1}]$ because $PG^\mathrm{up}(b)\in V^\mathrm{up}(\lambda )_\mathbb{Z} [q^{\pm 1}]$ and $u_{w\lambda }\in V^{\mathrm{low}}(\lambda )_\mathbb{Z} [q^{\pm 1}]$. Hence $\psi$ belongs to $A_q(\mathfrak{g})_\mathbb{Z} [q^{\pm 1}]$.

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The $\mathbb{Q}(q)$-algebra anti-automorphism $\varphi$ of $U_q(\mathfrak{g})$ induces a $\mathbb{Q}(q)$-linear automorphism $\varphi ^*$ of $A_q(\mathfrak{g})$ by

$$\begin{equation*} \bigl (\varphi ^* \psi \bigr )(x)=\psi \bigl (\varphi (x)\bigr )\quad \text{for any $x\in U_q(\mathfrak{g})$.} \end{equation*}$$

We have

$$\begin{equation*} \varphi ^*\bigl (\Phi (u\mathop{\otimes }v^{\mspace {1mu}\mathrm{r}})\bigr )=\Phi (v\mathop{\otimes }u^{\mspace {1mu}\mathrm{r}}), \end{equation*}$$

and

$$\begin{equation*} \operatorname {wt}_\mathrm{l}(\varphi ^*\psi )=\operatorname {wt}_{\mspace {1mu}\mathrm{r}}(\psi )\quad \text{and}\quad \operatorname {wt}_{\mspace {1mu}\mathrm{r}}(\varphi ^*\psi )=\operatorname {wt}_\mathrm{l}(\psi ). \end{equation*}$$

It is obvious that $\varphi ^*$ preserves $A_q(\mathfrak{g})_\mathbb{Z} [q^{\pm 1}]$, $L^\mathrm{up}(A_q(\mathfrak{g}))$, and ${\mathbf{B}}^\mathrm{up}(A_q(\mathfrak{g}))$.

Proposition 8.1.4.

$$\begin{equation*} \varphi ^*(\psi \theta )= q^{(\operatorname {wt}_{\mspace {1mu}\mathrm{r}}(\psi ),\operatorname {wt}_{\mspace {1mu}\mathrm{r}}(\theta ))-(\operatorname {wt}_\mathrm{l}(\psi ),\operatorname {wt}_\mathrm{l}(\theta ))} (\varphi ^*\psi )(\varphi ^*\theta ). \end{equation*}$$

In order to prove this proposition, we prepare a sublemma.

Let $\xi$ be the $\mathbb{Q}(q)$-algebra automorphism of ${U_q(\mathfrak{g})}$ given by

$$\begin{equation*} \xi (e_i)=q_i^{-1}t_ie_i, \qquad \xi (f_i)=q_if_it_i^{-1},\qquad \xi (q^h)=q^h. \end{equation*}$$

We can easily see

$$\begin{eqnarray*} &&(\xi \mathop{\otimes }\xi )\circ \Delta _+=\Delta _-\circ \xi . \end{eqnarray*}$$

Let $\xi ^*$ be the automorphism of $A_q(\mathfrak{g})$ given by

$$\begin{equation*} (\xi ^*\psi )(x)=\psi (\xi (x))\quad \text{for $\psi \in A_q(\mathfrak{g})$ and $x\in {U_q(\mathfrak{g})}$.} \end{equation*}$$

Sublemma 8.1.5.

We have

$$\begin{eqnarray*} &&\xi ^*(\psi )=q^{A(\operatorname {wt}_\mathrm{l}(\psi ),\operatorname {wt}_{\mspace {1mu}\mathrm{r}}(\psi ))}\psi , \end{eqnarray*}$$

where $A(\lambda ,\mu )=\dfrac{1}{2}\bigl ((\mu ,\mu )-(\lambda ,\lambda )\bigr )$.

Proof.

Let us show that, for each $x$, the following equality,

$$\begin{eqnarray} &&\psi (\xi (x))=q^{A(\operatorname {wt}_\mathrm{l}(\psi ),\operatorname {wt}_{\mspace {1mu}\mathrm{r}}(\psi ))}\psi (x), {\tag{8.2}} \end{eqnarray}$$

holds for any $\psi$.

The equality 8.2 is obviously true for $x=q^h$. If 8.2 is true for $x$, then

$$\begin{eqnarray*} &&\xi ^*(\psi )(xe_i)=\psi \bigl (\xi (xe_i)\bigr )=\psi \bigl (\xi (x)e_it_i)q_i\\ &&=q^{(\alpha _i,\operatorname {wt}_\mathrm{l}(\psi ))+(\alpha _i ,\alpha _i)/2}\psi (\xi (x)e_i)\\ &&=q^{(\alpha _i,\operatorname {wt}_\mathrm{l}(\psi ))+(\alpha _i , \alpha _i)/2}\bigl (\xi ^*(e_i\psi )\bigr )(x)\\ &&=q^{(\alpha _i,\operatorname {wt}_\mathrm{l}(\psi ))+(\alpha _i , \alpha _i)/2+A(\operatorname {wt}_\mathrm{l}(\psi )+\alpha _i,\operatorname {wt}_{\mspace {1mu}\mathrm{r}}(\psi ))} (e_i\psi )(x). \end{eqnarray*}$$

Since $\Vert \lambda +\alpha _i\Vert ^2 =\Vert \lambda \Vert ^2+2(\alpha _i,\lambda )+ \Vert \alpha _i\Vert ^2$, 8.2 holds for $xe_i$. Similarly if 8.2 holds for $x$, then it holds for $xf_i$.

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Proof of Proposition 8.1.4.

We have

$$\begin{eqnarray*} &&(\varphi \circ \varphi )\circ \Delta _-=\Delta _+\circ \varphi . \end{eqnarray*}$$

Hence, we have

$$\begin{align*} \langle \varphi ^*(\psi \theta ), x\rangle &=\langle \psi \theta ,\varphi (x)\rangle \\ &=\langle \psi \mathop{\otimes }\theta ,\Delta _+(\varphi (x))\rangle \\ &=\langle \psi \mathop{\otimes }\theta ,(\varphi \mathop{\otimes }\varphi )\circ \Delta _-(x)\rangle \\ &=\langle \varphi ^*(\psi )\mathop{\otimes }\varphi ^*(\theta ),\;\Delta _-(x)\rangle . \end{align*}$$

It follows that

$$\begin{align*} \langle \xi ^*(\varphi ^*(\psi \theta )),x\rangle &=\langle \varphi ^*(\psi \theta ), \xi (x)\rangle =\langle \varphi ^*(\psi )\mathop{\otimes }\varphi ^*(\theta ),\Delta _-(\xi (x))\rangle \\ &=\langle \varphi ^*(\psi )\mathop{\otimes }\varphi ^*(\theta ),(\xi \mathop{\otimes }\xi )\circ \Delta _+x \rangle \\ &=\langle \xi ^*\varphi ^*(\psi )\mathop{\otimes }\xi ^*\varphi ^*(\theta ),\Delta _+x \rangle \\ &=\langle \bigl (\xi ^*\varphi ^*(\psi )\bigr )\bigl (\xi ^*\varphi ^*(\theta )\bigr ),x\rangle \\ &=q^{A(\operatorname {wt}_{\mspace {1mu}\mathrm{r}}(\psi ),\operatorname {wt}_\mathrm{l}(\psi ))+A(\operatorname {wt}_{\mspace {1mu}\mathrm{r}}(\theta ),\operatorname {wt}_\mathrm{l}(\theta ))} \langle (\varphi ^*\psi )\,(\varphi ^*\theta ) ,\,x\rangle . \end{align*}$$

Therefore we obtain

$$\begin{equation*} \varphi ^*(\psi \theta )=q^c(\varphi ^*\psi )\,(\varphi ^*\theta ) \end{equation*}$$

with

$$\begin{eqnarray*} &&c=A(\operatorname {wt}_{\mspace {1mu}\mathrm{r}}(\psi ),\operatorname {wt}_\mathrm{l}(\psi ))+A(\operatorname {wt}_{\mspace {1mu}\mathrm{r}}(\theta ),\operatorname {wt}_\mathrm{l}(\theta ))\\ &&\hspace{4ex}-A(\operatorname {wt}_{\mspace {1mu}\mathrm{r}}(\psi )+\operatorname {wt}_{\mspace {1mu}\mathrm{r}}(\theta ),\operatorname {wt}_\mathrm{l}(\psi )+\operatorname {wt}_\mathrm{l}(\theta ))\\ &&\hspace{2ex}=(\operatorname {wt}_{\mspace {1mu}\mathrm{r}}(\psi ),\operatorname {wt}_{\mspace {1mu}\mathrm{r}}(\theta ))-(\operatorname {wt}_\mathrm{l}(\psi ),\operatorname {wt}_\mathrm{l}(\theta )). \end{eqnarray*}$$■

8.2. Unipotent quantum coordinate ring

Let us endow $U_q^+(\mathfrak{g}) \mathop{\otimes }U_q^+(\mathfrak{g})$ with the algebra structure defined by

$$\begin{equation*} (x_1 \mathop{\otimes }x_2) \cdot (y_1 \mathop{\otimes }y_2) = q^{-(\operatorname {wt}(x_2),\operatorname {wt}(y_1))}(x_1y_1 \mathop{\otimes }x_2y_2). \end{equation*}$$

Let $\Delta _\mathfrak{n}$ be the algebra homomorphism $U_q^+(\mathfrak{g}) \to U_q^+(\mathfrak{g}) \mathop{\otimes }U_q^+(\mathfrak{g})$ given by

$$\begin{equation*} \Delta _\mathfrak{n}(e_i) = e_i \mathop{\otimes }1 + 1 \mathop{\otimes }e_i. \end{equation*}$$

Set

$$\begin{equation*} A_q(\mathfrak{n}) = \mathop{\textstyle \bigoplus }\limits _{\beta \in \mathsf{Q}^-} A_q(\mathfrak{n})_\beta \quad \text{ where } A_q(\mathfrak{n})_\beta \mathbin{:=}(U^+_q(\mathfrak{g})_{-\beta })^*. \end{equation*}$$

Defining the bilinear form $\langle \ \cdot \ , \ \cdot \ \rangle \colon (A_q(\mathfrak{n}) \mathop{\otimes }A_q(\mathfrak{n})) \times (U^+_q(\mathfrak{g}) \mathop{\otimes }U^+_q(\mathfrak{g})) \to \mathbb{Q}(q)$ by

$$\begin{equation*} \langle \psi \mathop{\otimes }\theta , x \mathop{\otimes }y \rangle =\theta (x) \psi (y), \end{equation*}$$

we get an algebra structure on $A_q(\mathfrak{n})$ given by

$$\begin{equation*} (\psi \cdot \theta )(x) = \langle \psi \mathop{\otimes }\theta , \Delta _\mathfrak{n}(x) \rangle = \theta (x_{(1)})\psi (x_{(2)}), \end{equation*}$$

where $\Delta _\mathfrak{n}(x)=x_{(1)} \mathop{\otimes }x_{(2)}$.

Since $U_q^+(\mathfrak{g})$ has a $U_q^+(\mathfrak{g})$-bimodule structure, so does $A_q(\mathfrak{n})$.

We define the $\mathbb{Z} [q^{\pm 1}]$-form of $A_q(\mathfrak{n})$ by

$$\begin{eqnarray*} &&A_q(\mathfrak{n})_\mathbb{Z} [q^{\pm 1}]=\left\{\psi \in A_q(\mathfrak{n}) \mid \psi \bigl (U_q^+(\mathfrak{g})_\mathbb{Z} [q^{\pm 1}]\bigr )\subset \mathbb{Z} [q^{\pm 1}] \right\}, \end{eqnarray*}$$

and define the bar-involution $-$ on $A_q(\mathfrak{n})$ by

$$\begin{equation*} \overline{\psi }(x)=\overline{\psi (\overline{x})} \quad \text{for $\psi \in A_q(\mathfrak{n})$ and $x\in U_q^+(\mathfrak{g})$.} \end{equation*}$$

Note that the bar-involution is not a ring homomorphism but it satisfies

$$\begin{eqnarray*} &&\overline{\psi \;\theta \;}=q^{(\operatorname {wt}(\psi ),\operatorname {wt}(\theta ))}\;\overline{\theta }\;\overline{\psi }\quad \text{for any $\psi $, $\theta \in A_q(\mathfrak{n})$.} \end{eqnarray*}$$

For $i\in I$, we denote by $e_i^*$ the right action of $e_i$ on $A_q(\mathfrak{n})$.

Lemma 8.2.1.

For $u,v \in A_q(\mathfrak{n})$, we have $q$-boson relations

$$\begin{align*} e_i(uv) = (e_iu)v + q^{(\alpha _i,\operatorname {wt}(u))} u (e_iv) \ \text{ and } \ e_i^*(uv) = u(e_i^*v) + q^{(\alpha _i,\operatorname {wt}(v))} (e_i^*u)v. \end{align*}$$

Proof.

$$\begin{equation*} \langle e_i(uv), x\rangle =\langle uv, xe_i\rangle = \langle u\mathop{\otimes }v,\; \Delta _\mathfrak{n}(xe_i)\rangle . \end{equation*}$$ If we set $\Delta _\mathfrak{n}x=x_{(1)}\mathop{\otimes }x_{(2)}$, then we have

$$\begin{equation*} \Delta _\mathfrak{n}(xe_i)=(x_{(1)}\mathop{\otimes }x_{(2)})(e_i\mathop{\otimes }1+1\mathop{\otimes }e_i) =q^{-(\alpha _i,\operatorname {wt}(x_{(2)}))}(x_{(1)}e_i)\mathop{\otimes }x_{(2)}+x_{(1)}\mathop{\otimes }(x_{(2)}e_i). \end{equation*}$$

Hence, we have

$$\begin{align*} &\langle u\mathop{\otimes }v,\; \Delta _\mathfrak{n}(xe_i)\rangle =q^{-(\alpha _i,\operatorname {wt}(x_{(2)}))}u(x_{(2)})v(x_{(1)}e_i)+u(x_{(2)}e_i)v(x_{(1)})\\ &\quad =q^{(\alpha _i,\operatorname {wt}(u))}u(x_{(2)})\cdot (e_iv)(x_{(1)})+(e_iu)(x_{(2)}) \cdot v(x_{(1)})\\ &\quad =\langle q^{(\alpha _i,\operatorname {wt}(u))}u\mathop{\otimes }(e_iv)+(e_iu)\mathop{\otimes }v,\,\Delta _\mathfrak{n}x\rangle . \end{align*}$$

The second identity follows in a similar way.

■

We define the map $\iota \colon U_q^-(\mathfrak{g}) \to A_q(\mathfrak{n})$ by

$$\begin{equation*} \langle \iota (u),x \rangle = (u,\varphi (x)) \quad \text{ for any } u \in U_q^-(\mathfrak{g}) \text{ and } x \in U_q^+(\mathfrak{g}). \end{equation*}$$

Since $(\ ,\ )$ is a non-degenerate bilinear form on $U^-_q(\mathfrak{g})$, $\iota$ is injective. The relation

$$\begin{equation*} \langle \iota (e_i'u), x \rangle = (e_i'u,\varphi (x)) = (u,f_i \varphi (x)) =(u, \varphi (xe_i))=\langle \iota (u),x e_i \rangle = \langle e_i \iota (u),x \rangle \end{equation*}$$

implies that

$$\begin{eqnarray*} \iota (e_i'u) = e_i \iota (u). \end{eqnarray*}$$

Lemma 8.2.2.

$\iota$ is an algebra isomorphism.

Proof.

The map $\iota$ is an algebra homomorphism because $e_i'$ and $e_i$ both satisfy the same $q$-boson relation.

■

Hence, the algebra $A_q(\mathfrak{n})$ has an upper crystal basis $(L^\mathrm{up}(A_q(\mathfrak{n})),B(A_q(\mathfrak{n})))$ such that $B(A_q(\mathfrak{n})) \simeq B(\infty )$. Furthermore, $A_q(\mathfrak{n})$ has an upper global basis

$$\begin{equation*} {\mathbf{B}}^\mathrm{up}(A_q(\mathfrak{n}))=\{ G^\mathrm{up}(b) \}_{ \ b \in B(A_q(\mathfrak{n}))} \end{equation*}$$

induced by the balanced triple $\bigl (\mathbb{Q}\mathop{\otimes }A_q(\mathfrak{n})_\mathbb{Z} [q^{\pm 1}], L^\mathrm{up}(A_q(\mathfrak{n})), \overline{L^\mathrm{up}(A_q(\mathfrak{n}))}\bigr )$ (see 1.3).

There exists an injective map

$$\begin{eqnarray*} &&\overline{\iota }_\lambda \colon B(\lambda )\to B(\infty ) \end{eqnarray*}$$

induced by the $U_q^+(\mathfrak{g})$-linear homomorphism $\iota _\lambda \colon V(\lambda )\to A_q(\mathfrak{n})$ given by

$$\begin{equation*} v\longmapsto \big ( U_q^+(\mathfrak{g})\ni a\mapsto (av,u_\lambda )\big ). \end{equation*}$$

The map $\overline{\iota }_\lambda$ commutes with $\tilde{e}_i$. We have

$$\begin{equation*} G^{\mathrm{low}}_\lambda (b)=G^{\mathrm{low}}(\overline{\iota }_\lambda (b))u_\lambda \quad \text{and}\quad \iota _\lambda G_\lambda ^\mathrm{up}(b)=G^\mathrm{up}(\overline{\iota }_\lambda (b))\quad \text{for any $b\in B(\lambda )$.} \end{equation*}$$

Remark 8.2.3.

Note that the multiplication on $A_q(\mathfrak{n})$ given in 11 is different from ours. Indeed, by denoting the product of $\psi$ and $\phi$ in 11, Section 4.2 by $\psi \cdot \phi$, for $x \in U_q^+(\mathfrak{g})$, we have

$$\begin{eqnarray*} (\psi \cdot \phi ) (x) = \psi (x^{(1)}) \phi (x^{(2)}), \end{eqnarray*}$$

where $\Delta _+(x)=x^{(1)}q^{h_{(1)}} \otimes x^{(2)}q^{h_{(2)}}$ for $x^{(1)}, x^{(2)} \in U_q^{+}(\mathfrak{g}), \ h_{(1)}, h_{(2)} \in \mathsf{P}^\vee$. By Lemma 8.5.3 below, we have

$$\begin{eqnarray*} (\psi \cdot \phi ) (x) &&= \psi \bigl (q^{(\operatorname {wt}(x_{(1)}),\operatorname {wt}(x_{(2)}))} (x_{(2)}) \bigr )\phi \bigl (x_{(1)} \bigr )\\ &&=q^{(\operatorname {wt}(x_{(1)},\operatorname {wt}(x_{(2)}))} \psi (x_{(2)}) \phi (x_{(1)}) = q^{(\operatorname {wt}(\psi ),\operatorname {wt}(\phi ))} (\psi \phi )(x) \end{eqnarray*}$$

for $x \in U_q^+(\mathfrak{g})$, where $\Delta _\mathfrak{n}(x)=x_{(1)} \mathop{\otimes }x_{(2)}$. In particular, we have a $\mathbb{Q}(q)$-algebra isomorphism from $(A_q(\mathfrak{n}), \cdot )$ to $A_q(\mathfrak{n})$ given by

$$\begin{eqnarray} x \mapsto q^{-\frac{1}{2}(\beta , \beta )} x \quad \ \text{for} \ x \in A_q(\mathfrak{n})_{\beta }. {\tag{8.3}} \end{eqnarray}$$

Note also that the bar-involution $-$ is a ring anti-isomorphism between $A_q(\mathfrak{n})$ and $(A_q(\mathfrak{n}), \cdot )$.

8.3. Modified quantum enveloping algebra

For the materials in this subsection we refer the reader to 1932. We denote by ${\mathrm{Mod}}(\mathfrak{g},\mathsf{P})$ the category of left $U_q(\mathfrak{g})$-modules with the weight space decomposition. Let $({\mathrm{forget}})$ be the functor from ${\mathrm{Mod}}(\mathfrak{g},\mathsf{P})$ to the category of vector spaces over $\mathbb{Q}(q)$, forgetting the $U_q(\mathfrak{g})$-module structure.

Let us denote by $\mathscr{R}$ the endomorphism ring of $({\mathrm{forget}})$. Note that $\mathscr{R}$ contains $U_q(\mathfrak{g})$. For $\eta \in \mathsf{P}$, let $a_\eta \in \mathscr{R}$ denote the projector $M \to M_\eta$ to the weight space of weight $\eta$. Then the defining relation of $a_\eta$ (as a left $U_q(\mathfrak{g})$-module) is

$$\begin{equation*} q^h a_\eta = q^{\langle h,\eta \rangle }a_\eta . \end{equation*}$$

We have

$$\begin{align*} a_\eta a_\zeta =\delta _{\eta ,\zeta } a_\eta , \quad a_\eta P = P a_{\eta -\xi } \quad \text{for $\xi \in \mathsf{Q}$ and $P \in U_q(\mathfrak{g})_\xi $.} \end{align*}$$

Then $\mathscr{R}$ is isomorphic to $\displaystyle \prod _{\eta \in \mathsf{P}} U_q(\mathfrak{g})a_\eta$. We set

$$\begin{equation*} \widetilde{U}_q(\mathfrak{g})\mathbin{:=}\mathop{\textstyle \bigoplus }\limits _{\eta \in \mathsf{P}} U_q(\mathfrak{g})a_\eta \subset \mathscr{R}. \end{equation*}$$

Then $\widetilde{U}_q(\mathfrak{g})$ is a subalgebra of $\mathscr{R}$. We call it the modified quantum enveloping algebra. Note that any $U_q(\mathfrak{g})$-module in ${\mathrm{Mod}}(\mathfrak{g},\mathsf{P})$ has a natural $\widetilde{U}_q(\mathfrak{g})$-module structure.

The (anti-)automorphisms $*$, $\varphi$, and $\bar{ \ \ }$ of $U_q(\mathfrak{g})$ extend to the ones of $\widetilde{U}_q(\mathfrak{g})$ by

$$\begin{equation*} a_\eta ^* = a_{-\eta }, \quad \varphi (a_\eta )=a_\eta , \quad \overline{a}_\eta =a_\eta . \end{equation*}$$

For a dominant integral weight $\lambda \in \mathsf{P}^+$, let us denote by $V(\lambda )$ (resp. $V(-\lambda )$) the irreducible module with highest (resp. lowest) weight $\lambda$ (resp. $-\lambda$). Let $u_\lambda$ (resp. $u_{-\lambda }$) be the highest (resp. lowest) weight vector.

For $\lambda \in \mathsf{P}^+$, $\mu \in \mathsf{P}^- \mathbin{:=}- \mathsf{P}^+$, we set

$$\begin{equation*} V(\lambda ,\mu ) \mathbin{:=}V(\lambda ) \otimes _{_-}\mspace {-1mu}V(\mu ). \end{equation*}$$

Then $V(\lambda ,\mu )$ is generated by $u_\lambda \otimes _{_-}\mspace {-1mu}u_\mu$ as a $U_q(\mathfrak{g})$-module, and the defining relation of $u_\lambda \otimes _{_-}\mspace {-1mu}u_\mu$ is

$$\begin{align*} & q^h(u_\lambda \otimes _{_-}\mspace {-1mu}u_\mu )=q^{\langle h,\lambda +\mu \rangle }(u_\lambda \otimes _{_-}\mspace {-1mu}u_\mu ), \\ & e_i^{1-\langle h_i,\mu \rangle }(u_\lambda \otimes _{_-}\mspace {-1mu}u_\mu )=0, \quad f_i^{1+\langle h_i,\lambda \rangle }(u_\lambda \otimes _{_-}\mspace {-1mu}u_\mu )=0. \end{align*}$$

Let us define the $\mathbb{Q}$-linear automorphism $\bar{ \ \ }$ of $V(\lambda ,\mu )$ by

$$\begin{equation*} \overline{P(u_\lambda \otimes _{_-}\mspace {-1mu}u_\mu )} = \overline{P}(u_\lambda \otimes _{_-}\mspace {-1mu}u_\mu ). \end{equation*}$$

We set

(i)

$L^{\mathrm{low}}(\lambda ,\mu ) \mathbin{:=}L^{\mathrm{low}}(\lambda ) \mathop{\otimes }_{\mathbf{A}_0} L^{\mathrm{low}}(\mu )$,

(ii)

$V(\lambda ,\mu )_\mathbb{Z} [q^{\pm 1}]\mathbin{:=}V(\lambda )_\mathbb{Z} [q^{\pm 1}]\mathop{\otimes }_\mathbb{Z} [q^{\pm 1}]V(\mu )_\mathbb{Z} [q^{\pm 1}]$,

(iii)

$B(\lambda ,\mu ) \mathbin{:=}B(\lambda ) \mathop{\otimes }B(\mu )$.

Proposition 8.3.1 (32).

$(L^{\mathrm{low}}(\lambda ,\mu ),B(\lambda ,\mu ))$ is a lower crystal basis of $V(\lambda ,\mu )$. Furthermore, $\bigl (\mathbb{Q}\mathop{\otimes }V(\lambda ,\mu )_\mathbb{Z} [q^{\pm 1}],\,L^{\mathrm{low}}(\lambda ,\mu ),\,\overline{L^{\mathrm{low}}(\lambda ,\mu )}\bigr )$ is balanced, and there exists a lower global basis ${\mathbf{B}}^{\mathrm{low}}(V(\lambda ,\mu ))$ obtained from the lower crystal basis $(L^{\mathrm{low}}(\lambda ,\mu ),B(\lambda ,\mu ))$.

Theorem 8.3.2 (32).

The algebra $\widetilde{U}_q(\mathfrak{g})$ has a lower crystal basis $(L^{\mathrm{low}}(\widetilde{U}_q(\mathfrak{g})),B(\widetilde{U}_q(\mathfrak{g})))$ satisfying the following properties:

(i)

$L^{\mathrm{low}}(\widetilde{U}_q(\mathfrak{g})) = \mathop{\textstyle \bigoplus }\limits _{\lambda \in \mathsf{P}} L^{\mathrm{low}}(\widetilde{U}_q(\mathfrak{g})a_\lambda )$ and $B(\widetilde{U}_q(\mathfrak{g})) = \bigsqcup _{\lambda \in \mathsf{P}} B(\widetilde{U}_q(\mathfrak{g})a_\lambda )$, where

•

$L^{\mathrm{low}}(\widetilde{U}_q(\mathfrak{g})a_\lambda ) = L^{\mathrm{low}}(\widetilde{U}_q(\mathfrak{g})) \cap U_q(\mathfrak{g})a_\lambda$ and

•

$B(\widetilde{U}_q(\mathfrak{g})a_\lambda ) = B(\widetilde{U}_q(\mathfrak{g})) \cap \big ( L^{\mathrm{low}}(\widetilde{U}_q(\mathfrak{g})a_\lambda )/qL^{\mathrm{low}}(\widetilde{U}_q(\mathfrak{g})a_\lambda ) \big )$.

(ii)

Set $\widetilde{U}_q(\mathfrak{g})_\mathbb{Z} [q^{\pm 1}]:=\mathop{\textstyle \bigoplus }\limits _{\eta \in \mathsf{P}} U_q(\mathfrak{g})_\mathbb{Z} [q^{\pm 1}]a_\eta$. Then $\bigl (\mathbb{Q}\mathop{\otimes }\widetilde{U}_q(\mathfrak{g})_\mathbb{Z} [q^{\pm 1}],\, L^{\mathrm{low}}(\widetilde{U}_q(\mathfrak{g})),\,\overline{ L^{\mathrm{low}}(\widetilde{U}_q(\mathfrak{g}))}\bigr )$ is balanced, and $\widetilde{U}_q(\mathfrak{g})$ has the lower global basis ${\mathbf{B}}^{\mathrm{low}}(\widetilde{U}_q(\mathfrak{g}))\mathbin{:=}\{ G^{\mathrm{low}}(b)\mid b \in B(\widetilde{U}_q(\mathfrak{g}))\}$.

(iii)

For any $\lambda \in \mathsf{P}^+$ and $\mu \in \mathsf{P}^-$, let$$\begin{equation*} \Psi _{\lambda ,\mu }\colon U_q(\mathfrak{g})a_{\lambda +\mu } \to V(\lambda ,\mu ) \end{equation*}$$

be the $U_q(\mathfrak{g})$-linear map $a_{\lambda +\mu } \longmapsto u_\lambda \mathop{\otimes }u_\mu$. Then we have$$\begin{equation*} \Psi _{\lambda ,\mu }\bigl (L(\widetilde{U}_q(\mathfrak{g})a_{\lambda +\mu })\bigr )= L^{\mathrm{low}}(\lambda ,\mu ). \end{equation*}$$

(iv)

Let $\overline{\Psi }_{\lambda ,\mu }$ be the induced homomorphism$$\begin{equation*} L^{\mathrm{low}}(\widetilde{U}_q(\mathfrak{g})a_{\lambda +\mu })/qL^{\mathrm{low}}(\widetilde{U}_q(\mathfrak{g})a_{\lambda +\mu }) \longrightarrow L^{\mathrm{low}}(\lambda ,\mu )/qL^{\mathrm{low}}(\lambda ,\mu ) . \end{equation*}$$

Then we have

(a)

$\{ b \in B(\widetilde{U}_q(\mathfrak{g})a_{\lambda +\mu }) \ | \ \overline{\Psi }_{\lambda ,\mu }b \ne 0\} \overset{\sim }{\longrightarrow } B(\lambda ,\mu )$,

(b)

$\Psi _{\lambda ,\mu }\bigl (G^{\mathrm{low}}(b)\bigr )= G^{\mathrm{low}}(\overline{\Psi }_{\lambda ,\mu }(b))$ for any $b \in B(\widetilde{U}_q(\mathfrak{g})a_{\lambda +\mu })$.

(v)

$B(\widetilde{U}_q(\mathfrak{g}))$ has a structure of crystal such that the injective map induced by (iv) (a)$$\begin{equation*} B(\lambda ,\mu ) \to B(\widetilde{U}_q(\mathfrak{g})a_{\lambda +\mu }) \subset B(\widetilde{U}_q(\mathfrak{g})) \end{equation*}$$

is a strict embedding of crystals for any $\lambda \in \mathsf{P}^+$ and $\mu \in \mathsf{P}^-$.

For $\lambda \in \mathsf{P}$, take any $\zeta \in \mathsf{P}^+$ and $\eta \in \mathsf{P}^-$ such that $\lambda = \zeta +\eta$. Then $B(\zeta ) \mathop{\otimes }B(\eta )$ is embedded into $B(\widetilde{U}_q(\mathfrak{g})a_\lambda )$.

For $\mu \in \mathsf{P}$, let $T_\mu =\{ t_\mu \}$ be the crystal with

$$\begin{equation*} \operatorname {wt}(t_\mu ) = \mu , \quad \varepsilon _i(t_\mu ) = \varphi _i(t_\mu )=-\infty , \quad \tilde{e}_i (t_\mu ) =\tilde{f}_i (t_\mu ) =0. \end{equation*}$$

Since we have

$$\begin{equation*} B(\zeta ) \hookrightarrow B(\infty ) \mathop{\otimes }T_\zeta , \ B(\eta ) \hookrightarrow T_\eta \mathop{\otimes }B(-\infty ), \text{ and } T_\zeta \mathop{\otimes }T_\eta \simeq T_{\zeta +\eta }, \end{equation*}$$

$B(\zeta ) \mathop{\otimes }B(\eta )$ is embedded into the crystal $B(\infty ) \mathop{\otimes }T_\lambda \mathop{\otimes }B(-\infty )$. Taking $\zeta \to \infty$ and $\eta \to -\infty$, we have

Lemma 8.3.3 (19).

For any $\lambda \in \mathsf{P}$, we have a canonical crystal isomorphism

$$\begin{equation*} B(\widetilde{U}_q(\mathfrak{g})a_{\lambda }) \simeq B(\infty ) \mathop{\otimes }T_\lambda \mathop{\otimes }B(-\infty ). \end{equation*}$$

Hence we identify

$$\begin{equation*} B(\widetilde{U}_q(\mathfrak{g}))=\bigsqcup _{\lambda \in \mathsf{P}}B(\infty ) \mathop{\otimes }T_\lambda \mathop{\otimes }B(-\infty ). \end{equation*}$$

For $\xi \in \mathsf{Q}_-$ and $\eta \in \mathsf{Q}_+$, we shall denote by

$$\begin{align*} U^-_q(\mathfrak{g})_{>\xi }\mathbin{:=}\mathop{\textstyle \bigoplus }\limits _{\xi '\in \mathsf{Q}_-\cap (\xi +\mathsf{Q}_+)\setminus \{\xi \}}U^-_q(\mathfrak{g})_{\xi '}, \qquad U_q^+(\mathfrak{g})_{<\eta }\mathbin{:=}\mathop{\textstyle \bigoplus }\limits _{\eta '\in \mathsf{Q}_+\cap (\eta +\mathsf{Q}_-)\setminus \{\eta \}}U_q^+(\mathfrak{g})_{\eta '}. \end{align*}$$

Then for any $\lambda \in \mathsf{P}$, $b_-\in B(\infty )_\xi$, and $b_+\in B(-\infty )_\eta$, we have

$$\begin{eqnarray} G^{\mathrm{low}}(b_-\mathop{\otimes }t_\lambda \mathop{\otimes }b_+) -G^{\mathrm{low}}(b_-) G^{\mathrm{low}}(b_+)a_\lambda \in U^-_q(\mathfrak{g})_{>\xi }U_q^+(\mathfrak{g})_{<\eta }a_\lambda {\tag{8.4}} \end{eqnarray}$$

19, (3.1.1). In particular, we have

$$\begin{eqnarray*} G^{\mathrm{low}}(b_\infty \mathop{\otimes }t_\lambda \mathop{\otimes }b_+ ) = G^{\mathrm{low}}(b_+)a_\lambda \quad \text{and} \ G^{\mathrm{low}}(b_- \mathop{\otimes }t_\lambda \mathop{\otimes }b_{-\infty }) =G^{\mathrm{low}}(b_-)a_\lambda . \end{eqnarray*}$$

Theorem 8.3.4 (19).

(i)

$L^{\mathrm{low}}(\widetilde{U}_q(\mathfrak{g}))$ is invariant under the anti-automorphisms $*$ and $\varphi$.

(ii)

$B(\widetilde{U}_q(\mathfrak{g}))^* = \varphi (B(\widetilde{U}_q(\mathfrak{g}))) = B(\widetilde{U}_q(\mathfrak{g}))$.

(iii)

$\bigl (G^{\mathrm{low}}(b)\bigr )^*=G^{\mathrm{low}}(b^*)$ and $\varphi (G^{\mathrm{low}}(b))=G^{\mathrm{low}}(\varphi (b))$ for $b \in B(\widetilde{U}_q(\mathfrak{g}))$.

Corollary 8.3.5 (19).

For $b_1 \in B(\infty )$, $b_2 \in B(-\infty )$, we have

(1)

$(b_1 \mathop{\otimes }t_\mu \mathop{\otimes }b_2)^* = b_1^* \mathop{\otimes }t_{-\mu -\operatorname {wt}(b_1)-\operatorname {wt}(b_2)} \mathop{\otimes }b_2^*$.

(2)

$\varphi (b_1 \mathop{\otimes }t_\mu \mathop{\otimes }b_2) = \varphi (b_2) \mathop{\otimes }t_{\mu +\operatorname {wt}(b_1)+\operatorname {wt}(b_2)} \mathop{\otimes }\varphi (b_1)$.

We define, for $b \in B$ with $B=B(\widetilde{U}_q(\mathfrak{g})), B(\infty )$, or $B(-\infty )$,

$$\begin{align*} &\varepsilon _i^*(b) = \varepsilon _i(b^*), \ \varphi _i^*(b) = \varphi _i(b^*), \ \operatorname {wt}^*(b)=\operatorname {wt}(b^*), \\ &\tilde{e}_i^*(b)= \tilde{e}_i(b^*)^*, \text{ and } \tilde{f}_i^*(b) = \tilde{f}_i(b^*)^*. \end{align*}$$

This defines another crystal structure on $\widetilde{U}_q(\mathfrak{g})$: For $b_1 \in B(\infty )$, $b_2 \in B(-\infty )$, and $\eta \in \mathsf{P}$, we have

$$\begin{align*} \varepsilon ^*_i(b_1 \mathop{\otimes }t_\eta \mathop{\otimes }b_2) & = \max ( \varepsilon ^*_i(b_1), \varphi ^*_i(b_2)+\langle h_i, \eta \rangle ), \\ \varphi _i^*(b_1 \mathop{\otimes }t_\eta \mathop{\otimes }b_2) & = \max ( \varepsilon ^*_i(b_1)-\langle h_i, \eta \rangle , \varphi ^*_i(b_2)), \\ &= \varepsilon _i^*(b_1 \mathop{\otimes }t_\eta \mathop{\otimes }b_2) + \langle h_i,\operatorname {wt}^*(b_1 \mathop{\otimes }t_\eta \mathop{\otimes }b_2)\rangle , \\ \operatorname {wt}^*(b_1 \mathop{\otimes }t_\eta \mathop{\otimes }b_2) & = -\eta , \\ \tilde{e}^*_i(b_1 \mathop{\otimes }t_\eta \mathop{\otimes }b_2) & = \begin{cases} (\tilde{e}^*_ib_1) \mathop{\otimes }t_{\eta -\alpha _i} \mathop{\otimes }b_2 & \text{ if } \varepsilon ^*_i(b_1) \ge \varphi ^*_i(b_2)+\langle h_i,\eta \rangle , \\ b_1 \mathop{\otimes }t_{\eta -\alpha _i} \mathop{\otimes }( \tilde{e}^*_ib_2) & \text{ if } \varepsilon ^*_i(b_1) < \varphi ^*_i(b_2)+\langle h_i,\eta \rangle , \end{cases} \\ \tilde{f}^*_i(b_1 \mathop{\otimes }t_\eta \mathop{\otimes }b_2) & = \begin{cases} (\tilde{f}^*_ib_1) \mathop{\otimes }t_{\eta +\alpha _i} \mathop{\otimes }b_2 & \text{ if } \varepsilon ^*_i(b_1) > \varphi ^*_i(b_2)+\langle h_i,\eta \rangle , \\ b_1 \mathop{\otimes }t_{\eta +\alpha _i} \mathop{\otimes }( \tilde{f}^*_ib_2) & \text{ if } \varepsilon ^*_i(b_1) \le \varphi ^*_i(b_2)+\langle h_i,\eta \rangle . \end{cases} \end{align*}$$

In particular, we have

$$\begin{equation*} \tilde{e}_i\circ \varphi =\varphi \circ \tilde{f}^*_i\quad \text{and}\quad \tilde{f}_i\circ \varphi =\varphi \circ \tilde{e}^*_i\quad \text{for every $i\in I$.} \end{equation*}$$

8.4. Relationship of $A_q(\mathfrak{g})$ and $\widetilde{U}_q(\mathfrak{g})$

There exists a canonical pairing $A_q(\mathfrak{g}) \times \widetilde{U}_q(\mathfrak{g})\to \mathbb{Q}(q)$ by

$$\begin{equation*} \langle \psi , x a_\mu \rangle = \delta _{\mu ,\operatorname {wt}_{\operatorname {l}}(\psi )} \psi (x) \quad \text{ for any $\psi \in A_q(\mathfrak{g})$, $x\in U_q(\mathfrak{g})$, and $\mu \in \mathsf{P}$.} \end{equation*}$$

Theorem 8.4.1 (19).

There exists a bi-crystal embedding

$$\begin{equation*} \overline{\iota }_\mathfrak{g}\colon B(A_q(\mathfrak{g})) \xrightarrow {\,\hspace{2ex}\,}B(\widetilde{U}_q(\mathfrak{g})) \end{equation*}$$

which satisfies

$$\begin{equation*} \langle G^\mathrm{up}(b),\varphi (G^{\mathrm{low}}(b')) \rangle = \delta _{\,\overline{\iota }_\mathfrak{g}(b),b'} \end{equation*}$$

for any $b \in B(A_q(\mathfrak{g}))$ and $b' \in B(\widetilde{U}_q(\mathfrak{g}))$.

8.5. Relationship of $A_q(\mathfrak{g})$ and $A_q(\mathfrak{n})$

Definition 8.5.1.

Let $p_\mathfrak{n}\colon A_q(\mathfrak{g}) \to A_q(\mathfrak{n})$ be the homomorphism induced by $U_q^+(\mathfrak{g}) \to U_q(\mathfrak{g})$,

$$\begin{equation*} \langle p_\mathfrak{n}(\psi ),x \rangle = \psi (x) \quad \text{for any $x \in U_q^+(\mathfrak{g})$.} \end{equation*}$$

Then we have

$$\begin{equation*} \operatorname {wt}(p_\mathfrak{n}(\psi ))=\operatorname {wt}_{\operatorname {l}}(\psi )-\operatorname {wt}_{\mspace {1mu}\mathrm{r}}(\psi ). \end{equation*}$$

It is obvious that $p_\mathfrak{n}$ sends all $\Phi (u_{w\lambda }\mathop{\otimes }u_{w\lambda }^{\mspace {1mu}\mathrm{r}})$ ($\lambda \in \mathsf{P}^+$ and $w\in W$) to $1$. Note that $\overline{\iota }_\mathfrak{g}(u_{w\lambda }\mathop{\otimes }u_{w\lambda }^{\mspace {1mu}\mathrm{r}})=b_\infty \mathop{\otimes }t_{w\lambda }\mathop{\otimes }b_{-\infty }\in B(\widetilde{U}_q(\mathfrak{g}))$.

Proposition 8.5.2.

For $b \in B(A_q(\mathfrak{g}))$, set

$$\begin{equation*} {\overline{\iota }_\mathfrak{g}}(b)=b_1 \mathop{\otimes }t_\zeta \mathop{\otimes }b_2 \in B(\infty ) \mathop{\otimes }T_\zeta \mathop{\otimes }B(-\infty )\subset B(\widetilde{U}_q(\mathfrak{g})) \end{equation*}$$

$(\zeta \in \mathsf{P})$. Then we have

$$\begin{equation*} p_\mathfrak{n}(G^\mathrm{up}(b)) = \delta _{b_2,\,b_{-\infty }}G^\mathrm{up}(b_1). \end{equation*}$$

Proof.

Set $\eta \mathbin{:=}\operatorname {wt}(b_1)+\zeta +\operatorname {wt}(b_2)=\operatorname {wt}_\mathrm{l}(b)$. Then for any $\tilde{b} \in B(\infty )$, we have

$$\begin{align*} & \big \langle p_\mathfrak{n}(G^\mathrm{up}(b)), \varphi (G^{\mathrm{low}}(\tilde{b})) \big \rangle = \big \langle G^\mathrm{up}(b), G^{\mathrm{low}}(\varphi (\tilde{b}))a_{\eta } \big \rangle \\ &\quad = \big \langle G^\mathrm{up}(b), G^{\mathrm{low}}(b_\infty \mathop{\otimes }t_\eta \mathop{\otimes }\varphi (\tilde{b})) \big \rangle = \big \langle G^\mathrm{up}(b), \varphi (G^{\mathrm{low}}(\tilde{b} \mathop{\otimes }t_{\eta -\operatorname {wt}(\tilde{b})} \mathop{\otimes }b_{-\infty })) \big \rangle \\ &\quad = \delta ({\overline{\iota }_\mathfrak{g}}(b) = \tilde{b} \mathop{\otimes }t_{\eta -\operatorname {wt}(\tilde{b}) } \mathop{\otimes }b_{-\infty } ) =\delta (b_2=b_{-\infty },b_1 = \tilde{b}). \end{align*}$$■

Hence the map $p_\mathfrak{n}$ sends the upper global basis of $A_q(\mathfrak{g})$ to the upper global basis of $A_q(\mathfrak{n})$ or zero. Thus we have a map

$$\begin{equation*} \overline{p}_\mathfrak{n}\colon B(A_q(\mathfrak{g})) \to B(A_q(\mathfrak{n})) \bigsqcup \{ 0 \}. \end{equation*}$$

Although the map $p_\mathfrak{n}$ is not an algebra homomorphism, it preserves the multiplications up to a power of $q$, as we will see below.

Lemma 8.5.3.

For $x \in U_q^+(\mathfrak{g})$, if $\Delta _\mathfrak{n}(x)=x_{(1)} \mathop{\otimes }x_{(2)}$, then

$$\begin{align} \Delta _+(x)=q^{\operatorname {wt}(x_{(1)})} x_{(2)} \mathop{\otimes }x_{(1)}. {\tag{8.5}} \end{align}$$

Proof.

Assume that 8.5 holds for $x \in U_q^+(\mathfrak{g})$. Note that

$$\begin{equation*} \Delta _\mathfrak{n}(e_ix) = (e_i \mathop{\otimes }1+1 \mathop{\otimes }e_i)(x_{(1)} \mathop{\otimes }x_{(2)}) = e_ix_{(1)} \mathop{\otimes }x_{(2)} + q^{-(\alpha _i,\operatorname {wt}(x_{(1)}))}x_{(1)} \mathop{\otimes }(e_ix_{(2)}). \end{equation*}$$

On the other hand, we have

$$\begin{align*} \Delta _+(e_ix)&=(e_i \mathop{\otimes }1+ q^{\alpha _i} \mathop{\otimes }e_i)(q^{\operatorname {wt}(x_{(1)})} x_{(2)} \mathop{\otimes }x_{(1)}) \\ &=(e_i q^{\operatorname {wt}(x_1)}) x_{(2)} \mathop{\otimes }x_{(1)} + ( q^{\alpha _i+\operatorname {wt}(x_{(1)})} x_{(2)}) \mathop{\otimes }(e_ix_{(1)}) \\ & = q^{-(\alpha _i,\operatorname {wt}(x_{(1)}))}(q^{\operatorname {wt}(x_{(1)})}e_ix_{(2)}) \mathop{\otimes }x_{(1)}+ (q^{\operatorname {wt}(e_ix_{(1)})}x_{(2)}) \mathop{\otimes }(e_ix_{(1)}). \end{align*}$$

Hence 8.5 holds for $e_ix$.

■

Proposition 8.5.4.

For $\psi ,\theta \in A_q(\mathfrak{g})$, we have

$$\begin{equation*} p_\mathfrak{n}(\psi \theta )= q^{(\operatorname {wt}_{\mspace {1mu}\mathrm{r}}(\psi ),\operatorname {wt}_{\mspace {1mu}\mathrm{r}}(\theta )-\operatorname {wt}_{\operatorname {l}}(\theta ))}p_\mathfrak{n}(\psi )p_\mathfrak{n}(\theta ). \end{equation*}$$

Proof.

For $x \in U_q^+(\mathfrak{g})$, set $\Delta _\mathfrak{n}(x)=x_{(1)} \mathop{\otimes }x_{(2)}$. Then, we have

$$\begin{align*} \langle p_\mathfrak{n}(\psi \theta ),x \rangle & = \langle \psi \theta ,x \rangle = \langle \psi \mathop{\otimes }\theta , q^{\operatorname {wt}(x_{(1)})} x_{(2)} \mathop{\otimes }x_{(1)} \rangle = \langle \psi , q^{\operatorname {wt}(x_{(1)})} x_{(2)} \rangle \langle \theta , x_{(1)} \rangle \\ &= q^{(\operatorname {wt}_{\mspace {1mu}\mathrm{r}}(\psi ),\operatorname {wt}(x_{(1)}))} \langle \psi , x_{(2)} \rangle \langle \theta , x_{(1)} \rangle \\ &= q^{(\operatorname {wt}_{\mspace {1mu}\mathrm{r}}(\psi ),\operatorname {wt}(x_{(1)}))} \langle p_\mathfrak{n}(\psi ) , x_{(2)} \rangle \langle p_\mathfrak{n}(\theta ) , x_{(1)} \rangle \\ &\underset{\mathrm{(a)}}{=} q^{(\operatorname {wt}_{\mspace {1mu}\mathrm{r}}(\psi ),\operatorname {wt}_{\mspace {1mu}\mathrm{r}}(\theta )-\operatorname {wt}_{\operatorname {l}}(\theta ))} \langle p_\mathfrak{n}(\psi )\mathop{\otimes }p_\mathfrak{n}(\theta ),\Delta _\mathfrak{n}(x) \rangle \\ &= q^{(\operatorname {wt}_{\mspace {1mu}\mathrm{r}}(\psi ),\operatorname {wt}_{\mspace {1mu}\mathrm{r}}(\theta )-\operatorname {wt}_{\operatorname {l}}(\theta ))} \langle p_\mathfrak{n}(\psi )p_\mathfrak{n}(\theta ),x \rangle . \end{align*}$$

Here, we used $\operatorname {wt}(x_{(1)})=-\operatorname {wt}\bigl (p_\mathfrak{n}(\theta )\bigr )$ in (a).

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8.6. Global basis of $\widetilde{U}_q(\mathfrak{g})$ and tensor products of $U_q(\mathfrak{g})$-modules in $\mathcal{O}_{\mathrm{int}}(\mathfrak{g})$

Let $V$ be an integrable $U_q(\mathfrak{g})$-module with a bar-involution $-$; that is, there is a $\mathbb{Q}$-linear automorphism $-$ satisfying $\overline{P v} = \overline{P} \overline{v}$ for all $P \in U_q(\mathfrak{g})$ and for all $v \in V$. Then, for any $\lambda \in \mathsf{P}^+$, there exists a unique bar-involution $-$ on $V(\lambda )\otimes _{_-}\mspace {-1mu}V$ satisfying

$$\begin{equation*} \text{$\overline{(u_\lambda \otimes _{_-}\mspace {-1mu}v)}=u_\lambda \otimes _{_-}\mspace {-1mu}\overline{v}$ for any $v\in V$.} \end{equation*}$$

Indeed, there exists $\Xi \in {\mathbf{{1}}}+\prod _{\beta \in \mathsf{Q}_+\setminus \{0\}}U_q^+(\mathfrak{g})_\beta \mathop{\otimes }U^-_q(\mathfrak{g})_{-\beta }$, which defines a bar-involution by setting

$$\begin{equation*} \overline{u\otimes _{_-}\mspace {-1mu}v}:=\Xi \bigl (\overline{u}\otimes _{_-}\mspace {-1mu}\overline{v}\bigr ) \end{equation*}$$

(see 33, Chapter 4). Assume that $V$ has a lower crystal basis $\bigl (L(V),B(V)\bigr )$ and an $\mathbf{A}$-form $V_\mathbf{A}$ such that $\bigl (V_\mathbf{A}, L(V),\overline{L(V)}\bigr )$ is balanced. Then we have

Proposition 8.6.1.

The triple $\Bigl ( V(\lambda )_\mathbf{A}\mathop{\otimes }_\mathbf{A}V_\mathbf{A}, L(\lambda )\mathop{\otimes }_{\mathbf{A}_0} L(V), \overline{ L(\lambda )\mathop{\otimes }_{\mathbf{A}_0} L(V)} \Bigr )$ in $V(\lambda )\otimes _{_-}\mspace {-1mu}V$ is balanced.

Note that $u_\lambda \otimes _{_-}\mspace {-1mu}G^{\mathrm{low}}(b)$ is a lower global basis for any $b\in B(V)$, i.e., $G^{\mathrm{low}}(u_\lambda \mathop{\otimes }b)=u_\lambda \otimes _{_-}\mspace {-1mu}G^{\mathrm{low}}(b)$.

In particular, it applies to $V(\lambda )\otimes _{_-}\mspace {-1mu}V(\mu )$. Moreover, we have the following proposition.

Proposition 8.6.2.

Let $\lambda ,\mu \in \mathsf{P}^+$ and $w \in W$. Then for any $b \in B(\widetilde{U}_q(\mathfrak{g})a_{\lambda +w\mu })$, $G^{\mathrm{low}}(b)(u_\lambda \otimes _{_-}\mspace {-1mu}u_{w\mu })$ vanishes or is a member of the lower global basis of $V(\lambda ) \otimes _{_-}\mspace {-1mu}V(\mu )$.

Hence we have a crystal morphism

$$\begin{align} \pi _{\lambda ,w\mu }\colon B(\widetilde{U}_q(\mathfrak{g})a_{\lambda +w\mu }) \to B(\lambda ) \mathop{\otimes }B(\mu ) {\tag{8.6}} \end{align}$$

by $G^{\mathrm{low}}(b)(u_\lambda \otimes _{_-}\mspace {-1mu}u_{w\mu })=G^{\mathrm{low}}(\pi _{\lambda ,w\mu }(b))$.

Similarly, we have a bar-involution $-$ on $V\otimes _{_+}\mspace {-1mu}V(\lambda )$ such that

$$\begin{equation*} \text{$\overline{(v\otimes _{_+}\mspace {-1mu}u_\lambda )}=\overline{v}\otimes _{_+}\mspace {-1mu}u_\lambda $ for any $v\in V$.} \end{equation*}$$

Hence if $V$ has an upper crystal basis $(L^\mathrm{up}(V),B(V))$ and an $\mathbf{A}$-form $V_\mathbf{A}$ such that $\bigl (V_\mathbf{A}, L^\mathrm{up}(V), \overline{L^\mathrm{up}(V)} \bigr )$ is balanced, then $V\otimes _{_+}\mspace {-1mu}V(\lambda )$ has an upper global basis. Note that $G^\mathrm{up}(b)\otimes _{_+}\mspace {-1mu}u_\lambda$ is a member of the upper global basis for $b\in B(V)$.

In particular for $\lambda ,\mu \in \mathsf{P}$, $V(\lambda )\otimes _{_-}\mspace {-1mu}V(\mu )$ has a lower global basis and $V(\lambda )\otimes _{_+}\mspace {-1mu}V(\mu )$ has an upper global basis.

The bilinear form

$$\begin{eqnarray*} (\mdsmblkcircle ,\mdsmblkcircle )\colon \Bigl (V(\lambda )\otimes _{_-}\mspace {-1mu}V(\mu )\Bigr ) \times \Bigl (V(\lambda )\otimes _{_+}\mspace {-1mu}V(\mu )\Bigr )\to \mathbf{k} \end{eqnarray*}$$

defined by $(u\otimes _{_-}\mspace {-1mu}v,u'\otimes _{_+}\mspace {-1mu}v')=(u,u')(v,v')$, $u,u'\in V(\lambda )$, $v,v'\in V(\mu )$ satisfies

$$\begin{eqnarray*} \text{$(ax,y)=(x,\varphi (a)y)$ for any $x\in V(\lambda )\otimes _{_-}\mspace {-1mu}V(\mu )$, $y\in V(\lambda )\otimes _{_+}\mspace {-1mu}V(\mu )$, $a\in U_q(\mathfrak{g})$.} \end{eqnarray*}$$

With respect to this bilinear form, the lower global basis of $V(\lambda )\otimes _{_-}\mspace {-1mu}V(\mu )$ and the upper global basis of $V(\lambda )\otimes _{_+}\mspace {-1mu}V(\mu )$ are the dual bases of each other.

9. Quantum minors and $T$-systems

9.1. Quantum minors

Using the isomorphism $\Phi$ in 8.1, for each $\lambda \in \mathsf{P}^+$ and $\mu , \zeta \in W \lambda$, we define the elements

$$\begin{equation*} \Delta (\mu ,\zeta )\mathbin{:=}\Phi (u_\mu \mathop{\otimes }u_\zeta ^{\mspace {1mu}\mathrm{r}}) \in A_q(\mathfrak{g}) \end{equation*}$$

and

$$\begin{equation*} \mathrm{D}(\mu ,\zeta )\mathbin{:=}p_\mathfrak{n}(\Delta (\mu ,\zeta )) \in A_q(\mathfrak{n}). \end{equation*}$$

The element $\Delta (\mu ,\zeta )$ is called a (generalized) quantum minor and $\mathrm{D}(\mu ,\zeta )$ is called a unipotent quantum minor.

Lemma 9.1.1.

$\Delta (\mu ,\zeta )$ is a member of the upper global basis of $A_q(\mathfrak{g})$. Moreover, $\mathrm{D}(\mu ,\zeta )$ is either a member of the upper global basis of $A_q(\mathfrak{n})$ or zero.

Proof.

Our assertions follow from Proposition 8.1.3 and Proposition 8.5.2.

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Lemma 9.1.2 (3, (9.13)).

For $u,v \in W$ and $\lambda ,\mu \in \mathsf{P}^+$, we have

$$\begin{equation*} \Delta (u\lambda ,v\lambda )\Delta (u\mu ,v\mu ) = \Delta \bigl (u (\lambda +\mu ),v(\lambda +\mu )\bigr ). \end{equation*}$$

By Proposition 8.5.4, we have the following corollary.

Corollary 9.1.3.

For $u,v \in W$ and $\lambda ,\mu \in \mathsf{P}^+$, we have

$$\begin{equation*} \mathrm{D}(u\lambda ,v\lambda )\mathrm{D}(u\mu ,v\mu ) = q^{-(v\lambda ,v\mu -u\mu )}\mathrm{D}\bigl (u (\lambda +\mu ),v(\lambda +\mu )\bigr ). \end{equation*}$$

Note that

$$\begin{equation*} \mathrm{D}(\mu ,\mu )=1 \quad \text{ for $\mu \in W\lambda $ }. \end{equation*}$$

Then $\mathrm{D}(\mu ,\zeta ) \ne 0$ if and only if $\mu \preceq \zeta$. Recall that for $\mu ,\zeta$ in the same $W$-orbit, we say that $\mu \preceq \zeta$ if there exists a sequence $\{ \beta _k \}_{1 \le k \le l}$ of positive real roots such that, defining $\lambda _0=\zeta$, $\lambda _k=s_{\beta _k}\lambda _{k-1}$ $(1 \le k \le l)$, we have $(\beta _k,\lambda _{k-1}) \ge 0$ and $\lambda _l=\mu$.

More precisely, we have the following lemma.

Lemma 9.1.4.

Let $\lambda \in \mathsf{P}^+$ and $\mu ,\zeta \in W \lambda$. Then the following conditions are equivalent:

(i)

$\mathrm{D}(\mu ,\zeta )$ is an element of the upper global basis of $A_q(\mathfrak{n})$,

(ii)

$\mathrm{D}(\mu ,\zeta ) \ne 0$,

(iii)

$u_\mu \in U_q^-(\mathfrak{g})u_\zeta$,

(iv)

$u_\zeta \in U_q^+(\mathfrak{g})u_\mu$,

(v)

$\mu \preceq \zeta$,

(vi)

for any $w \in W$ such that $\mu = w \lambda$, there exists $u \le w$ (in the Bruhat order) such that $\zeta =u \lambda$,

(vii)

there exist $u,v \in W$ such that $\mu = w \lambda$, $\zeta =u\lambda$, and $u \le w$.

Proof.

(i) and (ii) are equivalent by Lemma 9.1.1. The equivalence of (ii), (iii), and (iv) is obvious. The equivalence of (v), (vi), and (vii) is well known. The equivalence of (iv) and (vi) is proved in 18.

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For any $u\in A_q(\mathfrak{n})\setminus \{0\}$ and $i\in I$, we set

$$\begin{align*} \varepsilon _i(u)&\mathbin{:=}\max \left\{n\in \mathbb{Z}_{\ge 0} \mid e_i^nu\not =0 \right\},\\ \varepsilon ^*_i(u)&\mathbin{:=}\max \left\{n\in \mathbb{Z}_{\ge 0} \mid e_i^{*\,n}u\not =0 \right\}. \end{align*}$$

Then for any $b\in B(A_q(\mathfrak{n}))$, we have

$$\begin{equation*} \text{$\varepsilon _i(G^\mathrm{up}(b))=\varepsilon _i(b)$ and $\varepsilon ^*_i(G^\mathrm{up}(b))=\varepsilon ^*_i(b)$.} \end{equation*}$$

Lemma 9.1.5.

Let $\lambda \in \mathsf{P}^+$, $\mu ,\zeta \in W \lambda$ such that $\mu \preceq \zeta$ and $i \in I$.

(i)

If $n \mathbin{:=}\langle h_i,\mu \rangle \ge 0$, then$$\begin{equation*} \varepsilon _i(\mathrm{D}(\mu ,\zeta ))=0 \ \ \text{ and } \ \ e_i^{(n)}\mathrm{D}(s_i\mu ,\zeta )=\mathrm{D}(\mu ,\zeta ). \end{equation*}$$

(ii)

If $\langle h_i,\mu \rangle \le 0$ and $s_i\mu \preceq \zeta$, then $\varepsilon _i(\mathrm{D}(\mu ,\zeta ))=-\langle h_i,\mu \rangle$.

(iii)

If $m \mathbin{:=}-\langle h_i,\zeta \rangle \ge 0$, then$$\begin{equation*} \varepsilon ^*_i(\mathrm{D}(\mu ,\zeta ))=0 \ \ \text{ and } \ \ {e_i^*}^{(m)} \mathrm{D}(\mu ,s_i\zeta )= \mathrm{D}(\mu ,\zeta ). \end{equation*}$$

(iv)

If $\langle h_i,\zeta \rangle \ge 0$ and $\mu \preceq s_i\zeta$, then $\varepsilon ^*_i(\mathrm{D}(\mu ,\zeta ))=\langle h_i,\zeta \rangle$.

Proof.

We have $\varepsilon _i\bigl (\Delta (\mu ,\zeta )\bigr )=\max (-\langle h_i,\mu \rangle , 0)$ and $\varepsilon _i^*\bigl (\Delta (\mu ,\zeta )\bigr )=\max (\langle h_i,\zeta \rangle , 0)$. Moreover, $p_\mathfrak{n}$ commutes with $e_i^{(n)}$ and ${e_i^*}^{(n)}$.

Let us show (ii). Set $\ell =-\langle h_i,\mu \rangle$. Then we have $e_i^{\ell +1}\Delta (\mu ,\zeta )=0$, which implies $e_i^{\ell +1}\mathrm{D}(\mu ,\zeta )=0$. Hence $\varepsilon _i(\mathrm{D}(\mu ,\zeta ))\le \ell$. We have

$$\begin{equation*} e_i^{(\ell )}\Delta (\mu ,\zeta )=\Delta (s_i\mu ,\zeta ). \end{equation*}$$

Hence we have $e_i^{(\ell )}\mathrm{D}(\mu ,\zeta )=\mathrm{D}(s_i\mu ,\zeta )$. By the assumption $s_i\mu \preceq \zeta$, $\mathrm{D}(s_i\mu ,\zeta )$ does not vanish. Hence we have $\varepsilon _i(\mathrm{D}(\mu ,\zeta ))\ge \ell$.

The other statements can be proved similarly.

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Proposition 9.1.6 (3, (10.2)).

Let $\lambda ,\mu \in \mathsf{P}^+$ and $s,t,s',t' \in W$ such that $\ell (s's)=\ell (s')+\ell (s)$ and $\ell (t't)=\ell (t')+\ell (t)$. Then we have

(i)

$\Delta (s's\lambda ,t'\lambda )\Delta (s'\mu ,t't\mu )=q^{(s\lambda ,\mu )-(\lambda ,t\mu )}\Delta (s'\mu ,t't\mu )\Delta (s's\lambda ,t'\lambda )$.

(ii)

If we assume further that $s's\lambda \preceq t'\lambda$ and $s'\mu \preceq t't\mu$, then we have$$\begin{align} & \mathrm{D}(s's\lambda ,t'\lambda )\mathrm{D}(s'\mu ,t't\mu ) = q^{(s's\lambda +t'\lambda ,\;s'\mu -t't\mu )} \mathrm{D}(s'\mu ,t't\mu )\mathrm{D}(s's\lambda ,t'\lambda ), {\tag{9.1}} \end{align}$$

or equivalently$$\begin{align} & q^{(t'\lambda ,\;t't\mu -s'\mu )}\mathrm{D}(s's\lambda ,t'\lambda )\mathrm{D}(s'\mu ,t't\mu ) =q^{(s'\mu -t't\mu ,\;s's\lambda )}\mathrm{D}(s'\mu ,t't\mu )\mathrm{D}(s's\lambda ,t'\lambda ). {\tag{9.2}} \end{align}$$

Note that (ii) follows from Proposition 8.5.4 and (i). Note also that both sides of 9.2 are bar-invariant, and hence they are members of the upper global basis as seen by Corollary 4.1.5.

Proposition 9.1.7.

For $\lambda ,\mu \in \mathsf{P}^+$ and $s,t\in W$, set ${\overline{\iota }_\mathfrak{g}}\bigl (u_{s\lambda }\mathop{\otimes }(u_\lambda )^{\mspace {1mu}\mathrm{r}}\bigr )= b_-\mathop{\otimes }t_\lambda \mathop{\otimes }b_{-\infty }$ and ${\overline{\iota }_\mathfrak{g}}\bigl (u_{\mu }\mathop{\otimes }(u_{t\mu })^{\mspace {1mu}\mathrm{r}}\bigr )=b_{\infty }\mathop{\otimes }t_{t\mu }\mathop{\otimes }b_+$ with $b_{\mp }\in B(\pm \infty )$. Then we have

$$\begin{equation*} \Delta (s\lambda ,\lambda )\Delta (\mu ,t\mu )=G^\mathrm{up}\bigl ({\overline{\iota }_\mathfrak{g}}^{-1} (b_-\mathop{\otimes }t_{\lambda +t\mu }\mathop{\otimes }b_+)\bigr ). \end{equation*}$$

Proof.

Recall that there is a pairing $(\mdsmblkcircle ,\mdsmblkcircle ): \bigl (V(\lambda )\otimes _{_-}\mspace {-1mu}V(\mu )\bigr )\times \bigl (V(\lambda )\otimes _{_+}\mspace {-1mu}V(\mu )\bigr )\to \mathbb{Q}(q)$ defined by $(u\otimes _{_-}\mspace {-1mu}v,u'\otimes _{_+}\mspace {-1mu}v')=(u,u')(v,v')$. It satisfies

$$\begin{equation*} \bigl (P(u\otimes _{_-}\mspace {-1mu}v), u'\otimes _{_+}\mspace {-1mu}v'\bigr )=\bigl (u\otimes _{_-}\mspace {-1mu}v, \varphi (P)(u'\otimes _{_+}\mspace {-1mu}v')\bigr )\quad \text{for any} \ P \in U_q(\mathfrak{g}). \end{equation*}$$

For $u,u'\in V(\lambda )$ and $v,v'\in V(\mu )$, we have

$$\begin{align*} \langle \Phi (u\mathop{\otimes }u'^{\mspace {1mu}\mathrm{r}})\Phi (v\mathop{\otimes }v'^{\mspace {1mu}\mathrm{r}}), P\rangle &= \bigl (u'\otimes _{_-}\mspace {-1mu}v', P(u\otimes _{_+}\mspace {-1mu}v)\bigr )\\ &=\bigl (\varphi (P)(u'\otimes _{_-}\mspace {-1mu}v'),u\otimes _{_+}\mspace {-1mu}v\bigr ). \end{align*}$$

Hence for $P\in U_q(\mathfrak{g})$, we have

$$\begin{align*} \langle \Delta (s\lambda ,\lambda )\Delta (\mu ,t\mu ), Pa_{\zeta }\rangle &=\delta (\zeta =s\lambda +\mu ) \bigl (\varphi (P)(u_{\lambda }\otimes _{_-}\mspace {-1mu}u_{t\mu }),u_{s\lambda }\otimes _{_+}\mspace {-1mu}u_{\mu }\bigr ). \end{align*}$$

If $Pa_\zeta =G^{\mathrm{low}}(\varphi (b))$ for $b\in B(\widetilde{U}_q(\mathfrak{g}))$, then we have

$$\begin{equation*} \langle \Delta (s\lambda ,\lambda )\Delta (\mu ,t\mu ),\varphi (G^{\mathrm{low}}(b))\rangle =\delta (\zeta =s\lambda +\mu ) \bigl (G^{\mathrm{low}}(b)(u_{\lambda }\otimes _{_-}\mspace {-1mu}u_{t\mu }),u_{s\lambda }\otimes _{_+}\mspace {-1mu}u_{\mu }\bigr ). \end{equation*}$$

The element $G^{\mathrm{low}}(b)(u_{\lambda }\otimes _{_-}\mspace {-1mu}u_{t\mu })$ vanishes or is a global basis of $V(\lambda )\otimes _{_-}\mspace {-1mu}V(\mu )$ by Proposition 8.6.2. Since $u_{s\lambda }\otimes _{_+}\mspace {-1mu}u_{\mu }$ is a member of the upper global basis of $V(\lambda )\otimes _{_+}\mspace {-1mu}V(\mu )$, we have

$$\begin{equation*} \langle \Delta (s\lambda ,\lambda )\Delta (\mu ,t\mu ),\varphi (G^{\mathrm{low}}(b))\rangle =\delta (\zeta =s\lambda +\mu ) \delta \bigl (\pi _{\lambda ,t\mu }(b)=u_{s\lambda }\mathop{\otimes }u_{\mu }\bigr ). \end{equation*}$$

Here $\pi _{\lambda ,t\mu }\colon B(\widetilde{U}_q(\mathfrak{g})a_{\lambda +t\mu })\to B(\lambda )\mathop{\otimes }B(\mu )$ is the crystal morphism given in 8.6.

Hence we obtain

$$\begin{equation*} \Delta (s\lambda ,\lambda )\Delta (\mu ,t\mu )=G^\mathrm{up}( {\overline{\iota }_\mathfrak{g}}^{-1}( b)), \end{equation*}$$

where $b\in B(\widetilde{U}_q(\mathfrak{g}))$ is a unique element such that

$$\begin{equation*} \bigl (G^{\mathrm{low}}(b)(u_{\lambda }\otimes _{_-}\mspace {-1mu}u_{s\mu }), u_{s\lambda }\otimes _{_+}\mspace {-1mu}u_{\mu }\bigr )=\nobreakspace 1 \end{equation*}$$

. On the other hand, we have $G^{\mathrm{low}}(b_+)u_{t\mu }=u_{\mu }$ and $G^{{\mathrm{low}}}(b_-)u_\lambda =u_{s\lambda }$. The last equality implies $\varphi (G^{{\mathrm{low}}}(b_-))u_{s\lambda }=u_\lambda$ because

$$\begin{equation*} (\varphi (G^{{\mathrm{low}}}(b_-))u_{s\lambda },u_\lambda )=(u_{s\lambda },G^{{\mathrm{low}}}(b_-)u_\lambda )=(u_{s\lambda },u_{s\lambda })=1. \end{equation*}$$

As seen in 8.4, we have

$$\begin{equation*} G^{{\mathrm{low}}}(b_-)G^{\mathrm{low}}(b_+)a_{\lambda +t\mu }-G^{\mathrm{low}}(b_-\mathop{\otimes }t_{\lambda +t\mu }\mathop{\otimes }b_+) \in U^-_q(\mathfrak{g})_{>s\lambda -\lambda }U_q^+(\mathfrak{g})_{<\mu -t\mu }a_{\lambda +t\mu }. \end{equation*}$$

Hence we obtain

$$\begin{eqnarray*} &&\bigl (G^{\mathrm{low}}(b_-\mathop{\otimes }t_{\lambda +t\mu }\mathop{\otimes }b_+)(u_{\lambda }\otimes _{_-}\mspace {-1mu}u_{t\mu }),\; u_{s\lambda }\otimes _{_+}\mspace {-1mu}u_{\mu }\bigr )\\ &&\hspace{10ex} =\bigl (G^{{\mathrm{low}}}(b_-)G^{\mathrm{low}}(b_+)(u_{\lambda }\otimes _{_-}\mspace {-1mu}u_{t\mu }),\; u_{s\lambda }\otimes _{_+}\mspace {-1mu}u_{\mu }\bigr )\\ &&\hspace{20ex} =\bigl (G^{\mathrm{low}}(b_+)(u_{\lambda }\otimes _{_-}\mspace {-1mu}u_{t\mu }),\; \varphi (G^{\mathrm{low}}(b_-))(u_{s\lambda }\otimes _{_+}\mspace {-1mu}u_{\mu })\bigr )=1. \end{eqnarray*}$$

In the last equality, we used $G^{\mathrm{low}}(b_+)(u_{\lambda }\otimes _{_-}\mspace {-1mu}u_{t\mu }) =u_{\lambda }\otimes _{_-}\mspace {-1mu}(G^{\mathrm{low}}(b_+)u_{t\mu })=u_{\lambda }\otimes _{_-}\mspace {-1mu}u_\mu$ and $\varphi (G^{\mathrm{low}}(b_-))(u_{s\lambda }\otimes _{_+}\mspace {-1mu}u_{\mu }) =\bigl (\varphi (G^{\mathrm{low}}(b_-))u_{s\lambda })\otimes _{_+}\mspace {-1mu}u_{\mu }=u_{\lambda }\otimes _{_+}\mspace {-1mu}u_\mu$.

Hence we conclude that $b=b_-\mathop{\otimes }t_{\lambda +t\mu }\mathop{\otimes }b_+$.

■

Let

$$\begin{eqnarray*} &&\iota _{\lambda ,\mu }\colon V(\lambda +\mu )\rightarrowtail V(\lambda )\mathop{\otimes }V(\mu ) \end{eqnarray*}$$

be the canonical embedding and

$$\begin{eqnarray*} &&\overline{\iota }_{\lambda ,\mu }\colon B(\lambda +\mu )\rightarrowtail B(\lambda )\mathop{\otimes }B(\mu ) \end{eqnarray*}$$

the induced crystal embedding.

Lemma 9.1.8.

For $\lambda ,\mu \in \mathsf{P}^+$ and $x,y \in W$ such that $x \ge y$, we have

$$\begin{equation*} u_{x\lambda } \otimes u_{y \mu } \in \overline{\iota }_{\lambda ,\mu } (B(\lambda +\mu )) \subset B(\lambda ) \otimes B(\mu ). \end{equation*}$$

Proof.

Let us show by induction on $\ell (x)$ the length of $x$ in $W$. We may assume that $x \ne 1$. Then there exists $i \in I$ such that $s_i x < x$. If $s_iy<y$, then $s_ix \ge s_iy$ and $\tilde{e}_i^{\max }(u_{x\lambda } \otimes u_{y\mu })= u_{s_ix\lambda } \otimes u_{s_iy\mu }$. If $s_iy>y$, then $s_ix\ge y$ and $\tilde{e}_i^{\max }(u_{x\lambda } \otimes u_{y\mu })= u_{s_ix\lambda } \otimes u_{y\mu }$. In both cases, $u_{x\lambda } \otimes u_{y \mu }$ is connected with an element of $\overline{\iota }_{\lambda ,\mu }(B(\lambda +\mu ))$.

■

Lemma 9.1.9.

For $\lambda , \mu \in \mathsf{P}^+$ and $w\in W$, we have

$$\begin{equation*} \Delta (w\lambda ,\lambda )\Delta (\mu ,\mu )= G^\mathrm{up}\bigl (\overline{\iota }_{\lambda ,\mu }^{-1}(u_{w\lambda }\mathop{\otimes }u_\mu )\mathop{\otimes }u_{\lambda +\mu }{}^{\mspace {1mu}\mathrm{r}}\bigr ). \end{equation*}$$

Proof.

We have

$$\begin{eqnarray*} &&{\overline{\iota }_\mathfrak{g}}(u_{w\lambda }\mathop{\otimes }u_\lambda ^{\mspace {1mu}\mathrm{r}}\bigr )=b_{w\lambda }\mathop{\otimes }t_{\lambda }\mathop{\otimes }b_{-\infty },\\ &&{\overline{\iota }_\mathfrak{g}}(u_{\mu }\mathop{\otimes }u_\mu ^{\mspace {1mu}\mathrm{r}}\bigr )=b_{\infty }\mathop{\otimes }t_{\mu }\mathop{\otimes }b_{-\infty }, \end{eqnarray*}$$

where $b_{w\lambda }\mathbin{:=}\overline{\iota }_\lambda (u_{w\lambda })$. Hence Proposition 9.1.7 implies that

$$\begin{equation*} \Delta (w\lambda ,\lambda )\Delta (\mu ,\mu )=G^\mathrm{up}\bigl ({\overline{\iota }_\mathfrak{g}}^{-1}(b_{w\lambda }\mathop{\otimes }t_{\lambda +\mu }\mathop{\otimes }b_{-\infty })\bigr ). \end{equation*}$$

Then, ${\overline{\iota }_\mathfrak{g}}\bigl (\overline{\iota }_{\lambda ,\mu }^{-1}(u_{w\lambda }\mathop{\otimes }u_\mu )\mathop{\otimes }u_{\lambda +\mu }{}^{\mspace {1mu}\mathrm{r}}\bigr )=b_{w\lambda }\mathop{\otimes }t_{\lambda +\mu }\mathop{\otimes }b_{-\infty }$ gives the desired result.

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9.2. $T$-system

In this subsection, we recall the $T$-system among the (unipotent) quantum minors for later use (see 25 for $T$-system).

Proposition 9.2.1 (11, Proposition 3.2).

Assume that the Kac–Moody algebra $\mathfrak{g}$ is of symmetric type. Assume that $u,v \in W$ and $i \in I$ satisfy $u < us_i$ and $v < vs_i$. Then

$$\begin{align*} & \Delta (us_i\varpi _i,vs_i\varpi _i)\Delta (u\varpi _i,v\varpi _i) = q^{-1}\Delta (u s_i \varpi _i,v\varpi _i)\Delta ( u\varpi _i, v s_i\varpi _i) +\Delta (u\lambda ,v\lambda ), \\ & \Delta (u\varpi _i,v\varpi _i)\Delta (us_i\varpi _i,vs_i\varpi _i) = q\Delta ( u\varpi _i\ ,vs_i\varpi _i)\Delta (u s_i \varpi _i,v\varpi _i) + \Delta (u\lambda ,v\lambda ), \end{align*}$$

and

$$\begin{align*} & q^{(vs_i\varpi _i,v\varpi _i-u\varpi _i)}\mathrm{D}(us_i\varpi _i,vs_i\varpi _i)\mathrm{D}(u\varpi _i,v\varpi _i) \\ & \quad = q^{-1+(v\varpi _i,vs_i\varpi _i-u\varpi _i)}\mathrm{D}(us_i\varpi _i,v\varpi _i)\mathrm{D}(u\varpi _i,vs_i\varpi _i) +\mathrm{D}(u\lambda ,v\lambda ) \\ & \quad = q^{-1+(vs_i\varpi _i,v\varpi _i-us_i\varpi _i)}\mathrm{D}(u\varpi _i,vs_i\varpi _i)\mathrm{D}(us_i\varpi _i,v\varpi _i) + \mathrm{D}(u\lambda ,v\lambda ), \\ & q^{(v\varpi _i,vs_i\varpi _i-us_i\varpi _i)}\mathrm{D}(u\varpi _i,v\varpi _i)\mathrm{D}(us_i\varpi _i,vs_i\varpi _i) \\ & \quad = q^{1+(vs_i\varpi _i,v\varpi _i-us_i\varpi _i)}\mathrm{D}(u\varpi _i,vs_i\varpi _i)\mathrm{D}(us_i\varpi _i,v\varpi _i) + \mathrm{D}(u\lambda ,v\lambda ) \\ & \quad = q^{1+(v\varpi _i,vs_i\varpi _i-u\varpi _i)}\mathrm{D}(us_i\varpi _i,v\varpi _i)\mathrm{D}(u\varpi _i,vs_i\varpi _i) + \mathrm{D}(u\lambda ,v\lambda ), \end{align*}$$

where $\lambda =s_i\varpi _i+\varpi _i$.

Note that the difference of $\lambda$ and $\displaystyle -\sum _{j \ne i}a_{j,i}\varpi _j$ are $W$-invariant. Hence we have $\displaystyle \mathrm{D}(u\lambda ,v\lambda ) = \prod _{j \ne i}\mathrm{D}(u\varpi _j,v\varpi _j)^{-a_{j,i}}$ from Corollary 9.1.3, by disregarding a power of $q$.

9.3. Revisit of crystal bases and global bases

In order to prove Theorem 9.3.3 below, we first investigate the upper crystal lattice of $\mathbf{D}_\varphi V$ induced by an upper crystal lattice of $V\in \mathcal{O}_{\mathrm{int}}(\mathfrak{g})$.

Let $V$ be a $U_q(\mathfrak{g})$-module in $\mathcal{O}_{\mathrm{int}}(\mathfrak{g})$. Let $L^\mathrm{up}$ be an upper crystal lattice of $V$. Then we have (see Lemma 1.3.1)

$$\begin{equation*} \mathop{\textstyle \bigoplus }\limits _{\xi \in \mathsf{P}} q^{(\xi ,\xi )/2}(L^\mathrm{up})_\xi \text{ is a lower crystal lattice of $V$.} \end{equation*}$$

Recall that, for $\lambda \in \mathsf{P}^+$, the upper crystal lattice $L^\mathrm{up}(\lambda )$ and the lower crystal lattice $L^{\mathrm{low}}(\lambda )$ of $V(\lambda )$ are related by

$$\begin{align} L^\mathrm{up}(\lambda ) = \mathop{\textstyle \bigoplus }\limits _{\xi \in \mathsf{P}} q^{((\lambda ,\lambda )-(\xi ,\xi ))/2}L^{\mathrm{low}}(\lambda )_\xi \subset L^{\mathrm{low}}(\lambda ). \tag{9.3} \end{align}$$

Write

$$\begin{equation*} V \simeq \mathop{\textstyle \bigoplus }\limits _{\lambda \in \mathsf{P}^+} E_\lambda \otimes V(\lambda ) \end{equation*}$$

with finite-dimensional $\mathbb{Q}(q)$-vector spaces $E_\lambda$. Accordingly, we have a canonical decomposition

$$\begin{equation*} L^\mathrm{up}\simeq \mathop{\textstyle \bigoplus }\limits _{\lambda \in \mathsf{P}^+} C_\lambda \otimes _{\mathbf{A}_0} L^\mathrm{up}(\lambda ), \end{equation*}$$

where $C_\lambda \subset E_\lambda$ is an $\mathbf{A}_0$-lattice of $E_\lambda$.

On the other hand, we have

$$\begin{equation*} \mathbf{D}_\varphi V \simeq \mathop{\textstyle \bigoplus }\limits _{\lambda \in \mathsf{P}^+} E^*_\lambda \otimes V(\lambda ). \end{equation*}$$

Note that we have

$$\begin{equation*} \Phi _V \left( (a \otimes u) \otimes (b \otimes v)^{\mspace {1mu}\mathrm{r}}\right) = \langle a,b \rangle \Phi _\lambda (u \otimes v^{\mspace {1mu}\mathrm{r}}) \quad \text{for $u,v \in V(\lambda )$ and $a \in E_\lambda $, $b \in E^*_\lambda $.} \end{equation*}$$

We define the induced upper crystal lattice $\mathbf{D}_\varphi L^\mathrm{up}$ of $\mathbf{D}_\varphi V$ by

$$\begin{equation*} \mathbf{D}_\varphi L^\mathrm{up}\mathbin{:=}\mathop{\textstyle \bigoplus }\limits _{\lambda \in \mathsf{P}^+} C_\lambda ^\vee \otimes _{\mathbf{A}_0} L^\mathrm{up}(\lambda ) \subset \mathbf{D}_\varphi V, \end{equation*}$$

where $C_\lambda ^\vee \mathbin{:=}\left\{ u \in E_\lambda ^* \mid \langle u,C_\lambda \rangle \subset \mathbf{A}_0 \right\}$. Then we have

$$\begin{equation*} \Phi _V \left(L^\mathrm{up}\otimes (\mathbf{D}_\varphi L^\mathrm{up})^{\mspace {1mu}\mathrm{r}}\right) \subset L^\mathrm{up}(A_q(\mathfrak{g})). \end{equation*}$$

Indeed, we have

$$\begin{equation*} \mathbf{D}_\varphi L^\mathrm{up}=\left\{ u \in \mathbf{D}_\varphi V \mid \Phi _V(L^\mathrm{up}\otimes u^{\mspace {1mu}\mathrm{r}}) \subset L^\mathrm{up}(A_q(\mathfrak{g})) \right\}. \end{equation*}$$

Since $(L^\mathrm{up}(\lambda ))^\vee = L^{\mathrm{low}}(\lambda )$, we have

$$\begin{equation*} (L^\mathrm{up})^\vee = \mathop{\textstyle \bigoplus }\limits _{\lambda \in \mathsf{P}^+} C_\lambda ^\vee \otimes _{\mathbf{A}_0} L^{\mathrm{low}}(\lambda ). \end{equation*}$$

The properties $L^\mathrm{up}(\lambda ) \subset L^{\mathrm{low}}(\lambda )$ and $L^\mathrm{up}(\lambda )_\lambda = L^{\mathrm{low}}(\lambda )_\lambda$ imply the following lemma.

Lemma 9.3.1.

$\mathbf{D}_\varphi L^\mathrm{up}$ is the largest upper crystal lattice of $\mathbf{D}_\varphi V$ contained in the lower crystal lattice $(L^\mathrm{up})^\vee$.

Let $\lambda , \mu \in \mathsf{P}^+$. Then $\bigl (L^\mathrm{up}(\lambda )\otimes _{_+}\mspace {-1mu}L^\mathrm{up}(\mu )\bigr )^\vee =L^{\mathrm{low}}(\lambda )\otimes _{_-}\mspace {-1mu}L^{\mathrm{low}}(\mu )$ is a lower crystal lattice of $\mathbf{D}_\varphi \bigl (V(\lambda )\otimes _{_+}\mspace {-1mu}V(\mu )\bigr )\simeq V(\lambda )\otimes _{_-}\mspace {-1mu}V(\mu )$. Let $\Xi _{\lambda ,\mu }\colon V(\lambda )\otimes _{_+}\mspace {-1mu}V(\mu )\similarrightarrow V(\lambda )\otimes _{_-}\mspace {-1mu}V(\mu ) \simeq \mathbf{D}_\varphi \bigl (V(\lambda )\otimes _{_+}\mspace {-1mu}V(\mu )\bigr )$ be the $U_q(\mathfrak{g})$-module isomorphism defined by

$$\begin{equation*} \Xi _{\lambda ,\mu }(u\otimes _{_+}\mspace {-1mu}v)=q^{(\lambda ,\mu )-(\xi ,\eta )}\bigl (u\otimes _{_-}\mspace {-1mu}v\bigr )\quad \text{for $u\in V(\lambda )_\xi $ and $v\in V(\mu )_\eta $.} \end{equation*}$$

Then

$$\begin{eqnarray*} \widetilde{L}&\mathbin{:=}&\Xi _{\lambda ,\mu }\bigl (L^\mathrm{up}(\lambda )\otimes _{_+}\mspace {-1mu}L^\mathrm{up}(\mu )\bigr )\\ &\;=&\mathop{\textstyle \bigoplus }\limits _{\xi ,\eta \in \mathsf{P}} q^{(\lambda ,\mu )-(\xi ,\eta )}L^\mathrm{up}(\lambda )_\xi \otimes _{_-}\mspace {-1mu}L^\mathrm{up}(\mu )_\eta \end{eqnarray*}$$

is an upper crystal lattice of $V(\lambda )\otimes _{_-}\mspace {-1mu}V(\mu )$. Since we have $(\lambda ,\mu )-(\xi ,\eta )\ge 0$ for any $\xi \in \operatorname {wt}\bigl (V(\lambda )\bigr )$ and $\eta \in \operatorname {wt}\bigl (V(\mu )\bigr )$, Lemma 9.3.1 implies that

$$\begin{eqnarray} \widetilde{L}\subset \mathbf{D}_\varphi \bigl (L^\mathrm{up}(\lambda )\otimes _{_+}\mspace {-1mu}L^\mathrm{up}(\mu )\bigr ). {\tag{9.4}} \end{eqnarray}$$

Lemma 9.3.2.

Let $\lambda , \mu \in \mathsf{P}^+$ and $x_1,x_2,y_1,y_2 \in W$ such that $x_k \ge y_k$ $(k=1,2)$. Then we have

$$\begin{equation} \begin{aligned} & q^{(\lambda ,\mu )-(x_2\lambda ,y_2\mu )} \Delta (x_1\lambda ,x_2\lambda )\Delta (y_1\mu ,y_2\mu ) \\ & \qquad \equiv G^\mathrm{up}( \overline{\iota }^{-1}_{\lambda ,\mu }(u_{x_1\lambda } \otimes u_{y_1 \mu }) \otimes \overline{\iota }^{-1}_{\lambda ,\mu }(u_{x_2\lambda } \otimes u_{y_2 \mu })^{\mspace {1mu}\mathrm{r}}) \quad {\mathrm{mod}} \ q L^\mathrm{up}(A_q(\mathfrak{g})). \end{aligned} {\tag{9.5}} \end{equation}$$

Proof.

By the definition, we have

$$\begin{equation*} \Delta (x_1\lambda ,x_2\lambda )\Delta (y_1\mu ,y_2\mu ) =\Phi _{V(\lambda )\otimes _{_+}\mspace {-1mu}V(\mu )}\bigl ((u_{x_1\lambda }\otimes _{_+}\mspace {-1mu}u_{y_1\mu })\mathop{\otimes }(u_{x_2\lambda }\otimes _{_-}\mspace {-1mu}u_{y_2\mu })^{\mspace {1mu}\mathrm{r}}\bigr ). \end{equation*}$$

Hence we have

$$\begin{eqnarray*} &&q^{(\lambda ,\mu )-(x_2\lambda ,y_2\mu )} \Delta (x_1\lambda ,x_2\lambda )\Delta (y_1\mu ,y_2\mu )\\ &&\hspace{3ex}=\Phi _{V(\lambda )\otimes _{_+}\mspace {-1mu}V(\mu )}\bigl ((u_{x_1\lambda }\otimes _{_+}\mspace {-1mu}u_{y_1\mu })\mathop{\otimes }q^{(\lambda ,\mu )-(x_2\lambda ,y_2\mu )}(u_{x_2\lambda }\otimes _{_-}\mspace {-1mu}u_{y_2\mu })^{\mspace {1mu}\mathrm{r}}\bigr )\\ &&\hspace{6ex}=\Phi _{V(\lambda )\otimes _{_+}\mspace {-1mu}V(\mu )}\bigl ((u_{x_1\lambda }\otimes _{_+}\mspace {-1mu}u_{y_1\mu })\mathop{\otimes }\bigl (\Xi _{\lambda ,\mu }(u_{x_2\lambda }\otimes _{_+}\mspace {-1mu}u_{y_2\mu })\bigr )^{\mspace {1mu}\mathrm{r}}\bigr ). \end{eqnarray*}$$

The right-hand side of 9.5 can be calculated as follows. Let us take $v_k\in L^\mathrm{up}(\lambda +\mu )$ such that $\iota _{\lambda ,\mu }(v_k)-u_{x_k\lambda }\otimes _{_+}\mspace {-1mu}u_{y_k\mu } \in qL^\mathrm{up}(\lambda )\otimes _{_+}\mspace {-1mu}L^\mathrm{up}(\mu )$ for $k=1,2$. Here $\iota _{\lambda ,\mu } \colon V(\lambda +\mu )\to V(\lambda )\otimes _{_+}\mspace {-1mu}V(\mu )$ denotes the canonical $U_q(\mathfrak{g})$-module homomorphism and such a $v_k$ exists by Lemma 9.1.8.

Then we have

$$\begin{eqnarray*} &&G^\mathrm{up}\Bigl (\overline{\iota }_{\lambda ,\mu }^{\ -1}(u_{x_1\lambda }\mathop{\otimes }u_{y_1\mu }) \mathop{\otimes }\bigl (\overline{\iota }_{\lambda ,\mu }^{\ -1}(u_{x_2\lambda }\mathop{\otimes }u_{y_2\mu })\bigr )^{\mspace {1mu}\mathrm{r}}\Bigl )\\ &&\hspace{5ex}\equiv \Phi _{\lambda +\mu }(v_1\mathop{\otimes }v_2^{\mspace {1mu}\mathrm{r}})\mod qL^\mathrm{up}(A_q(\mathfrak{g}))\\ &&\hspace{6ex}=\Phi _{V(\lambda )\otimes _{_+}\mspace {-1mu}V(\mu )}\bigl (\iota _{\lambda ,\mu }(v_1)\mathop{\otimes }\bigl (\Xi _{\lambda ,\mu }\iota _{\lambda ,\mu }(v_2)\bigr )^{\mspace {1mu}\mathrm{r}}\bigr ). \end{eqnarray*}$$

The last equality follows from $(v_2, u) = (\Xi _{\lambda ,\mu }\iota _{\lambda ,\mu }(v_2), \iota _{\lambda ,\mu } (u))$ for all $u \in V(\lambda +\mu )$.

On the other hand, we have

$$\begin{equation*} \iota _{\lambda ,\mu }(v_1)\equiv u_{x_1\lambda }\otimes _{_+}\mspace {-1mu}u_{y_1\mu }\mod qL^\mathrm{up}(\lambda )\otimes _{_+}\mspace {-1mu}L^\mathrm{up}(\mu ) \end{equation*}$$

and

$$\begin{equation*} \Xi _{\lambda ,\mu }\bigl (\iota _{\lambda ,\mu }(v_2)\bigr )\equiv \Xi _{\lambda ,\mu }(u_{x_2\lambda }\otimes _{_+}\mspace {-1mu}u_{y_2\mu })\mod q\widetilde{L}. \end{equation*}$$

Hence

$$\begin{eqnarray*} &&\Phi _{V(\lambda )\otimes _{_+}\mspace {-1mu}V(\mu )}\bigl ((u_{x_1\lambda }\otimes _{_+}\mspace {-1mu}u_{y_1\mu })\mathop{\otimes }\Xi _{\lambda ,\mu }(u_{x_2\lambda }\otimes _{_+}\mspace {-1mu}u_{y_2\mu })^{\mspace {1mu}\mathrm{r}}\bigr )\\ &&\hspace{10ex}\equiv \Phi _{V(\lambda )\otimes _{_+}\mspace {-1mu}V(\mu )}\bigl ((\iota _{\lambda ,\mu }(v_1)\mathop{\otimes }(\Xi _{\lambda ,\mu }\iota _{\lambda ,\mu }(v_2))^{\mspace {1mu}\mathrm{r}}\bigr )\mod qL^\mathrm{up}(A_q(\mathfrak{g})) \end{eqnarray*}$$

by 9.4, as desired.

■

Theorem 9.3.3.

Let $\lambda \in \mathsf{P}^+$ and $x,y \in W$ such that $x \ge y$. Then we have

$$\begin{equation*} \mathrm{D}(x\lambda ,y\lambda )\mathrm{D}(y\lambda ,\lambda ) \equiv \mathrm{D}(x\lambda ,\lambda ) \quad {\mathrm{mod}} \ qL^\mathrm{up}(A_q(\mathfrak{n})). \end{equation*}$$

Proof.

Applying $p_\mathfrak{n}$ to 9.5, we have

$$\begin{equation*} \begin{aligned} & \mathrm{D}(x\lambda ,y\lambda )\mathrm{D}(y\lambda ,\lambda ) \\ & \qquad \equiv p_\mathfrak{n}\left( G^\mathrm{up}( \overline{\iota }^{-1}_{\lambda ,\lambda }(u_{x\lambda } \otimes u_{y \lambda }) \otimes \overline{\iota }^{-1}_{\lambda ,\lambda }(u_{y\lambda } \otimes u_{\lambda })^{\mspace {1mu}\mathrm{r}}) \right) \quad {\mathrm{mod}} \ q L^\mathrm{up}(A_q(\mathfrak{n})). \end{aligned} \end{equation*}$$

Hence the desired result follows from Proposition 8.5.2, Proposition 8.5.4, and Lemma 9.3.4 below.

■

Lemma 9.3.4.

Let $\lambda \in \mathsf{P}^+$ and $x,y\in W$ such that $x\ge y$. Then we have

$$\begin{equation*} {\overline{\iota }_\mathfrak{g}}\Bigl ( \overline{\iota }_{\lambda ,\lambda }^{\ -1}(u_{x\lambda }\mathop{\otimes }u_{y\lambda }) \mathop{\otimes }\bigl (\overline{\iota }_{\lambda ,\lambda }^{\ -1}(u_{y\lambda }\mathop{\otimes }u_{\lambda })\bigr )^{\mspace {1mu}\mathrm{r}}\Bigr ) =\overline{\iota }_\lambda (u_{x\lambda })\mathop{\otimes }t_{y\lambda +\lambda }\mathop{\otimes }b_{-\infty }. \end{equation*}$$

Proof.

We shall argue by induction on $\ell (x)$. We set $b_{x\lambda }=\overline{\iota }_\lambda (u_{x\lambda })$. Since the case $x=1$ is obvious, assume that $x\not =1$. Take $i\in I$ such that $x'\mathbin{:=}s_ix<x$.

(a)

First assume that $s_iy>y$. Then we have $y\le x'$. Hence by the induction hypothesis,$$\begin{eqnarray} &&\overline{\iota }_\mathfrak{g}\Bigl ( \overline{\iota }_{\lambda ,\lambda }^{\ -1}(u_{x'\lambda }\mathop{\otimes }u_{y\lambda }) \mathop{\otimes }\bigl (\overline{\iota }_{\lambda ,\lambda }^{\ -1}(u_{y\lambda }\mathop{\otimes }u_{\lambda })\bigr )^{\mspace {1mu}\mathrm{r}}\Bigr ) =b_{x'\lambda }\mathop{\otimes }t_{y\lambda +\lambda }\mathop{\otimes }b_{-\infty }. {\tag{9.6}} \end{eqnarray}$$

We have $\varphi _i(u_{x'\lambda })=\langle h_i,x'\lambda \rangle$ and $\varphi _i(b_{x'\lambda }\mathop{\otimes }t_{y\lambda +\lambda }\mathop{\otimes }b_{-\infty }) =\varphi _i(b_{x'\lambda }\mathop{\otimes }t_{y\lambda +\lambda })=\langle h_i,x'\lambda \rangle +\langle h_i,y\lambda \rangle \ge \langle h_i,x'\lambda \rangle$. Hence, applying $\tilde{f}_i^{\langle h_i,x'\lambda \rangle }$ to 9.6, we obtain$$\begin{eqnarray*} &&\overline{\iota }_\mathfrak{g}\Bigl ( \overline{\iota }_{\lambda ,\lambda }^{\ -1}(u_{x\lambda }\mathop{\otimes }u_{y\lambda }) \mathop{\otimes }\bigl (\overline{\iota }_{\lambda ,\lambda }^{\ -1}(u_{y\lambda }\mathop{\otimes }u_{\lambda })\bigr )^{\mspace {1mu}\mathrm{r}}\Bigr ) =b_{x\lambda }\mathop{\otimes }t_{y\lambda +\lambda }\mathop{\otimes }b_{-\infty }. \end{eqnarray*}$$

(b)

Assume that $y'\mathbin{:=}s_iy<y$. Then we have $y'\le x'$, and the induction hypothesis implies that$$\begin{equation*} \overline{\iota }_\mathfrak{g}\Bigl ( \overline{\iota }_{\lambda ,\lambda }^{\ -1}(u_{x'\lambda }\mathop{\otimes }u_{y'\lambda }) \mathop{\otimes }\bigl (\overline{\iota }_{\lambda ,\lambda }^{\ -1}(u_{y'\lambda }\mathop{\otimes }u_{\lambda })\bigr )^{\mspace {1mu}\mathrm{r}}\Bigr ) =b_{x'\lambda }\mathop{\otimes }t_{y'\lambda +\lambda }\mathop{\otimes }b_{-\infty }. \end{equation*}$$

Apply $\tilde{e}_i^{*\;\langle h_iy'\lambda \rangle }\tilde{f}_i^{\langle h_i,x'\lambda +y'\lambda \rangle }$ to both sides. Then the left-hand side yields$$\begin{equation*} \overline{\iota }_\mathfrak{g}\Bigl ( \overline{\iota }_{\lambda ,\lambda }^{\ -1}(u_{x\lambda }\mathop{\otimes }u_{y\lambda }) \mathop{\otimes }\bigl (\overline{\iota }_{\lambda ,\lambda }^{\ -1}(u_{y\lambda }\mathop{\otimes }u_{\lambda })\bigr )^{\mspace {1mu}\mathrm{r}}\Bigr ). \end{equation*}$$

Since $\varphi _i(b_{x'\lambda }\mathop{\otimes }t_{y'\lambda +\lambda })=\langle h_i,x'\lambda \rangle +\langle h_i,y'\lambda +\lambda \rangle \ge \langle h_i,x'\lambda +y'\lambda \rangle$, the right-hand side yields$$\begin{align*} &\tilde{e}_i^{*\;\langle h_i,y'\lambda \rangle }\tilde{f}_i^{\langle h_i,x'\lambda +y'\lambda \rangle } \bigl (b_{x'\lambda }\mathop{\otimes }t_{y'\lambda +\lambda }\mathop{\otimes }b_{-\infty }\bigr )\\ &\hspace{5ex}=\tilde{e}_i^{*\;\langle h_i,y'\lambda \rangle }\bigl ((\tilde{f}_i^{\langle h_i,x'\lambda +y'\lambda \rangle } b_{x'\lambda })\mathop{\otimes }t_{y'\lambda +\lambda }\mathop{\otimes }b_{-\infty }\bigr )\\ &\hspace{5ex}=\tilde{e}_i^{*\;\langle h_i,y'\lambda \rangle }\bigl ((\tilde{f}_i^{\langle h_i,y'\lambda \rangle } b_{x\lambda })\mathop{\otimes }t_{y'\lambda +\lambda }\mathop{\otimes }b_{-\infty }\bigr ). \end{align*}$$

Since $\varepsilon _i^*(b_{x\lambda })=-\varphi _i(b_{x\lambda }) = \langle h_i, \lambda \rangle$ and $\tilde{f}_i^{\langle h_i,y'\lambda \rangle }b_{x\lambda }=\tilde{f}_i^{*\;\langle h_i,y'\lambda \rangle }b_{x\lambda }$, we have$$\begin{align*} \tilde{e}_i^{*\;\langle h_i,y'\lambda \rangle }\bigl ((\tilde{f}_i^{\langle h_i,y'\lambda \rangle } b_{x\lambda })\mathop{\otimes }t_{y'\lambda +\lambda }\mathop{\otimes }b_{-\infty }\bigr )&=b_{x\lambda }\mathop{\otimes }t_{y\lambda +\lambda }\mathop{\otimes }b_{-\infty }. \end{align*}$$■

9.4. Generalized $T$-system

The $T$-system in Section 9.2 can be interpreted as a system of equations among the three products of elements in ${\mathbf{B}}^\mathrm{up}(A_q(\mathfrak{g}))$ or ${\mathbf{B}}^\mathrm{up}(A_q(\mathfrak{n}))$. In this subsection, we introduce another among the three products of elements in ${\mathbf{B}}^\mathrm{up}(A_q(\mathfrak{g}))$, called a generalized $T$-system.

Proposition 9.4.1.

Let $\mu \in W \varpi _i$, and set $b = \overline{\iota }_{\varpi _i}(u_\mu ) \in B(\infty )$. Then we have

$$\begin{equation} \begin{aligned} \Delta (\mu ,s_i\varpi _i)\Delta (\varpi _i,\varpi _i) & = q_i^{-1} G^\mathrm{up}\left( \overline{\iota }^{-1}_{\varpi _i,\varpi _i}(u_\mu \otimes u_{\varpi _i}) \otimes \big ( \overline{\iota }^{-1}_{\varpi _i,\varpi _i}(u_{s_i\varpi _i} \otimes u_{\varpi _i}) \big )^{\mspace {1mu}\mathrm{r}}\right) \\ & \qquad +G^\mathrm{up}\left( \overline{\iota }_{\varpi _i+s_i\varpi _i}^{-1}(\widetilde{e^*_i}b) \otimes u^{\mspace {1mu}\mathrm{r}}_{\varpi _i+s_i\varpi _i} \right). \end{aligned} {\tag{9.7}} \end{equation}$$

Note that if $\mu = \varpi _i$, then $b=1$ and the last term in 9.7 vanishes. If $\mu \ne \varpi _i$, then $\varepsilon _i^*(b)=1$ and $\overline{\iota }_{\varpi _i+s_i\varpi _i}^{-1}(\tilde{e}_i^* b) \in B(\varpi _i+s_i\varpi _i)$, $u_\mu \otimes u_{\varpi _i} \in \overline{\iota }_{\varpi _i,\varpi _i}B(2\varpi _i)$.

Proof.

In the sequel, we omit $\overline{\iota }^{-1}_{\varpi _i,\varpi _i}$ for the sake of simplicity. Set

$$\begin{equation*} u = \Delta (\mu ,s_i\varpi _i)\Delta (\varpi _i,\varpi _i)- q_i^{-1} G^\mathrm{up}\left( (u_\mu \otimes u_{\varpi _i}) \otimes (u_{s_i\varpi _i} \otimes u_{\varpi _i})^{\mspace {1mu}\mathrm{r}}\right). \end{equation*}$$

Then $\operatorname {wt}_{\mspace {1mu}\mathrm{r}}(u)=\lambda \mathbin{:=}\varpi _i+s_i\varpi _i$.

It is obvious that we have $uf_j=0$ for $j \ne i$. Since $\tilde{e}_i(u_{s_i\varpi _i} \otimes u_{\varpi _i})= u_{\varpi _i} \otimes u_{\varpi _i}$, we have

$$\begin{align*} G^\mathrm{up}\left( (u_\mu \otimes u_{\varpi _i}) \otimes (u_{s_i\varpi _i} \otimes u_{\varpi _i})^{\mspace {1mu}\mathrm{r}}\right)f_i & = G^\mathrm{up}\left( (u_\mu \otimes u_{\varpi _i}) \otimes (u_{\varpi _i} \otimes u_{\varpi _i})^{\mspace {1mu}\mathrm{r}}\right) \\ & = \Delta (\mu ,\varpi _i)\Delta (\varpi _i,\varpi _i) \\ & = G^\mathrm{up}(u_\mu \otimes u^{\mspace {1mu}\mathrm{r}}_{\varpi _i})G^\mathrm{up}(u_{\varpi _i} \otimes u^{\mspace {1mu}\mathrm{r}}_{\varpi _i}). \end{align*}$$

Here the second equality follows from Lemma 9.1.9 and the third follows from Proposition 8.1.3. On the other hand, we have

$$\begin{align*} \left( \Delta (\mu ,s_i\varpi _i)\Delta (\varpi _i,\varpi _i) \right) f_i & = \left( \Delta (\mu ,s_i\varpi _i)f_i \right) \left(\Delta (\varpi _i,\varpi _i)t_i^{-1}\right) \\ & = q_i^{-1}\Delta (\mu ,\varpi _i)\Delta (\varpi _i,\varpi _i). \end{align*}$$

Hence we have $uf_i=0$. Thus, $u$ is a lowest weight vector of weight $\lambda$ with respect to the right action of $U_q(\mathfrak{g})$. Therefore there exists some $v \in V(\lambda )$ such that

$$\begin{equation*} u=\Phi (v \otimes u_\lambda ^{\mspace {1mu}\mathrm{r}}).{} \end{equation*}$$

Hence we have $p_\mathfrak{n}(u)=\iota _\lambda (v) \in A_q(\mathfrak{n})$. On the other hand, we have

$$\begin{align*} p_\mathfrak{n}\left( \Delta (\mu ,s_i\varpi _i)\Delta (\varpi _i,\varpi _i) \right) & = p_\mathfrak{n}\left(\Delta (\mu ,s_i\varpi _i) \right) p_\mathfrak{n}\left(\Delta (\varpi _i,\varpi _i)\right) \\ & = \mathrm{D}(\mu ,s_i\varpi _i)=G^{\mathrm{up}}(\tilde{e}_i^*b) \\ & = \iota _\lambda \left(G_\lambda ^\mathrm{up}\big (\overline{\iota }_\lambda ^{-1}(\tilde{e}_i^* b) \big ) \right). \end{align*}$$

Note that since $\varepsilon _i^*(\tilde{e}_i^*b) =0$ and $\varepsilon _j^*(\tilde{e}_i^*b) \le -\langle h_j,\alpha _i \rangle$ for $j \ne i$, we have $\tilde{e}_i^*b \in \overline{\iota }_\lambda (B(\lambda ))$.

Hence in order to prove our assertion, it is enough to show that

$$\begin{equation*} p_\mathfrak{n}\left( G^\mathrm{up}\big ( (u_\mu \otimes u_{\varpi _i}) \otimes (u_{s_i\varpi _i} \otimes u_{\varpi _i})^{\mspace {1mu}\mathrm{r}}\big ) \right) =0. \end{equation*}$$

This follows from Proposition 8.5.2 and

$$\begin{align} \overline{\iota }_\mathfrak{g}\left( (u_\mu \otimes u_{\varpi _i}) \otimes (u_{s_i\varpi _i} \otimes u_{\varpi _i})^{\mspace {1mu}\mathrm{r}}\right) = b \otimes t_\lambda \otimes \tilde{e}_ib_{-\infty }. {\tag{9.8}} \end{align}$$

Let us prove 9.8. Since

$$\begin{equation*} (u_\mu \otimes u_{\varpi _i}) \otimes (u_{s_i\varpi _i} \otimes u_{\varpi _i})^{\mspace {1mu}\mathrm{r}}= \tilde{e}_i^* \bigl ((u_\mu \otimes u_{\varpi _i}) \otimes (u_{\varpi _i} \otimes u_{\varpi _i})^{\mspace {1mu}\mathrm{r}}\bigr ), \end{equation*}$$

the left-hand side of 9.8 is equal to

$$\begin{equation*} \tilde{e}_i^*\left(\overline{\iota }_\mathfrak{g}\big ( (u_\mu \otimes u_{\varpi _i}) \otimes (u_{\varpi _i} \otimes u_{\varpi _i})^{\mspace {1mu}\mathrm{r}}\big )\right) = \tilde{e}_i^*(b \otimes t_{2\varpi _i}\otimes b_{-\infty }). \end{equation*}$$

Since $\varepsilon _i^*(b)=1 < \langle h_i,2\varpi _i \rangle =2$, we obtain

$$\begin{align*} &\tilde{e}_i^*\left(b \otimes t_{2\varpi _i}\otimes b_{-\infty }\right) = b \otimes t_{2\varpi _i-\alpha _i}\otimes \tilde{e}_i^* b_{-\infty } =b \otimes t_\lambda \otimes \tilde{e}_ib_{-\infty }. \end{align*}$$■

10. KLR algebras and their modules

10.1. Chevalley and Kashiwara operators

Let us recall the definition of several functors on modules over KLR algebras which are used to categorify $U^-_q(\mathfrak{g})_{\mathbb{Z}[q^\pm 1]}^\vee$.

Definition 10.1.1.

Let $\beta \in \mathsf{Q}^+$.

(i)

For $i \in I$ and $1 \le a \le |\beta |$, set$$\begin{equation*} e_a(i)=\sum _{\nu \in I^\beta ,\nu _a=i}e(\nu )\in R(\beta ). \end{equation*}$$

(ii)

We take conventions$$\begin{align*} &E_iM=e_1(i)M, \\ &E^*_iM=e_{|\beta |}(i)M, \end{align*}$$

which are functors from $R(\beta ) \text{-gmod}$ to $R(\beta -\alpha _i) \text{-gmod}$.

(iii)

For a simple module $M$, we set$$\begin{align*} &\varepsilon _i(M)=\max \left\{n\in \mathbb{Z}_{\ge 0} \mid E_i^nM\not =0 \right\},\\ &\varepsilon ^*_i(M)=\max \left\{n\in \mathbb{Z}_{\ge 0} \mid E_i^{*\;n}M\not =0 \right\},\\ & \widetilde{F}_iM = q_i^{\varepsilon _i(M)} L(i) \mathbin{\text{$\nabla $}}M, \\ & \widetilde{F}^*_iM = q_i^{\varepsilon ^*_i(M)} M \mathbin{\text{$\nabla $}}L(i), \\ & \widetilde{E}_iM = q_i^{1-\varepsilon _i(M)} {\operatorname {soc}}(E_iM) \simeq q_i^{\varepsilon _i(M)-1} {\operatorname {hd}}(E_iM), \\ & \tilde{E}^*_iM = q_i^{1-\varepsilon ^*_i(M)} {\operatorname {soc}}(E^*_iM) \simeq q_i^{\varepsilon ^*_i(M)-1} {\operatorname {hd}}(E_i^*M),\\ &\tilde{E}_i^{\,\max } M=\tilde{E}_i^{\varepsilon _i(M)}M\quad \text{and}\quad \tilde{E}_i^{*\;\max } M=\tilde{E}_i^{*\;\varepsilon ^*_i(M)}M. \end{align*}$$

(iv)

For $i \in I$ and $n \in \mathbb{Z}_{\ge 0}$, we set$$\begin{equation*} L(i^n)=q_i^{n(n-1)/2} \underbrace{L(i)\circ \cdots \circ L(i)}_n. \end{equation*}$$

Here $L(i)$ denotes the $R(\alpha _i)$-module $R(\alpha _i)/R(\alpha _i)x_1$. Then $L(i^n)$ is a self-dual real simple $R(n\alpha _i)$-module.

Note that, under the isomorphism in Theorem 2.1.2, the functors $E_i$ and $E_i^*$ correspond to the linear operators $e_i$ and $e_i^*$ on $A_q(\mathfrak{n})_{\mathbb{Z}[q^{\pm 1}]} = \iota (U^-_q(\mathfrak{g})_{\mathbb{Z} [q^{\pm 1}]}^\vee ) \subset A_q(\mathfrak{n})$, respectively. Note also that, for a simple $R(\beta )$-module $S$, we have $\tilde{E}_i \widetilde{F}_i S \simeq S$, and $\widetilde{F}_i \tilde{E}_i S \simeq S$ if $\varepsilon _i(M) >0$.

In the course of proving the following propositions, we use the following notations:

$$\begin{align} \overline{Q}_{i,j}(x_a,x_{a+1},x_{a+2}) \mathbin{:=}\dfrac{Q_{i,j}(x_a,x_{a+1})-Q_{i,j}(x_{a+2},x_{a+1})}{x_a-x_{a+2}}. {\tag{10.1}} \end{align}$$ Then we have

$$\begin{equation*} \tau _{a+1}\tau _a\tau _{a+1}-\tau _{a}\tau _{a+1}\tau _{a} =\sum _{i,j\in I}\overline{Q}_{i,j}(x_a,x_{a+1},x_{a+2}) e_a(i)e_{a+1}(j)e_{a+2}(i). \end{equation*}$$

Proposition 10.1.2.

Let $\beta \in \mathsf{Q}^+$ with $n=|\beta |$. Assume that an $R(\beta )$-module $M$ satisfies $E_iM=0$. Then the left $R(\alpha _i)$-module homomorphism $R(\alpha _i) \mathop{\otimes }M \longrightarrow q^{(\alpha _i,\beta )}M \circ R(\alpha _i)$ given by

$$\begin{align} e(i) \mathop{\otimes }u \longmapsto \tau _1 \cdots \tau _n(u \mathop{\otimes }e(i)) {\tag{10.2}} \end{align}$$

extends uniquely to an $(R(\alpha _i+\beta ),R(\alpha _i))$-bilinear homomorphism

$$\begin{align} R(\alpha _i) \circ M \longrightarrow q^{(\alpha _i,\beta )}M \circ R(\alpha _i). {\tag{10.3}} \end{align}$$

Proof.

(i)

First note that, for $1 \le a \le n$,$$\begin{equation} \tau _1 \cdots \tau _{a-1}e_a(i)\tau _{a+1}\cdots \tau _n\big ( u \mathop{\otimes }e(i) \big ) = \tau _{a+1}\cdots \tau _n\big (e_1(i)\tau _1 \cdots \tau _{a-1}(u \mathop{\otimes }e(i)) \big )=0 {\tag{10.4}} \end{equation}$$

since $E_iM =0$.

(ii)

In order to see that 10.3 is a well-defined $R(\alpha _i+\beta )$-linear homomorphism, it is enough to show that 10.2 is $R(\beta )$-linear.

(a)

Commutation with $x_a\in R(\beta )$ $(1\le a \le n)$: We have$$\begin{align*} x_{a+1}\tau _1 \cdots \tau _n\big (u \mathop{\otimes }e(i)\big ) &= \tau _1 \cdots \tau _{a-1}x_{a+1}\tau _{a} \cdots \tau _n\big (u \mathop{\otimes }e(i)\big ) \\ &= \tau _1 \cdots \tau _{a-1}\big ( \tau _{a} x_{a}+ e_a(i) \big ) \tau _{a+1}\cdots \tau _n\big (u \mathop{\otimes }e(i) \big ) \\ &= \tau _1 \cdots \tau _n x_{a}\big (u \mathop{\otimes }e(i)\big ) \end{align*}$$

by 10.4.

(b)

Commutation with $\tau _a\in R(\beta )$ $(1\le a < n)$: We have$$\begin{align*} &\tau _{a+1}\tau _1 \cdots \tau _n\big (u \mathop{\otimes }e(i)\big ) \\ & \quad = \tau _1 \cdots \tau _{a-1}(\tau _{a+1}\tau _{a}\tau _{a+1})\tau _{a+2} \cdots \tau _n\big (u \mathop{\otimes }e(i)\big ) \\ & \quad = \tau _1 \cdots \tau _{a-1}\big (\tau _{a}\tau _{a+1}\tau _{a}{+}\sum _{j}\overline{Q}_{i,j}(x_{a},x_{a+1},x_{a+2})e_a(i)e_{a+1}(j)\!\big )\tau _{a+2}\cdots \tau _n\big (u \mathop{\otimes }e(i)\big ) \\ & \quad = \tau _1 \cdots \tau _n \tau _{a}\big (u \mathop{\otimes }e(i)\big ) \\ & \qquad + \sum _{j}\tau _1 \cdots \tau _{a-1}\overline{Q}_{i,j}(x_{a},x_{a+1},x_{a+2})e_a(i)e_{a+1}(j)\tau _{a+2}\cdots \tau _n\big (u \mathop{\otimes }e(i) \big ). \end{align*}$$

The last term vanishes because $E_iM=0$ implies$$\begin{align*} & \tau _1 \cdots \tau _{a-1}f(x_{a},x_{a+1})g(x_{a+2})e_a(i)\tau _{a+2}\cdots \tau _n\big (u \mathop{\otimes }e(i) \big ) \\ & \qquad \quad = g(x_{a+2})\tau _{a+2}\cdots \tau _n e_1(i)\tau _1 \cdots \tau _{a-1}f(x_{a},x_{a+1})\big (u \mathop{\otimes }e(i) \big )=0 \end{align*}$$

for any polynomial $f(x_{a},x_{a+1})$ and $g(x_{a+2})$.

(iii)

Now let us show that 10.3 is right $R(\alpha _i)$-linear. By 10.4, we have$$\begin{align*} \tau _1 \cdots \tau _{a-1} x_a \tau _a \cdots \tau _{n}\big (u \mathop{\otimes }e(i) \big ) & = \tau _1 \cdots \tau _{a-1} \big (\tau _a x_{a+1}-e_{a}(i) \big )\tau _{a+1}\cdots \tau _{n} \big (u \mathop{\otimes }e(i) \big ) \\ & = \tau _1 \cdots \tau _{a} x_{a+1} \tau _{a+1} \cdots \tau _{n}\big (u \mathop{\otimes }e(i) \big ) \end{align*}$$

for $1 \le a \le n$. Therefore we have$$\begin{equation*} x_{1}\tau _1 \cdots \tau _n\big (u \mathop{\otimes }e(i)\big ) = \tau _1 \cdots \tau _n x_{n+1}\big (u \mathop{\otimes }e(i)\big )=\tau _1 \cdots \tau _n \big (u \mathop{\otimes }e(i)x_{1}\big ). \end{equation*}$$■

Recall that for $m,n\in \mathbb{Z}_{\ge 0}$, we denote by $w[{m,n}]$ the element of $\mathfrak{S}_{m+n}$ defined by

$$\begin{eqnarray} &&w[{m,n}](k)=\begin{cases} k+n&\text{if $1\le k\le m$,}\\ k-m&\text{if $m<k\le m+n$}.\end{cases} \tag{10.5} \end{eqnarray}$$

Set $\tau _{w[{m,n}]}:=\tau _{i_1} \cdots \tau _{i_r}$, where $s_{i_1} \cdots s_{i_r}$ is a reduced expression of $w[{m,n}]$. Note that $\tau _{w[{m,n}]}$ does not depend on the choice of reduced expression 14, Corollary 1.4.3.

Proposition 10.1.3.

Let $M \in R(\beta ) \text{-gmod}$ and $N \in R(\gamma ) \text{-gmod}$, and set $m=|\beta |$ and $n=|\gamma |$. If $E_iM=0$ for any $i \in \operatorname {supp}(\gamma )$, then

$$\begin{align*} v \mathop{\otimes }u \longmapsto \tau _{w[m,n]}(u \mathop{\otimes }v) \end{align*}$$

gives a well-defined $R(\beta +\gamma )$-linear homomorphism $N \circ M \longrightarrow q^{(\beta ,\gamma )} M \circ N$.

Proof.

The proceeding proposition implies that

$$\begin{equation*} v \mathop{\otimes }u \longmapsto \tau _{w[m,n]}(u \mathop{\otimes }v) \quad \text{ for } u \in M, v \in R(\gamma ) \end{equation*}$$

gives a well-defined $R(\beta +\gamma )$-linear homomorphism $R(\gamma ) \circ M \to M \circ R(\gamma )$. Hence it is enough to show that it is right $R(\gamma )$-linear. Since we know that it commutes with the right multiplication of $x_k$, it is enough to show that it commutes with the right multiplication of $\tau _k$. For this, we may assume that $n = 2$ and $k = 1$. Set $\gamma = \alpha _i +\alpha _j$.

Thus we have reduced the problem to the equality

$$\begin{align*} \tau _1(\tau _2\tau _1)\cdots (\tau _{m+1}\tau _m)\big ( u \mathop{\otimes }e(i) \mathop{\otimes }e(j) \big ) = (\tau _2\tau _1)\cdots (\tau _{m+1}\tau _m) \tau _{m+1}\big ( u \mathop{\otimes }e(i) \mathop{\otimes }e(j) \big ) \end{align*}$$

for $u \in M$, which is a consequence of

$$\begin{equation*} \begin{aligned} & (\tau _2\tau _1)\cdots (\tau _{a}\tau _{a-1})\tau _a(\tau _{a+1}\tau _{a})\cdots (\tau _{m+1}\tau _m)\big ( u \mathop{\otimes }e(i) \mathop{\otimes }e(j) \big ) \\ & \hspace{5ex} = (\tau _2\tau _1)\cdots (\tau _{a+1}\tau _{a}) \tau _{a+1}(\tau _{a+2}\tau _{a+1})\cdots (\tau _{m+1}\tau _m)\big ( u \mathop{\otimes }e(i) \mathop{\otimes }e(j) \big ) \end{aligned} \end{equation*}$$

for $1 \le a \le m$. Note that

$$\begin{align*} & \tau _a(\tau _{a+1}\tau _{a})\cdots (\tau _{m+1}\tau _m)\big ( u \mathop{\otimes }e(i) \mathop{\otimes }e(j) \big ) \\ & \qquad = \tau _a(\tau _{a+1}\tau _{a})e_{a+1}(i)e_{a+2}(j)(\tau _{a+2}\tau _{a+1})\cdots (\tau _{m+1}\tau _m)\big ( u \mathop{\otimes }e(i) \mathop{\otimes }e(j) \big ) \end{align*}$$

and

$$\begin{align*} & \tau _a(\tau _{a+1}\tau _{a})e_{a+1}(i)e_{a+2}(j) \\ & \qquad = (\tau _{a+1}\tau _{a})\tau _{a+1} e_{a+1}(i)e_{a+2}(j)-\overline{Q}_{ji}(x_{a},x_{a+1},x_{a+2})e_{a}(j)e_{a+1}(i)e_{a+2}(j). \end{align*}$$

Hence it is enough to show

$$\begin{align*} & (\tau _2\tau _1)\cdots (\tau _{a}\tau _{a-1})\overline{Q}_{j,i}(x_{a},x_{a+1},x_{a+2}) e_a(j) \\ & \qquad \qquad (\tau _{a+2}\tau _{a+1})\cdots (\tau _{m+1}\tau _{m})\big ( u \mathop{\otimes }e(i) \mathop{\otimes }e(j) \big )=0. \end{align*}$$

This follows from

$$\begin{align*} &(\tau _2\tau _1)\cdots (\tau _{a}\tau _{a-1})f(x_a)g(x_{a+1},x_{a+2}) e_a(j) (\tau _{a+2}\tau _{a+1})\cdots (\tau _{m+1}\tau _m) \big ( u\mathop{\otimes }e(i)\mathop{\otimes }e(j)\big )\\ & \qquad =(\tau _2\cdots \tau _a) (\tau _1\cdots \tau _{a-1})f(x_a)g(x_{a+1},x_{a+2}) e_a(j) \\ &\qquad \qquad \qquad (\tau _{a+2}\tau _{a+1})\cdots (\tau _{m+1}\tau _m) \big ( u\mathop{\otimes }e(i)\mathop{\otimes }e(j)\big )\\ &\qquad =(\tau _2\cdots \tau _a) g(x_{a+1},x_{a+2})(\tau _{a+2}\tau _{a+1})\cdots (\tau _{m+1}\tau _m)\\ &\qquad \qquad \qquad e_1(j)(\tau _1\cdots \tau _{a-1})f(x_a)\big ( u\mathop{\otimes }e(i)\mathop{\otimes }e(j)\big )\\ &\qquad =0 \end{align*}$$

for $1 \le a \le m$ and $f(x_a) \in {{\mathbf{k}}}[x_{a}]$, $g(x_{a+1},x_{a+2}) \in {{\mathbf{k}}}[x_{a+1},x_{a+2}]$.

■

Let $P(i^n)$ be a projective cover of $L(i^n)$. Define the functor

$$\begin{equation*} E_i^{(n)} : R(\beta ) \text{-Mod}\to R(\beta - n \alpha _i) \text{-Mod} \end{equation*}$$

by

$$\begin{equation*} E_i^{(n)}(M) := P(i^n)^\psi \mathop{\otimes }_{R(n\alpha _i)}E_i^nM, \end{equation*}$$

where $P(i^n)^\psi$ denotes the right $R(n\alpha _i)$-module obtained from the left $R(\beta )$-module $P(i^n)$ via the anti-automorphism $\psi$. We define the functor $E_i^{*\,(n)}$ in a similar way. Note that

$$\begin{equation*} E_i^n\simeq [n]_i ! E_i^{(n)}. \end{equation*}$$

Corollary 10.1.4.

Let $R$ be a symmetric KLR algebra. Let $i\in I$ and $M$ a simple module. Then we have

$$\begin{eqnarray*} &&\widetilde{\Lambda }\bigl (L(i),M\bigr )=\varepsilon _i(M),\\ &&\Lambda \bigl (L(i),M\bigr )=2\varepsilon _i(M)+\langle h_i,\operatorname {wt}(M)\rangle =\varepsilon _i(M)+\varphi _i(M).\\ \end{eqnarray*}$$

Proof.

Set $n=\varepsilon _i(M)$ and $M_0=E_i^{(n)}(M)$. Then the preceding proposition implies $\Lambda (L(i),M_0)=\bigl (\alpha _i,\operatorname {wt}(M_0)\bigr )$. Hence we have $\widetilde{\Lambda }(L(i),M_0)=0$, which implies

$$\begin{equation*} \widetilde{\Lambda }(L(i),M)=\widetilde{\Lambda }(L(i), L(i^n)\circ M_0) =\widetilde{\Lambda }\bigl (L(i), L(i^n)\bigr )+\widetilde{\Lambda }(L(i),M_0)=n. \end{equation*}$$■

Proposition 10.1.5.

Let $M$, $N$ be modules and $m,n \in \mathbb{Z}_{\ge 0}$.

(i)

If $E_i^{m+1}M=0$ and $E_i^{n+1}N=0$, then we have$$\begin{align*} E_i^{(m+n)} (M \circ N) \simeq q^{mn+n \langle h_i, \operatorname {wt}(M) \rangle } E_i^{(m)} M \circ E_i^{(n)}N. \end{align*}$$

(ii)

If $E_i^{*\,m+1}M=0$ and $E_i^{*\,n+1}N=0$, then we have$$\begin{align*} E_i^{*\,(m+n)} (M \circ N) \simeq q^{mn+m \langle h_i, \operatorname {wt}(N) \rangle } E_i^{*\,(m)} M \circ E_i^{*\,(n)} N. \end{align*}$$

Proof.

Our assertions follow from the shuffle lemma 21, Lemma 2.20.

■

The following corollaries are immediate consequences of Proposition 10.1.5.

Corollary 10.1.6.

Let $i\in I$, and let $M$ be a real simple module. Then $\tilde{E}_i^{\,\max }M$ is also real simple.

Corollary 10.1.7.

Let $i\in I$, and let $M$ be a simple module with $\varepsilon _i(M)=m$. Then we have $\tilde{E}_i^{m} M \simeq E_i^{(m)} M$.

Proposition 10.1.8.

Let $M$ and $N$ be simple modules. We assume that one of them is real. If $\varepsilon _i(M \mathbin{\text{$\nabla $}}N)=\varepsilon _i(M)$, then we have an isomorphism in $R\text{-gmod}$

$$\begin{equation*} \widetilde{E}^{\,\max }_i (M \mathbin{\text{$\nabla $}}N) \simeq (\widetilde{E}^{\,\max }_i M) \mathbin{\text{$\nabla $}}N. \end{equation*}$$

Similarly, if $\varepsilon ^*_i(N \mathbin{\text{$\nabla $}}M)=\varepsilon ^*_i(M)$, then we have

$$\begin{equation*} \widetilde{E}^{*\max }_i (N \mathbin{\text{$\nabla $}}M) \simeq (N \mathbin{\text{$\nabla $}}\widetilde{E}^{*\max }_i M). \end{equation*}$$

Proof.

Set $n=\varepsilon _i(M \mathbin{\text{$\nabla $}}N) = \varepsilon _i(M)$ and $M_0=\widetilde{E}^{\,\max }_i M$. Then $M_0$ or $N$ is real. Now we have

$$\begin{equation*} L(i^n) \otimes M_0 \otimes N \rightarrowtail E_i^n(M\mathbin{\text{$\nabla $}}N)\simeq L(i^n) \otimes \widetilde{E}^{\,\max }_i (M \mathbin{\text{$\nabla $}}N), \end{equation*}$$

which induces a non-zero map $M_0 \otimes N \to \widetilde{E}^{\,\max }_i (M \mathbin{\text{$\nabla $}}N)$. Hence we have a surjective map

$$\begin{equation*} M_0 \circ N \twoheadrightarrow \widetilde{E}^{\,\max }_i (M \mathbin{\text{$\nabla $}}N). \end{equation*}$$

Since $M_0$ or $N$ is real by Corollary 10.1.6, $M_0\circ N$ has a simple head and we obtain the desired result. A similar proof works for the second statement.

■

10.2. Determinantial modules and $T$-system

We will use the materials in Section 9 to obtain properties on the determinantial modules.

In the rest of this paper, we assume that $R$ is symmetric and the base field ${{\mathbf{k}}}$ is of characteristic $0$. Under this condition, the family of self-dual simple $R$-modules corresponds to the upper global basis of $A_q(\mathfrak{n})$ by Theorem 2.1.4.

Let $\operatorname {ch}$ be the map from $K(R \text{-gmod})$ to $A_q(\mathfrak{n})$ obtained by composing $\iota$ and the isomorphism 2.2 in Theorem 2.1.2.

Definition 10.2.1.

For $\lambda \in \mathsf{P}^+$ and $\mu ,\zeta \in W \lambda$ such that $\mu \preceq \zeta$, let $\mathsf{M}(\mu ,\zeta )$ be a simple $R(\zeta -\mu )$-module such that $\operatorname {ch}(\mathsf{M}(\mu ,\zeta ))=\mathrm{D}(\mu ,\zeta )$.

Since $\mathrm{D}(\mu ,\zeta )$ is a member of the upper global basis, such a module exists uniquely due to Theorem 2.1.4. The module $\mathsf{M}(\mu ,\zeta )$ is self-dual, and we call it the determinantial module.

Lemma 10.2.2.

$\mathsf{M}(\mu ,\zeta )$ is a real simple module.

Proof.

It follows from $\operatorname {ch}(\mathsf{M}(\mu ,\zeta ) \circ \mathsf{M}(\mu ,\zeta )) =\operatorname {ch}\big ( \mathsf{M}(\mu ,\zeta ) \big )^2=q^{-(\zeta ,\zeta -\mu )}\mathrm{D}(2\mu ,2\zeta )$ which is a member of the upper global basis up to a power of $q$. Here the last equality follows from Corollary 9.1.3.

■

Proposition 10.2.3.

Let $\lambda ,\mu \in \mathsf{P}^+$, and $s,s',t,t' \in W$ such that $\ell (s's)=\ell (s')+\ell (s)$, $\ell (t't)=\ell (t')+\ell (t)$, $s's \lambda \preceq t'\lambda$, and $s'\mu \preceq t't\mu$. Then

(i)

$\mathsf{M}(s's\lambda ,t'\lambda )$ and $\mathsf{M}(s'\mu ,t't\mu )$ commute,

(ii)

$\Lambda \bigl (\mathsf{M}(s's\lambda ,t'\lambda ),\mathsf{M}(s'\mu ,t't\mu )\bigr )= (s's\lambda +t'\lambda ,\;t't\mu -s'\mu )$,

(iii)

$\widetilde{\Lambda }\bigl (\mathsf{M}(s's\lambda ,t'\lambda ),\mathsf{M}(s'\mu ,t't\mu )\bigr )=(t'\lambda ,\;t't\mu -s'\mu )$,$\widetilde{\Lambda }\bigl (\mathsf{M}(s'\mu ,t't\mu ),\mathsf{M}(s's\lambda ,t'\lambda )\bigr )=(s'\mu -t't\mu ,\;s's\lambda )$.

Proof.

It is a consequence of Proposition 9.1.6 (ii) and Corollary 4.1.4.

■

Proposition 10.2.4.

Let $\lambda \in \mathsf{P}^+$, $\mu ,\zeta \in W \lambda$ such that $\mu \preceq \zeta$ and $i \in I$.

(i)

If $n \mathbin{:=}\langle h_i,\mu \rangle \ge 0$, then$$\begin{equation*} \varepsilon _i(\mathsf{M}(\mu ,\zeta ))=0 \ \ \text{ and } \ \ \mathsf{M}(s_i\mu ,\zeta ) \simeq \widetilde{F}_i^n\mathsf{M}(\mu ,\zeta ) \simeq L(i^n) \mathbin{\text{$\nabla $}}\mathsf{M}(\mu ,\zeta ) \ \text{in $R\text{-gmod}$.} \end{equation*}$$

(ii)

If $\langle h_i,\mu \rangle \le 0$ and $s_i\mu \preceq \zeta$, then $\varepsilon _i(\mathsf{M}(\mu ,\zeta ))=-\langle h_i,\mu \rangle$.

(iii)

If $m \mathbin{:=}-\langle h_i,\zeta \rangle \ge 0$, then$$\begin{equation*} \varepsilon ^*_i(\mathsf{M}(\mu ,\zeta ))=0 \ \ \text{ and } \ \ \mathsf{M}(\mu ,s_i\zeta ) \simeq \widetilde{F}_i^{*\,m}\mathsf{M}(\mu ,\zeta ) \simeq \mathsf{M}(\mu ,\zeta ) \mathbin{\text{$\nabla $}}L(i^m ) \ \text{ in $R\text{-gmod}$.} \end{equation*}$$

(iv)

If $\langle h_i,\zeta \rangle \ge 0$ and $\mu \preceq s_i\zeta$, then $\varepsilon ^*_i(\mathsf{M}(\mu ,\zeta ))=\langle h_i,\zeta \rangle$.

Proof.

It is a consequence of Lemma 9.1.5.

■

Proposition 10.2.5.

Assume that $u,v \in W$ and $i \in I$ satisfy $u<us_i$ and $v<vs_i \le u$.

(i)

We have exact sequences$$\begin{equation} \begin{aligned} & 0 \longrightarrow \mathsf{M}(u\lambda ,v\lambda ) \longrightarrow q^{(vs_i\varpi _i,v\varpi _i-u\varpi _i)}\mathsf{M}(us_i\varpi _i,vs_i\varpi _i) \circ \mathsf{M}(u\varpi _i,v\varpi _i) \\ & \qquad \quad \longrightarrow q^{-1+(v\varpi _i,vs_i\varpi _i-u\varpi _i)}\mathsf{M}(us_i\varpi _i,v\varpi _i) \circ \mathsf{M}(u\varpi _i,vs_i\varpi _i) \longrightarrow 0, \end{aligned} {\tag{10.6}} \end{equation}$$

and$$\begin{equation} \begin{aligned} & 0 \longrightarrow q^{1+(v\varpi _i,vs_i\varpi _i-u\varpi _i)} \mathsf{M}(us_i\varpi _i,v\varpi _i) \circ \mathsf{M}(u\varpi _i,vs_i\varpi _i) \\ & \quad \longrightarrow q^{(v\varpi _i,vs_i\varpi _i-us_i\varpi _i)} \mathsf{M}(u\varpi _i,v\varpi _i)\circ \mathsf{M}(us_i\varpi _i,vs_i\varpi _i) \longrightarrow \mathsf{M}(u\lambda ,v\lambda ) \longrightarrow 0, \end{aligned} {\tag{10.7}} \end{equation}$$

where $\lambda = s_i\varpi _i+\varpi _i$.

(ii)

$\operatorname {\mathfrak{d}}\bigl (\mathsf{M}(u\varpi _i,v\varpi _i), \mathsf{M}(us_i\varpi _i,vs_i\varpi _i)\bigr )=1$.

Proof.

Since the proof of 10.6 is similar, let us only prove 10.7. (Indeed, they are dual to each other.)

Set

$$\begin{eqnarray*} &&X=q^{(v\varpi _i,vs_i\varpi _i-u\varpi _i)} \mathsf{M}(us_i\varpi _i,v\varpi _i) \circ \mathsf{M}(u\varpi _i,vs_i\varpi _i) ,\\ &&Y=q^{(v\varpi _i,vs_i\varpi _i-us_i\varpi _i)} \mathsf{M}(u\varpi _i,v\varpi _i)\circ \mathsf{M}(us_i\varpi _i,vs_i\varpi _i),\\ &&Z=\mathsf{M}(u\lambda ,v\lambda ). \end{eqnarray*}$$

Then Proposition 9.2.1 implies that

$$\begin{equation*} \operatorname {ch}(Y)=\operatorname {ch}(qX)+\operatorname {ch}(Z). \end{equation*}$$

Since $X$ and $Z$ are simple and self-dual, our assertion follows from Lemma 3.2.19.

■

10.3. Generalized $T$-system on determinantial module

Theorem 10.3.1.

Let $\lambda \in \mathsf{P}^+$ and $\mu _1,\mu _2,\mu _3 \in W \lambda$ such that $\mu _1 \preceq \mu _2 \preceq \mu _3$. Then there exists a canonical epimorphism

$$\begin{equation*} \mathsf{M}(\mu _1,\mu _2) \circ \mathsf{M}(\mu _2,\mu _3) \twoheadrightarrow \mathsf{M}(\mu _1,\mu _3), \end{equation*}$$

which is equivalent to saying that $\mathsf{M}(\mu _1,\mu _2) \mathbin{\text{$\nabla $}}\mathsf{M}(\mu _2,\mu _3) \simeq \mathsf{M}(\mu _1,\mu _3)$.

In particular, we have

$$\begin{equation*} \widetilde{\Lambda }(\mathsf{M}(\mu _1,\mu _2),\mathsf{M}(\mu _2,\mu _3))=0\quad \text{and}\quad \Lambda (\mathsf{M}(\mu _1,\mu _2),\mathsf{M}(\mu _2,\mu _3)) = -(\mu _1-\mu _2,\mu _2-\mu _3). \end{equation*}$$

Proof.

(a)

Our assertion follows from Theorem 9.3.3 and Theorem 4.2.1 when $\mu _3=\lambda$.

(b)

We shall prove the general case by induction on $|\lambda -\mu _3|$. By (a), we may assume that $\mu _3 \ne \lambda$. Then there exists $i$ such that $\langle h_i,\mu _3\rangle <0$. The induction hypothesis yields that$$\begin{equation*} \mathsf{M}(\mu _1,\mu _2) \mathbin{\text{$\nabla $}}\mathsf{M}(\mu _2,s_i \mu _3) \simeq \mathsf{M}(\mu _1,s_i \mu _3). \end{equation*}$$

Since $\mu _1\preceq \mu _2\preceq \mu _3\preceq s_i\mu _3$, Proposition 10.2.4 (iv) gives$$\begin{equation*} \varepsilon _i^*\bigl (\mathsf{M}(\mu _2,s_i\mu _3)\bigr )=\varepsilon _i^*\bigl (\mathsf{M}(\mu _1,s_i\mu _3)\bigr )=-\langle h_i,\mu _3\rangle . \end{equation*}$$

Then Proposition 10.1.8 implies that$$\begin{equation*} \widetilde{E}^*_i{}^{\max }\bigl (\mathsf{M}(\mu _1,\mu _2)\mathbin{\text{$\nabla $}}\mathsf{M}(\mu _2,s_i\mu _3)\bigr )\simeq \mathsf{M}(\mu _1,\mu _2)\mathbin{\text{$\nabla $}}\bigl (\widetilde{E}^*_i{}^{\max } \mathsf{M}(\mu _2,s_i\mu _3)\bigr ), \end{equation*}$$

from which we obtain$$\begin{equation*} \mathsf{M}(\mu _1,\mu _3)\simeq \mathsf{M}(\mu _1,\mu _2)\mathbin{\text{$\nabla $}}\mathsf{M}(\mu _2,\mu _3). \end{equation*}$$

By Lemma 3.1.4, we have $\widetilde{\Lambda }\bigl (\mathsf{M}(\mu _1,\mu _2),\mathsf{M}(\mu _2,\mu _3)\bigr )=0$. Hence we obtain$$\begin{align*} &\Lambda \bigl (\mathsf{M}(\mu _1,\mu _2),\mathsf{M}(\mu _2,\mu _3)\bigr )=-\bigl (\operatorname {wt}(\mathsf{M}(\mu _1,\mu _2),\operatorname {wt}(\mathsf{M}(\mu _2,\mu _3))\bigr ). \end{align*}$$■

Proposition 10.3.2.

Let $i \in I$ and $x,y,z \in W$.

(i)

If $\ell (xy)=\ell (x)+\ell (y)$, $zs_i>z$, $xy\ge zs_i$, and $x \ge z$, then we have$$\begin{equation*} \operatorname {\mathfrak{d}}(\mathsf{M}(xy\varpi _i,zs_i\varpi _i),\mathsf{M}(x\varpi _i,z\varpi _i)) \le 1. \end{equation*}$$

(ii)

If $\ell (zy)=\ell (z)+\ell (y)$, $xs_i > x$, $xs_i\ge zy$, and $x \ge z$, then we have$$\begin{equation*} \operatorname {\mathfrak{d}}(\mathsf{M}(xs_i\varpi _i,zy\varpi _i),\mathsf{M}(x\varpi _i,z\varpi _i)) \le 1. \end{equation*}$$

Proof.

In the course of proof, we omit $\overline{\iota }_{{\varpi _i},{\varpi _i}}^{-1}$ for the sake of simplicity. If $y{\varpi _i}= {\varpi _i}$, then the assertion follows from Proposition 10.2.3 (i). Hence we may assume that $y' \mathbin{:=}ys_i < y$.

Let us show (i). By Proposition 9.4.1, we have

$$\begin{equation} \begin{aligned} \Delta (y{\varpi _i},s_i{\varpi _i})\Delta ({\varpi _i},{\varpi _i}) & =q^{-1}G^\mathrm{up}\left( (u_{y{\varpi _i}} \otimes u_{{\varpi _i}}) \otimes (u_{s_i{\varpi _i}} \otimes u_{{\varpi _i}})^{\mspace {1mu}\mathrm{r}}\right) \\ & \qquad +G^\mathrm{up}(\overline{\iota }_\lambda ^{-1}(\tilde{e}_i^*b) \otimes u_\lambda ^{\mspace {1mu}\mathrm{r}}), \end{aligned} {\tag{10.8}} \end{equation}$$

where $\lambda ={\varpi _i}+s_i{\varpi _i}$ and $b=\overline{\iota }_{\varpi _i}(u_{y{\varpi _i}}) \in B(\infty )$. Let $S_{z,\lambda }^*$ be the operator on $A_q(\mathfrak{g})$ given by the application of $e_{j_1}^{(a_1)}\cdots e_{j_t}^{(a_t)}$ from the right, where $z=s_{j_t}\cdots s_{j_1}$ is a reduced expression of $z$ and $a_k=\langle h_{j_k},s_{j_{k-1}}\cdots s_{j_1}\lambda \rangle$. Then applying $S_{z,\lambda }^*$ to 10.8, we obtain

$$\begin{align*} \Delta (y{\varpi _i},zs_i{\varpi _i})\Delta ({\varpi _i},z{\varpi _i}) & =q^{-1}G^\mathrm{up}\left( (u_{y{\varpi _i}} \otimes u_{{\varpi _i}} ) \otimes (u_{zs_i{\varpi _i}} \otimes u_{z{\varpi _i}})^{\mspace {1mu}\mathrm{r}}\right) \\ & \qquad +G^\mathrm{up}(\overline{\iota }_\lambda ^{-1}(\tilde{e}_i^*b) \otimes u_{z\lambda }^{\mspace {1mu}\mathrm{r}}). \end{align*}$$

Recall that $\mu \in \mathsf{P}$ is called $x$-dominant if $c_k\ge 0$. Here $x=s_{i_r} \cdots s_{i_1}$ is a reduced expression of $x$ and $c_k\mathbin{:=}\langle h_{i_k}, s_{i_{k-1}} \cdots s_{i_1} \mu \rangle$ ($1\le k\le r$). Recall that an element $v\in A_q(\mathfrak{g})$ with $\operatorname {wt}_\mathrm{l}(v)=\mu$ is called $x$-highest if $\mu$ is $x$-dominant and

$$\begin{equation*} \text{$f_{i_k}^{1+c_k}f_{i_{k-1}}^{(c_{k-1})}\cdots f_{i_1}^{(c_{1})}v=0$ for any $k$ ($1\le k\le r$).} \end{equation*}$$

If $v$ is $x$-highest, then $v$ is a linear combination of $x$-highest $G^\mathrm{up}(b)$’s. Moreover, $S_{x,\mu }G^\mathrm{up}(b)\mathbin{:=}f^{(c_r)}_{i_r}\cdots f^{(c_1)}_{i_1} G^\mathrm{up}(b)$ is either a member of the upper global basis or zero. Since $\Delta (y{\varpi _i},zs_i{\varpi _i})\Delta ({\varpi _i},z{\varpi _i})$ is $x$-highest of weight $\mu \mathbin{:=}y{\varpi _i}+{\varpi _i}$, we obtain

$$\begin{align*} \Delta (xy{\varpi _i},zs_i{\varpi _i})\Delta (x{\varpi _i},z{\varpi _i}) & =q^{-1}G^\mathrm{up}\left( (u_{xy{\varpi _i}} \otimes u_{x{\varpi _i}} ) \otimes (u_{zs_i{\varpi _i}} \otimes u_{z{\varpi _i}})^{\mspace {1mu}\mathrm{r}}\right) \\ & \qquad +S_{x,\mu }G^\mathrm{up}(\overline{\iota }_\lambda ^{-1}(\tilde{e}_i^*b) \otimes u_{z\lambda }^{\mspace {1mu}\mathrm{r}}). \end{align*}$$

Applying $p_\mathfrak{n}$, we obtain

$$\begin{align*} q^c \mathrm{D}(xy{\varpi _i},zs_i{\varpi _i})\mathrm{D}(x{\varpi _i},z{\varpi _i}) & =q^{-1}p_\mathfrak{n}G^\mathrm{up}\left( (u_{xy{\varpi _i}} \otimes u_{x{\varpi _i}} ) \otimes (u_{zs_i{\varpi _i}} \otimes u_{z{\varpi _i}})^{\mspace {1mu}\mathrm{r}}\right) \\ & \qquad +p_\mathfrak{n}S_{x,\mu }G^\mathrm{up}(\overline{\iota }_\lambda ^{-1}(\tilde{e}_i^*b) \otimes u_{z\lambda }^{\mspace {1mu}\mathrm{r}}) \end{align*}$$

for some integer $c$. Hence we obtain (i) by Lemma 3.2.19 (i).

(ii) is proved similarly. By applying $\varphi ^*$ to 10.8, we obtain

$$\begin{align*} &q^{(s_i{\varpi _i}_i,{\varpi _i}_i)-(y{\varpi _i}_i,{\varpi _i}_i)}\Delta (s_i{\varpi _i},y{\varpi _i})\Delta ({\varpi _i},{\varpi _i}) \\ &\quad =q^{-1}G^\mathrm{up}\left( (u_{s_i{\varpi _i}} \otimes u_{{\varpi _i}}) \otimes (u_{y{\varpi _i}} \otimes u_{{\varpi _i}})^{\mspace {1mu}\mathrm{r}}\right) \\ & \qquad +G^\mathrm{up}(u_\lambda \otimes (\overline{\iota }_\lambda ^{-1}\tilde{e}_i^*b)^{\mspace {1mu}\mathrm{r}}). \end{align*}$$

Here we used Proposition 8.1.4. Then the similar arguments as above show (ii).

■

Proposition 10.3.3.

Let $x \in W$ such that $xs_i > x$ and $x {\varpi _i}\ne {\varpi _i}$. Then we have

$$\begin{equation*} \operatorname {\mathfrak{d}}(\mathsf{M}(xs_i{\varpi _i},x{\varpi _i}),\mathsf{M}(x{\varpi _i},{\varpi _i})) =1. \end{equation*}$$

Proof.

By Proposition 10.3.2 (ii), we have $\operatorname {\mathfrak{d}}(\mathsf{M}(xs_i{\varpi _i},x{\varpi _i}),\mathsf{M}(x{\varpi _i},{\varpi _i})) \le 1$. Assuming $\operatorname {\mathfrak{d}}(\mathsf{M}(xs_i{\varpi _i},x{\varpi _i}),\mathsf{M}(x{\varpi _i},{\varpi _i}))=0$, let us derive a contradiction.

By Theorem 10.3.1 and the assumption, we have

$$\begin{equation*} \mathsf{M}(xs_i{\varpi _i},x{\varpi _i}) \circ \mathsf{M}(x{\varpi _i},{\varpi _i}) \simeq \mathsf{M}(xs_i{\varpi _i},{\varpi _i}). \end{equation*}$$

Hence we have

$$\begin{equation*} \varepsilon _j^*(\mathsf{M}(xs_i{\varpi _i},{\varpi _i}))= \varepsilon _j^*(\mathsf{M}(xs_i{\varpi _i},x{\varpi _i}))+\varepsilon _j^*(\mathsf{M}(x{\varpi _i},{\varpi _i})) \end{equation*}$$

for any $j \in I$. Since $xs_i{\varpi _i}\preceq x{\varpi _i}\preceq s_i{\varpi _i}$, Proposition 10.2.4 implies that

$$\begin{equation*} \varepsilon _j^*(\mathsf{M}(xs_i{\varpi _i},{\varpi _i}))=\varepsilon _j^*(\mathsf{M}(x{\varpi _i},{\varpi _i}))=\langle h_j,{\varpi _i}\rangle . \end{equation*}$$

It implies that

$$\begin{equation*} \varepsilon _j^*(\mathsf{M}(xs_i{\varpi _i},x{\varpi _i}))=0 \quad \text{ for any } j \in I. \end{equation*}$$

It is a contradiction since $\operatorname {wt}\bigl (\mathsf{M}(xs_i{\varpi _i},x{\varpi _i})\bigr )=xs_i{\varpi _i}-x{\varpi _i}$ does not vanish.

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11. Monoidal categorification of $A_q(\mathfrak{n}(w))$

11.1. Quantum cluster algebra structure on $A_q(\mathfrak{n}(w))$

In this subsection, we shall consider the Kac–Moody algebra $\mathfrak{g}$ associated with a symmetric Cartan matrix $\mathsf{A}=(a_{i,j})_{i,j\in I}$. We shall recall briefly the definition of the subalgebra $A_q(\mathfrak{n}(w))$ of $A_q(\mathfrak{g})$ and its quantum cluster algebra structure by using the results of 11 and 23. Remark that we bring the results in 11 through the isomorphism 8.3.

For a given $w \in W$, fix a reduced expression $\widetilde{w}=s_{i_r}\cdots s_{i_1}$.

For $s \in \{ 1,\ldots ,r \}$ and $j \in I$, we set

$$\begin{equation*} \begin{aligned} s_+&\mathbin{:=}\min ( \{k \ | \ s<k\le r,\; i_k=i_s \}\cup \{r+1\}),\\ s_-&\mathbin{:=}\max (\{k \ | \ 1\le k<s, \; i_k=i_s \}\cup \{0\}),\\ s^-(j)&\mathbin{:=}\max (\{k \ | \ 1\le k<s,\; i_k=j \}\cup \{0\}). \end{aligned} \end{equation*}$$

We set

$$\begin{eqnarray} \text{$u_k:=s_{i_1}\cdots s_{i_k}$ for $0\le k \le r$}, {\tag{11.1}} \end{eqnarray}$$

and

$$\begin{eqnarray*} \text{$\lambda _k\mathbin{:=}u_k\varpi _{i_k}$ for $1\le k \le r$}. \end{eqnarray*}$$

Note that $\lambda _{k_- }=u_{k-1}\varpi _{i_k}$, if $k_- >0$. For $0\le t\le s\le r$, we set

$$\begin{align*} \mathrm{D}(s,t)&= \begin{cases} \mathrm{D}(\lambda _s,\lambda _t) & \text{if $0<t$,}\\ \mathrm{D}(\lambda _s,\varpi _{i_s})& \text{if $0=t<s\le r$,}\\ {\mathbf{{1}}}&\text{if $t=s=0$.} \end{cases} \end{align*}$$

The $\mathbb{Q}(q)$-subalgebra of $A_q(\mathfrak{n})$ generated by $\mathrm{D}(i,i_-)$ ($1\le i\le r$) is independent of the choice of a reduced expression of $w$. We denote it by $A_q(\mathfrak{n}(w))$. Then every $\mathrm{D}(s,t)$ ($0\le t\le s\le r$) is contained in $A_q(\mathfrak{n}(w))$ 11, Corollary 12.4. The set ${\mathbf{B}}^\mathrm{up}\bigl (A_q(\mathfrak{n}(w))\bigr )\mathbin{:=}{\mathbf{B}}^\mathrm{up}\bigl (A_q(\mathfrak{g})\bigr )\cap A_q(\mathfrak{n}(w))$ is a basis of $A_q(\mathfrak{n}(w))$ as a $\mathbb{Q}(q)$-vector space 23, Theorem 4.2.5. We call it the upper global basis of $A_q(\mathfrak{n}(w))$. We denote by $A_q(\mathfrak{n}(w))_\mathbb{Z} [q^{\pm 1}]$ the $\mathbb{Z} [q^{\pm 1}]$-module generated by ${\mathbf{B}}^\mathrm{up}\bigl (A_q(\mathfrak{n}(w))$. Then it is a $\mathbb{Z} [q^{\pm 1}]$-subalgebra of $A_q(\mathfrak{n}(w))$ 23, Section 4.7.2. We set $A_{q^{1/2}}(\mathfrak{n}(w)) \mathbin{:=}\mathbb{Q}(q^{1/2}) \otimes _{\mathbb{Q}(q)} A_{q}(\mathfrak{n}(w))$.

Let ${J}=\{1,\ldots ,r\}$, ${{J}}_{\mathrm{f\mspace {.01mu}r}}\mathbin{:=}\left\{k\in {J} \mid k_+=r+1 \right\}$, and ${{J}}_{\mathrm{ex}}\mathbin{:=}{J}\setminus {{J}}_{\mathrm{f\mspace {.01mu}r}}$.

Definition 11.1.1.

We define the quiver $Q$ with the set of vertices $Q_0$ and the set of arrows $Q_1$ as follows:

$(Q_0)$

$Q_0={J}=\{1,\ldots ,r\}$,

$(Q_1)$

There are two types of arrows:$$\begin{eqnarray*} &&\begin{array}{lccl}\text{\textit{ordinary arrows}}&:&\text{$s\xrightarrow {\,\ |a_{i_s,i_t}|\ \,} t$} &\text{if $1\le s<t<s_+<t_+\le r+1$,}\\[5.0pt] \text{\textit{horizontal arrows}}&:& \text{$s\longrightarrow s_-$}& \text{if $1\le s_-<s\le r$.} \end{array} \end{eqnarray*}$$

Let $\widetilde{B} = (b_{i,j})$ be the integer-valued ${J}\times {{J}}_{\mathrm{ex}}$-matrix associated to the quiver $Q$ by 5.2.

Lemma 11.1.2.

Assume that $0 \le d\le b\le a\le c \le r$ and

•

$i_b=i_a$ when $b \ne 0$,

•

$i_d=i_c$ when $d \ne 0$.

Then $\mathrm{D}(a,b)$ and $\mathrm{D}(c,d)$ $q$-commute; that is, there exists $\lambda \in \mathbb{Z}$ such that

$$\begin{equation*} \mathrm{D}(a,b)\mathrm{D}(c,d)= q^{\lambda }\mathrm{D}(c,d)\mathrm{D}(a,b). \end{equation*}$$

Proof.

We may assume $a>0$. Let $u_k$ be as in 11.1. Take $s'=u_a$, $s=u_a^{-1}u_c$, $t'=u_d$, and $t=u_d^{-1}u_b$. Then we have

$$\begin{equation*} \mathrm{D}(s'\varpi _{i_a},t't\varpi _{i_a})=\mathrm{D}(a,b)\quad \text{and}\quad \mathrm{D}(s's\varpi _{i_c},t'\varpi _{i_c})=\mathrm{D}(c,d). \end{equation*}$$

From Proposition 9.1.6, our assertion follows.

■

Hence we have an integer-valued skew-symmetric matrix $L=(\lambda _{i,j})_{1 \le i,j \le r}$ such that

$$\begin{equation*} \mathrm{D}(i,0)\mathrm{D}(j,0)=q^{\lambda _{i,j}}\mathrm{D}(j,0)\mathrm{D}(i,0). \end{equation*}$$

Proposition 11.1.3 (11, Proposition 10.1).

The pair $(L,\widetilde{B})$ is compatible with $d=2$ in 5.3.

Theorem 11.1.4 (11, Theorem 12.3).

Let $\mathscr{A}_{q^{1/2}}([\mathscr{S}])$ be the quantum cluster algebra associated to the initial quantum seed $[\mathscr{S}]\mathbin{:=}( \{ q^{-(d_s,d_s)/4}\mathrm{D}(s,0) \}_{1 \le s \le r}, L, \widetilde{B})$. Then we have an isomorphism of $\mathbb{Q}(q^{1/2})$-algebras

$$\begin{equation*} \mathbb{Q}(q^{1/2}) \otimes _{\mathbb{Z}[q^{\pm 1/2}]} \mathscr{A}_{q^{1/2}}([\mathscr{S}]) \simeq A_{q^{1/2}}(\mathfrak{n}(w)), \end{equation*}$$

where $d_s \mathbin{:=}\lambda _s-\varpi _{i_s}=\operatorname {wt}(D(s,0))$ and $A_{q^{1/2}}(\mathfrak{n}(w)) \mathbin{:=}\mathbb{Q}(q^{1/2}) \otimes _{\mathbb{Q}(q)} A_{q}(\mathfrak{n}(w))$.

11.2. Admissible seeds in the monoidal category $\mathcal{C}_w$

For $0\le t\le s\le r$, we set $\mathsf{M}(s,t)=\mathsf{M}(\lambda _s,\lambda _t)$. It is a real simple module with $\operatorname {ch}(\mathsf{M}(s,t))=D(s,t)$.

Definition 11.2.1.

For $w \in W$, let $\mathcal{C}_w$ be the smallest monoidal abelian full subcategory of $R \text{-gmod}$ satisfying the following properties:

(i)

$\mathcal{C}_w$ is stable under the subquotients, extensions, and grading shifts,

(ii)

$\mathcal{C}_w$ contains $\mathsf{M}(s,s_-)$ for all $1 \le s \le \ell (w)$.

Then by 11, $M\in R\text{-gmod}$ belongs to $\mathcal{C}_w$ if and only if $\operatorname {ch}(M)$ belongs to $A_q(\mathfrak{n}(w))$. Hence we have a $\mathbb{Z} [q^{\pm 1}]$-algebra isomorphism

$$\begin{equation*} K(\mathcal{C}_w)\simeq A_q(\mathfrak{n}(w))_\mathbb{Z} [q^{\pm 1}]. \end{equation*}$$

We set

$$\begin{equation*} \Lambda \mathbin{:=}(\Lambda (\mathsf{M}(i,0),\mathsf{M}(j,0)))_{1 \le i,j\le r} \quad \text{ and } \quad D= (d_i)_{1 \le i \le r} \mathbin{:=}(\operatorname {wt}(\mathsf{M}(i,0)))_{1 \le i \le r}. \end{equation*}$$

Then, by Proposition 10.2.3, $\mathscr{S}\mathbin{:=}( \{ \mathsf{M}(k,0) \}_{1 \le k \le r}, -\Lambda , \widetilde{B}, D )$ is a quantum monoidal seed in $\mathcal{C}_w$. We are now ready to state the main theorem in this section.

Theorem 11.2.2.

The pair $\bigl (\{ \mathsf{M}(k,0) \}_{1 \le k \le r}, \widetilde{B}\bigr )$ is admissible.

As we already explained, combined with Theorem 7.1.3 and Corollary 7.1.4, this theorem implies the following theorem.

Theorem 11.2.3.

The category $\mathcal{C}_w$ is a monoidal categorification of the quantum cluster algebra $A_{q^{1/2}}(\mathfrak{n}(w))$.

In the course of proving Theorem 11.2.2, we omit grading shifts if there is no danger of confusion.

We shall start the proof of Theorem 11.2.2 by proving that, for each $s\in {{J}}_{\mathrm{ex}}$, there exists a simple module $X$ such that

(11.2)

(a)

there exists a surjective homomorphism (up to a grading shift)$$\begin{equation*} X \circ \mathsf{M}(s,0) \twoheadrightarrow \circ _{t;\; b_{t,s}> 0 } \mathsf{M}(t,0)^{\circ b_{t,s} }, \end{equation*}$$

(b)

there exists a surjective homomorphism (up to a grading shift)$$\begin{equation*} \mathsf{M}(s,0) \circ X \twoheadrightarrow \circ _{t;\;\ b_{t,s}<0} \mathsf{M}(t,0)^{\circ -b_{t,s}}, \end{equation*}$$

(c)

$\operatorname {\mathfrak{d}}(X,\mathsf{M}(s,0))=1$.

We set

$$\begin{eqnarray*} &&x\mathbin{:=}i_s\in I,\\ &&I_s\mathbin{:=}\left\{i_k \mid s<k<s_+ \right\}\subset I\setminus \{x\},\\ &&A\mathbin{:=}\displaystyle \mathop{\circ }\limits _{t<s < t_+ < s_+ } \mathsf{M}(t,0)^{\circ |a_{i_s,i_t}| } = \displaystyle \mathop{\circ } \limits _{y\in I_s} \mathsf{M}(s^-(y),0)^{\circ |a_{x,y}| }. \end{eqnarray*}$$

Then $A$ is a real simple module.

Now we claim that the following simple module $X$ satisfies the conditions in 11.2:

$$\begin{equation*} X \mathbin{:=}\mathsf{M}(s_+,s) \mathbin{\text{$\nabla $}}A. \end{equation*}$$

Let us show 11.2 (a). The incoming arrows to $s$ are

•

$t \xrightarrow {\,|a_{x,i_t}|\,}s$ for $1\le t<s<t_+<s_+$,

•

$s_+ \xrightarrow {\,\hspace{2ex}\,}s$.

Hence we have

$$\begin{equation*} \circ _{t;\; b_{t,s} > 0 } \mathsf{M}(t,0)^{\circ b_{t,s}} \simeq A\circ M(s_+,0). \end{equation*}$$

Then the morphism in (a) is obtained as the composition,

$$\begin{align} X \circ \mathsf{M}(s,0) \rightarrowtail A \circ \mathsf{M}(s_+,s) \circ \mathsf{M}(s,0) \twoheadrightarrow A \circ \mathsf{M}(s_+,0). {\tag{11.3}} \end{align}$$

Here the second epimorphism is given in Theorem 10.3.1, and Lemma 3.1.5 asserts that the composition 11.3 is non-zero and hence an epimorphism.

Let us show 11.2 (b). The outgoing arrows from $s$ are

•

$s \xrightarrow {\,\;|a_{x,i_t}|\;\,}t$ for $s<t<s_+<t_+\le r+1$.

•

$s \longrightarrow s_-$ if $s_- > 0$.

Hence we have

$$\begin{eqnarray} &&\mathop{\circ }_{t; b_{t,s}<0 } \mathsf{M}(t,0)^{\circ -b_{t,s}} \simeq \mathsf{M}(s_-,0)\circ \left(\mathop{\circ }_{y\in I_s} \mathsf{M}((s_+)^-(y),0)^{\circ -a_{x,y}}\right). \tag{11.4} \end{eqnarray}$$

Lemma 11.2.4.

There exists an epimorphism (up to a grading)

$$\begin{align*} \Omega : \mathsf{M}(s,0) \circ \mathsf{M}(s_+,s) \circ A \twoheadrightarrow \circ _{t;b_{t,s}<0} \mathsf{M}(t,0)^{\circ -b_{t,s}}. \end{align*}$$

Proof.

By the dual of Theorem 10.3.1 and the $T$-system 10.7 with $i=i_s$, $u=u_{s_+-1}$, and $v=u_{s-1}$, we have morphisms

$$\begin{eqnarray*} && \mathsf{M}(s,0) \rightarrowtail \mathsf{M}(s_-,0) \circ \mathsf{M}(s,s_-), \\ &&\mathsf{M}(s,s_-)\circ \mathsf{M}(s_+,s) \twoheadrightarrow \displaystyle \circ _{y \in I\setminus \{x\}} \mathsf{M}((s_+)^-(y),s^-(y))^{\circ -a_{x,y}}\\ &&\hspace{35ex}\simeq \displaystyle \circ _{y \in I_s} \mathsf{M}((s_+)^-(y),s^-(y))^{\circ -a_{x,y }}. \end{eqnarray*}$$

Here the last isomorphism follows from the fact that $(s_+)^-(y)=s^-(y)$ for any $y\not \in \{x\}\cup I_s=\left\{i_k \mid s\le k<s_+ \right\}$.

Thus we have a sequence of morphisms

$$\begin{eqnarray*} &&\mathsf{M}(s,0) \circ \mathsf{M}(s_+,s) \circ A\; \vcenter{\img[][31pt][11pt][{$\xymatrix{\ar@{>->}[r]^-{\varphi_1}&}$}]{Images/img5080f509bc2dc41072f1fe14e13896c5.svg}} \mathsf{M}(s_-,0) \circ \mathsf{M}(s,s_-) \circ \mathsf{M}(s_+,s) \circ A \\ &&\qquad \vcenter{\img[][30pt][11pt][{$\xymatrix@C=6ex{\ar@{->>}[r]^-{\varphi_2} &}$}]{Images/img02c490968dfee4ca96be6e63b6017e16.svg}}\mathsf{M}(s_-,0) \circ \left(\circ _{y \in I_s} \mathsf{M}((s_+)^-(y),s^-(y))^{\circ -a_{x,y}}\right) \circ A. \end{eqnarray*}$$

By Lemma 3.1.5 (i), the composition $\varphi \mathbin{:=}\varphi _2 \circ \varphi _1$ is non-zero.

Since $A= \displaystyle \circ _{ \substack{y\in I_s} } \mathsf{M}(s^-(y),0)^{\circ -a_{x,y} }$, Theorem 10.3.1 gives the morphisms

$$\begin{align*} \mathsf{M}(s,0) \circ \mathsf{M}(s_+,s) \circ A & \vcenter{\img[][20pt][11pt][{$\xymatrix@C=4ex{\ar[r]^-{\varphi}&}$}]{Images/img21b53a6115b801abd6028ab65545bfe8.svg}} \mathsf{M}(s_-,0) \circ \left(\circ _{y\in I_s} \mathsf{M}((s_+)^-(y),s^-(y))^{\circ -a_{x,y}}\right) \circ A \\ & \vcenter{\img[][20pt][12pt][{$\xymatrix@C=4ex{\ar@{->>}[r]^-{\phi} &}$}]{Images/imge4eb36948d822249fa41bdd73438ca8a.svg}}\mathsf{M}(s_-,0)\circ \left(\circ _{y\in I_s} \mathsf{M}((s_+)^-(y),0)^{\circ -a_{x,y}}\right)\\ &\qquad \quad \simeq \circ _{t; b_{t,s}<0} \mathsf{M}(t,0)^{\circ -b_{t,s}}. \end{align*}$$

Here we have used Lemma 3.2.22 to obtain the morphism $\phi$. Note that the module $\circ _{y \in I_s} \mathsf{M}((s_+)^-(y),s^-(y))^{\circ -a_{x,y}}$ is simple. By applying Lemma 3.1.5 once again, $\phi \circ \varphi$ is non-zero, and hence it is an epimorphism.

■

Lemma 11.2.5.

We have $\operatorname {\mathfrak{d}}(X,\mathsf{M}(s,0))=1$.

Proof.

Since $A$ and $\mathsf{M}(s,0)$ commute and $\operatorname {\mathfrak{d}}\big (\mathsf{M}(s_+,s),\mathsf{M}(s,0)\big )=1$ by Proposition 10.3.3, we have

$$\begin{equation*} \operatorname {\mathfrak{d}}\big ( X,\mathsf{M}(s,0) \big ) \le \operatorname {\mathfrak{d}}\big (\mathsf{M}(s_+,s),\mathsf{M}(s,0)\big )+ \operatorname {\mathfrak{d}}\big (A,\mathsf{M}(s,0)\big )\le 1 \end{equation*}$$

by Proposition 3.2.10 and Lemma 3.2.3. If $X$ and $\mathsf{M}(s,0)$ commute, then 11.2 (a) would imply that ${\mathrm{ch}} \left(\circ _{t;\ b_{t,s}>0} M(t,0)^{\circ b_{t,s}} \right)$ belongs to $K(R \text{-gmod})\,{\mathrm{ch}}(\mathsf{M}(s,0))$. It contradicts the result in 10 that all the ${\mathrm{ch}}(\mathsf{M}(k,0))$’s are prime at $q=1$.

■

Proposition 11.2.6.

The map $\Omega$ factors through $\mathsf{M}(s,0) \circ X$; that is,

$$\begin{align*} \vcenter{\img[][327pt][54pt][{$\xymatrix{ \M(s,0) \conv\M(s_+,s) \conv A \ar@{->>}[drr]^{\tau}\ar@{->>}[rrrr]^{\Omega} &&&& \displaystyle\conv_{t; b_{t,s}<0} \M(t,0)^{\circ-b_{t,s}}. \\ && \M(s,0) \conv X \ar@{->>}[urr]^{\overline{\Omega}} }$}]{Images/img80cc1a01723c9aeb9da8dac367a201ba.svg}} \end{align*}$$

Here $\tau$ is the canonical surjection.

Proof.

We have $1=\operatorname {\mathfrak{d}}\big (\mathsf{M}(s,0), \mathsf{M}(s_+,s)\mathbin{\text{$\nabla $}}A\big )$ by Lemma 11.2.5, and

$$\begin{equation*} \operatorname {\mathfrak{d}}\big (\mathsf{M}(s,0),\mathsf{M}(s_+,s)\big )+ \operatorname {\mathfrak{d}}\big (\mathsf{M}(s,0),A\big )=1 \end{equation*}$$

by Proposition 10.3.3 with $x=u_{s_+ -1}, \ i = i_s$. Hence $\mathsf{M}(s,0) \circ \mathsf{M}(s_+,s) \circ A$ has a simple head by Proposition 3.2.16 (iii).

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End of the proof of Theorem 11.2.2.

By the above arguments, we have proved the existence of $X$ which satisfies 11.2. By Proposition 3.2.17 and 11.2 (c), $\mathsf{M}(s,0) \circ X$ has composition length $2$. Moreover, it has a simple socle and simple head. On the other hand, taking the dual of 11.2 (a), we obtain a monomorphism

$$\begin{equation*} \mathop{\textstyle \bigodot }\limits _{t; b_{t,s}>0} \mathsf{M}(t,0)^{\text{$\odot $}b_{t,s}} \rightarrowtail \mathsf{M}(s,0)\circ X \end{equation*}$$

in $R\text{-mod}$. Together with 11.2 (b), there exists a short exact sequence in $R\text{-gmod}$:

$$\begin{equation*} 0 \to q^c\mathop{\textstyle \bigodot }\limits _{t;b_{t,s}>0} \mathsf{M}(t,0)^{\text{$\odot $}b_{t,s}} \to q^{\widetilde{\Lambda }(\mathsf{M}(s,0), X)} \mathsf{M}(s,0) \circ X \to \mathop{\textstyle \bigodot }\limits _{t;b_{t,s}<0} \mathsf{M}(t,0)^{\text{$\odot $}(-b_{t,s})} \to 0 \end{equation*}$$

for some $c\in \mathbb{Z}$. By Lemma 3.2.18 $c$ must be equal to $1$.

It remains to prove that $X$ commutes with $\mathsf{M}(k,0)$ ($k\not =s$). For any $k\in {J}$, we have

$$\begin{align*} \Lambda (\mathsf{M}(k,0),X)&=\Lambda (\mathsf{M}(k,0),\mathsf{M}(s,0)\mathbin{\text{$\nabla $}}X)-\Lambda (\mathsf{M}(k,0), \mathsf{M}(s,0))\\ &=\sum _{t;\;b_{t,s}<0}\Lambda (\mathsf{M}(k,0),\mathsf{M}(t,0))(-b_{t,s})-\Lambda (\mathsf{M}(k,0), \mathsf{M}(s,0)) \end{align*}$$

and

$$\begin{align*} \Lambda (X,\mathsf{M}(k,0))&=\Lambda (X\mathbin{\text{$\nabla $}}\mathsf{M}(s,0),\mathsf{M}(k,0))-\Lambda (\mathsf{M}(s,0), \mathsf{M}(k,0))\\ &=\sum _{t;\;b_{t,s}>0}\Lambda (\mathsf{M}(t,0),\mathsf{M}(k,0))b_{t,s}-\Lambda (\mathsf{M}(s,0), \mathsf{M}(k,0)). \end{align*}$$

Hence we have

$$\begin{align*} 2\operatorname {\mathfrak{d}}(\mathsf{M}(k,0),X)& =-2\operatorname {\mathfrak{d}}(\mathsf{M}(k,0),\mathsf{M}(s,0)) - \sum _{t;\;b_{t,s}<0}\Lambda (\mathsf{M}(k,0),\mathsf{M}(t,0))b_{t,s} \\ & \qquad \qquad \quad -\sum _{t;\;b_{t,s}>0}\Lambda (\mathsf{M}(k,0),\mathsf{M}( t ,0))b_{t,s} \\ & =- \sum _{1 \le t \le r}\Lambda (\mathsf{M}(k,0),\mathsf{M}(t,0))b_{t,s} \\ & = 2\delta _{k,s}. \end{align*}$$

We conclude that $X$ commutes with $M(k,0)$ if $k\not =s$. Thus we complete the proof of Theorem 11.2.2.

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As a corollary, we prove the following conjecture on the cluster monomials.

Theorem 11.2.7 (11, Conjecture 12.9, 23, Conjecture 1.1(2)).

Every cluster variable in $A_{q}(\mathfrak{n}(w))$ is a member of the upper global basis up to a power of $q^{1/2}$.

Theorem 11.2.2 also implies 11, Conjecture 12.7 in the refined form as follows.

Corollary 11.2.8.

$\mathbb{Z}[q^{\pm 1/2}]\mathop{\otimes }_{\mathbb{Z}[q^{\pm 1}]} A_{q}(\mathfrak{n}(w))_{\mathbb{Z}[q^{\pm 1}]}$ has a quantum cluster algebra structure associated with the initial quantum seed

$$\begin{equation*} [\mathscr{S}]=( \{ q^{-(d_i,d_i)/4}\mathrm{D}(i,0) \}_{1 \le i \le r}, L, \widetilde{B}); \end{equation*}$$

i.e.,

$$\begin{equation*} \mathbb{Z}[q^{\pm 1/2}]\mathop{\otimes }_{\mathbb{Z}[q^{\pm 1}]} A_{q}(\mathfrak{n}(w))_{\mathbb{Z}[q^{\pm 1}]} \simeq \mathscr{A}_{q^{1/2}}([\mathscr{S}]). \end{equation*}$$