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 $N$.

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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