Finite GK-dimensional pre-Nichols algebras of quantum linear spaces and of Cartan type

Abstract

We study pre-Nichols algebras of quantum linear spaces and of Cartan type with finite GK-dimension. We prove that except for a short list of exceptions involving only roots of order 2, 3, 4, 6, any such pre-Nichols algebra is a quotient of the distinguished pre-Nichols algebra introduced by Angiono generalizing the De Concini-Kac-Procesi quantum groups. There are two new examples, one of which can be thought of as $G_2$ at a third root of one.

1. Introduction

1.1. Overview

1.1.1. The problem

Let $\mathbb{k}$ be a field. Let $\operatorname {GK-dim}$ be an abbreviation of Gelfand-Kirillov dimension, see KL. In this paper we contribute to the ongoing program of classifying Hopf algebras with finite $\operatorname {GK-dim}$. See B+GL and references therein.

Let $H$ be a Hopf algebra and let ${}^{H}_{H}\mathcal{YD}$ be the category of Yetter-Drinfeld modules over $H$. Assume that $H$ is pointed (similar arguments apply more generally if its coradical is a Hopf subalgebra). Basic invariants of $H$ are

(i)

the group of grouplikes $\Gamma = G(H)$,

(ii)

the diagram $R = \oplus _{n \in \mathbb{N}_0} R^n$, a graded connected Hopf algebra in ${}^{\mathbb{k}\Gamma }_{\mathbb{k}\Gamma }\mathcal{YD}$,

(iii)

the infinitesimal braiding $V \coloneq R^1$, an object in ${}^{\mathbb{k}\Gamma }_{\mathbb{k}\Gamma }\mathcal{YD}$.

See AS3. Assume that $\Gamma$ has finite growth. In order to classify those $H$ with finite $\operatorname {GK-dim}$, one first needs to understand all such $R$ with finite $\operatorname {GK-dim}$. As a coalgebra $R$ is coradically graded and connected; in other words, it is strictly graded as in Sw, Section 11.2, see p. 232 and Lemma 11.2.1. Strictly graded Hopf algebras $R$ in ${}^{\mathbb{k}\Gamma }_{\mathbb{k}\Gamma }\mathcal{YD}$ with $R^1 \simeq V$ are called post-Nichols algebras of $V$; also, graded Hopf algebras $R$ in ${}^{\mathbb{k}\Gamma }_{\mathbb{k}\Gamma }\mathcal{YD}$ generated by $R^1 \simeq V$ are called pre-Nichols algebras of $V$. See §2.5.

The Nichols algebra $\mathscr{B}(V)$ is isomorphic to the subalgebra of $R$ generated by $V$; see A for an introduction to Nichols algebras. When $\operatorname {char}\mathbb{k}=0$ and $\dim H< \infty$ (thus $\Gamma$ is finite), it was conjectured in AS2 that $R = \mathscr{B}(V)$; this is known to be true when $\Gamma$ is abelian by An1. The validity of this conjecture says that the classification of the finite-dimensional Nichols algebras in ${}^{\mathbb{k}\Gamma }_{\mathbb{k}\Gamma }\mathcal{YD}$ is a substantial step towards the problem of classifying finite-dimensional pointed Hopf algebras with group $\Gamma$. When $\operatorname {char}\mathbb{k}> 0$ or $\dim H = \infty$, the conjecture fails to be true and the knowledge of the Nichols algebras is not enough. Thus, towards classifying pointed Hopf algebras with group $\Gamma$ and finite $\operatorname {GK-dim}$, we do not see how to avoid the consideration of the following questions:

(A)

classify all $V \in {}^{\mathbb{k}\Gamma }_{\mathbb{k}\Gamma }\mathcal{YD}$ such that $\mathscr{B}(V)$ has finite $\operatorname {GK-dim}$,

(B)

for such $V$ classify all post-Nichols algebras with finite $\operatorname {GK-dim}$.

Question (B) appears to be difficult to handle directly. However Lemma 2.2 below, proved in AAH3, reduces Question (B) for $V$ as in (A) to

(C)

classify all pre-Nichols algebras of $V^*$ with finite $\operatorname {GK-dim}$.

As usual it is more flexible to deal with classes of braided vector spaces rather than classes of groups $\Gamma$ and correspondingly pre-Nichols algebras as braided Hopf algebras; see §2.3 for unexplained vocabulary.

1.1.2. Eminent pre-Nichols algebras

For Question (C) we point out that all pre-Nichols algebras of $V$ form a poset $\mathfrak{Pre}(V)$ with $T(V)$ minimal and $\mathscr{B}(V)$ maximal; those with finite $\operatorname {GK-dim}$ form a saturated subposet $\mathfrak{Pre}_{\mathrm{fGK}}(V)$, cf. §2.5. When $\operatorname {char}\mathbb{k}=0$ and the braiding is the usual flip, the Nichols algebra is just the symmetric algebra and the pre-Nichols algebras with finite $\operatorname {GK-dim}$ are the universal enveloping algebras of the finite-dimensional $\mathbb{N}$-graded Lie algebras generated in degree one. Thus $\mathfrak{Pre}_{\mathrm{fGK}}(V)$ is hardly computable when $\dim V \geq 2$. Similar considerations are valid when the braiding is the super flip of a super vector space, see §2.9.2. But if $\dim V = 1$ and $\operatorname {char}\mathbb{k}= 0$, then $\mathfrak{Pre}_{\mathrm{fGK}}(V) = \mathfrak{Pre}(V)$ has obviously a minimal element. We introduce in this paper the notion of eminent pre-Nichols algebra as one that is a minimum in $\mathfrak{Pre}_{\mathrm{fGK}}(V)$. That is, a pre-Nichols algebra $\widehat{\mathscr{B}}$ of a braided vector $V$ is eminent if

(a)

$\operatorname {GK-dim}\widehat{\mathscr{B}} < \infty$;

(b)

if $\mathscr{B}$ is a pre-Nichols algebra of $V$ with $\operatorname {GK-dim}\mathscr{B} < \infty$, then there exists a morphism of pre-Nichols algebras $\widehat{\mathscr{B}} \to \mathscr{B}$, necessarily surjective.

The existence of an eminent pre-Nichols algebra $\widehat{\mathscr{B}}$ reduces Question (C) to the determination of all pre-Nichols algebra quotients of $\widehat{\mathscr{B}}$, that is its homogeneous Hopf ideals starting in degree (at least) 2. Presently there is no general recipe to decide whether a braided vector space admits an eminent pre-Nichols algebra. In this paper we shall show that many braided vector spaces of diagonal type have eminent pre-Nichols algebras.

1.1.3. Distinguished pre-Nichols algebras

From now on we assume that $\mathbb{k}$ is algebraically closed and $\operatorname {char}\mathbb{k}=0$. In this paper we deal with Question (C) for braided vector spaces $V$ of diagonal type, i.e. with braiding determined by a matrix $\mathfrak{q} = (q_{ij})_{i, j \in \mathbb{I}}$ with entries in $\mathbb{k}^{\times }$ where $\theta \in \mathbb{N}$ and $\mathbb{I} = \{1, \dots , \theta \}$. See §2.8 for precise definitions.

First we need to discuss Question (A) for this class. Finite-dimensional Nichols algebras of diagonal type, i.e. those with $\operatorname {GK-dim}= 0$, were classified in H1 through the notion of (generalized) root system. More generally the list of all Nichols algebras of diagonal type with finite root system is given in loc. cit. It was conjectured in AAH1, and verified in various cases R, AA1AAH2, that Nichols algebras of diagonal type with finite $\operatorname {GK-dim}$ are precisely those with finite root system. We recall this as Conjecture 2.6. We shall assume in a few proofs that Conjecture 2.6 is valid in dimensions $\leq 5$.

Let $\mathscr{B}(V) = T(V)/ \mathcal{J}(V)$ be a finite-dimensional Nichols algebra of diagonal type. The distinguished pre-Nichols algebra of $V$ introduced in An3 is the quotient $\widetilde{\mathscr{B}}(V) \coloneq T(V)/ \mathcal{I}(V)$, where $\mathcal{I}(V)$ is the ideal of $T(V)$ generated by the defining relations of $\mathcal{J}(V)$ given in An1 but excluding the powers of the Cartan root vectors and including the quantum Serre relations at Cartan vertices. Detailed presentations of $\mathcal{J}(V)$ and $\mathcal{I}(V)$ are available in AA2, §4. The notion of Cartan root requires the theory of Weyl groupoid that would led us too far from the goal of this paper. Indeed in Cartan type all roots are so and the distinguished pre-Nichols algebras are the positive parts of the quantum groups of DKP. Originally $\widetilde{\mathscr{B}}(V)$ was introduced as a tool for understanding the relations of $\mathscr{B}(V)$; several results on $\widetilde{\mathscr{B}}(V)$ were established in An3. The graded duals of the distinguished pre-Nichols algebras have been presented by generators and relations in AAR.

Unlike the notion of eminent pre-Nichols algebra, we lack at the moment a concise abstract definition of $\widetilde{\mathscr{B}}(V)$ that could be adapted beyond finite dimensional $\mathscr{B}(V)$ of diagonal type; but see §1.2.1 for quantum linear spaces.

In An3 the author and one of us asked whether a distinguished pre-Nichols algebra is eminent (in the terminology just introduced). Recall that the classification in H1 was organized in AA2 in various types: Cartan, standard, super, modular, super modular and UFO. Here we address the question above when $\mathscr{B}(V)$ is either a quantum linear space or of Cartan type. The recent paper ACS treats super and standard types, the remaining ones being the subject of work in progress.

1.2. The main results

In the present paper we focus on braided vector spaces of diagonal type of two kinds. Fix $V$ of diagonal type as in §2.8, with braiding given by the matrix $\mathfrak{q} = (q_{ij})_{i, j \in \mathbb{I}}$.

1.2.1. Quantum linear spaces

Here we assume that $\mathfrak{q}$ satisfies $q_{ij} q_{ji} = 1$ for all $i \neq j \in \mathbb{I}$. We extend the notion of distinguished pre-Nichols algebra to quantum linear spaces even when they are infinite-dimensional. Namely we define the distinguished pre-Nichols algebra $\widetilde{\mathscr{B}}(V)$ as the one presented by generators $(x_i)_{i \in \mathbb{I}}$ and relations $x_i x_j - q_{ij} x_j x_i$, for all $i \neq j \in \mathbb{I}$. If $1 \neq q_{ii}$ is a root of unit for all $i$, then $\dim \mathscr{B}(V) < \infty$, $V$ is of Cartan type $A_1 \times \cdots \times A_1$ and any root is Cartan, so this definition is consistent with the one given in An3 and discussed above.

We need some notation to state our first Theorem. Set

$$\begin{align} \begin{aligned} \mathbb{I}^{\infty } &= \{i \in \mathbb{I} \colon q_{ii} \notin \mathbb{G}_{\infty } \},& \mathbb{I}^N &= \{ i\in \mathbb{I} \colon \operatorname {ord}q_{ii} = N\},\ N \geq 1, \\ \mathbb{I}^t &= \bigcup _{N > 3} \mathbb{I}^N, & \mathbb{I}^{\pm } &= \{i \in \mathbb{I} \colon q_{ii} = \pm 1 \} = \mathbb{I}^1 \sqcup \mathbb{I}^2. \end{aligned}{\tag{1.1}} \end{align}$$

Thus $\mathbb{I} = \mathbb{I}^{\pm } \sqcup \mathbb{I}^3 \sqcup \mathbb{I}^t \sqcup \mathbb{I}^{\infty }$. For $\star \in \mathbb{N} \cup \{\pm , t,\infty \}$, let $V^{\star }$ be the subspace of $V$ spanned by $(x_i)_{i \in \mathbb{I}^{\star }}$ and $\mathfrak{q}^{\star }$ the restriction of $\mathfrak{q}$ to $V^{\star }$. Then

$$\begin{align*} V &= V^{\pm } \oplus V^{3} \oplus V^{t} \oplus V^{\infty }. \end{align*}$$

Theorem 1.1.

Assume that Conjecture 2.6 is true.

(a)

The distinguished pre-Nichols algebra $\widetilde{\mathscr{B}} (V^{\star })$ is eminent, $\star \in \{3, t,\infty \}$.

(b)

Let $\mathscr{B}$ be a finite GK-dimensional pre-Nichols algebra of $V$; let $\mathscr{B}^{\pm , 3}$, respectively $\mathscr{B}^t$, $\mathscr{B}^\infty$ be the subalgebra of $\mathscr{B}$ generated by $V^{\pm } \oplus V^{3}$, respectively $V^{t}$, $V^{\infty }$. Then there is a decomposition$$\begin{align} \mathscr{B} &\simeq \mathscr{B}^{\pm , 3} \underline{\otimes } \mathscr{B}^t \underline{\otimes } \mathscr{B}^\infty . {\tag{1.2}} \end{align}$$

(c)

Assume that $V$ has a basis $\{x_1, x_2\}$ with $x_1\in V^3$, $x_2\in V^1$. Then$$\begin{align*} \breve{\mathscr{B}}(V) = T(V)/ \langle (\operatorname {ad}_c x_{1})^4(x_2), \, (\operatorname {ad}_c x_{2})^2(x_1) \rangle \end{align*}$$

is an eminent pre-Nichols algebra of $V$ and has $\operatorname {GK-dim}= 6$.

See §2.1 for the meaning of $\underline{\otimes }$. Parts (a) and (b) follow from Proposition 3.2 whose proof assumes that Conjecture 2.6 is true. Part (c) is Proposition 3.3. Although $\breve{\mathscr{B}}(V)$ of part (c) is not the distinguished pre-Nichols algebra of the quantum plane $V$, it can be thought of as the distinguished one of the braided vector space of Cartan type $G_2$, but degenerated in the sense that the parameter is a primitive third root of unity. Via suitable bosonizations, $\breve{\mathscr{B}}(V)$ provides new examples of pointed Hopf algebras with finite $\operatorname {GK-dim}$.

Let $\mathscr{B}\in \mathfrak{Pre}_{\mathrm{fGK}}(V)$. By (a) and 1.2 we have a surjective map of pre-Nichols algebras $\mathscr{B}^{\pm , 3} \underline{\otimes } \widetilde{\mathscr{B}} (V^{t}) \underline{\otimes } \widetilde{\mathscr{B}} (V^{\infty }) \to \mathscr{B}$. Therefore it remains to understand $\mathscr{B}^{\pm , 3}$. Towards this, we know:

$\circ$

The pre-Nichols algebras of $V^{\pm }$ with finite $\operatorname {GK-dim}$ are (up to a twist) the enveloping superalgebras $U(\mathfrak{n})$, where $\mathfrak{n} = \oplus _{j\in \mathbb{N}} \mathfrak{n}^j$ is a finite-dimensional Lie superalgebra generated by $\mathfrak{n}^1 \simeq V$, see §2.9.2.

$\circ$

By Proposition 3.2, $\mathscr{B}^{2, 3} \simeq \mathscr{B}^2 \underline{\otimes } \mathscr{B}^3$.

$\circ$

For instance, if $V^{1} =0$, then there is a surjective map of pre-Nichols algebras $\mathscr{B}^{2}\underline{\otimes } \widetilde{\mathscr{B}} (V^{3}) \underline{\otimes } \widetilde{\mathscr{B}} (V^{t}) \underline{\otimes } \widetilde{\mathscr{B}} (V^{\infty }) \to \mathscr{B}$.

Towards $\mathscr{B}^{1, 3}$ we know Part (c) and §3.3. It is natural to ask:

Question 1.2.

Assume that $V = V^1 \oplus V^3$, $\dim V^1 = 1$ and $\dim V^3 = 2$. Is the distinguished pre-Nichols algebra $\widetilde{\mathscr{B}} (V)$ eminent?

1.2.2. Connected Cartan type

Here $\mathfrak{q}$ is of finite Cartan type, i.e.

$$\begin{align*} q_{ij}q_{ji} &= q_{ii}^{a_{ij}} , &-\operatorname {ord}q_{ii} < a_{ij} \leq 0, & & i \neq j \in \mathbb{I}, \end{align*}$$

where $\mathbf{a} = (a_{ij})_{i,j \in \mathbb{I}}$ is a Cartan matrix of finite type with connected Dynkin diagram. In §4 we recall the possibilities for such $\mathfrak{q}$. They depend on a root of unity $q$, whose order is denoted by $N$. In the following statement the symbols $x_{1112}, x_{2221}, x_{2112}, x_{1221}$ are defined in 2.1.

Theorem 1.3.

(a)

The distinguished pre-Nichols algebra $\widetilde{\mathscr{B}}_{\mathfrak{q}}$ is eminent except in the following cases: $A_2$ with $N=3$,$$\begin{align} \begin{aligned} A_{\theta },\quad \theta \geq 2,\quad N=2; & \quad & D_{\theta }, \quad \theta \geq 4, \quad N=2; & \quad & G_2, \quad N=4,6. \end{aligned} {\tag{1.3}} \end{align}$$

(b)

Suppose $\mathfrak{q}$ is of type $A_2$ with $N=3$. Then$$\begin{align*} \widehat{\mathscr{B}}=\mathbb{k}\langle x_1, x_2 | x_{1112}, x_{2221}, x_{2112}, x_{1221}\rangle \end{align*}$$

is an eminent pre-Nichols algebra of $\mathfrak{q}$, and $\operatorname {GK-dim}\widehat{\mathscr{B}} = 5$.

This answers (partially) a question in An3.

The proof of (a) is given in Lemmas 4.12, 4.13, 4.15, 4.16, 4.17, 4.18. For the cases listed in 1.3 the determination of the poset $\mathfrak{Pre}_{\mathrm{fGK}}(V)$ remains an open problem except for $G_2$, with $N=4,6$ that was solved in ACS. See Section 5 for partial results; answers to Questions 5.2, 5.5, 5.7, 5.9 and 5.11 would shed light on the issue. The proof of (b) is given in Proposition 4.11. The eminent pre-Nichols algebra $\widehat{\mathscr{B}}$ is introduced and studied in §4.2.2. There we show that $\widehat{\mathscr{B}}$ properly covers the distinguished pre-Nichols algebra $\widetilde{\mathscr{B}}(V)$, which has $\operatorname {GK-dim}\widetilde{\mathscr{B}}(V) =3$.

2. Preliminaries

2.1. Conventions

For $n \leq m \in \mathbb{N}_0$, put $\mathbb{I}_{n,m} = \{k \in \mathbb{N}_0 \colon n \leq k \leq m\}$ and $\mathbb{I}_m = \mathbb{I}_{1,m}$. Given a positive integer $N$, we denote by $\mathbb{G}_N$ the group of $N$-th roots of unity in $\mathbb{k}^{\times }$, and by $\mathbb{G}'_N \subset \mathbb{G}_N$ the subset of those of order $N$. The group of all roots of unity is denoted by $\mathbb{G}_{\infty }$ and $\mathbb{G}'_{\infty } \coloneq \mathbb{G}_{\infty } - \{1\}$.

The subalgebra generated by a subset $X$ of an associative algebra is denoted by $\mathbb{k}\langle X \rangle$.

All Hopf algebras are assumed to have bijective antipode. If $H$ is a Hopf algebra, the group of group-like elements is denoted by $G(H)$, while $\mathcal{P}(H)$ is the subspace of primitive elements. By $\operatorname {gr}H$ we mean the graded coalgebra associated to the coradical filtration.

If $A$ and $B$ are algebras in ${}^{H}_{H}\mathcal{YD}$, we denote by $A \underline{\otimes } B=(A \otimes B, \mu _{A \underline{\otimes } B})$ the algebra with multiplication $\mu _{A \underline{\otimes } B} = (\mu _A \otimes \mu _B)(\operatorname {id}_A \otimes c_{B,A} \otimes \operatorname {id}_B)$, where $\mu _A$ and $\mu _B$ are the multiplications of $A$ and $B$, respectively.

2.2. Gelfand-Kirillov dimension

We refer to KL for general information on this topic. The following useful statement is immediate from the definition of $\operatorname {GK-dim}$. Let $R$ be a ring and let $M = \oplus _{n \in \mathbb{N}_0} M_n$ be a direct sum of $R$-modules $M_n$ which are free of finite rank (we say $M$ is a locally finite graded $R$-module). The Poincaré series of $M$ is

$$\begin{align*} P_M \coloneq \sum _{n \in \mathbb{N}_0} \operatorname {rank}M_n\, X^n \in \mathbb{Z}[[X]]. \end{align*}$$

Lemma 2.1.

Let $\mathbb{L}$ and $\mathbb{F}$ be fields and let $T = \oplus _{n \in \mathbb{N}_0} T_n$ and $U = \oplus _{n \in \mathbb{N}_0} U_n$ be two locally finite graded algebras generated in degree one over $\mathbb{L}$ and $\mathbb{F}$ respectively. If $P_T = P_U$, then $\operatorname {GK-dim}T = \operatorname {GK-dim}U$.■

Actually KL, 12.6.2 shows that the Poincaré series of a graded finitely generated algebra provides its $\operatorname {GK-dim}$.

2.3. Braided Hopf algebras

A pair $(V,c)$ where $V$ is a vector space and $c \in GL(V^{\otimes 2})$ satisfies the braid equation

$$\begin{align*} (c\otimes \operatorname {id}) (\operatorname {id}\otimes c) (c\otimes \operatorname {id}) = (\operatorname {id}\otimes c) (c\otimes \operatorname {id}) (\operatorname {id}\otimes c) \end{align*}$$

is called a braided vector space. A braided vector space with compatible algebra and coalgebra structures as in T is called a braided Hopf algebra. For instance the tensor algebra $T(V)$ has a canonical structure of (graded connected) braided Hopf algebra such that the elements of degree 1 are primitive. Also the tensor coalgebra $T^c(V)$ becomes a braided Hopf algebra by the twisted shuffle product; see e.g. R, Proposition 9. There is a homogeneous morphism of braided Hopf algebras $\Omega \colon T(V) \to T^c(V)$ determined by $\Omega (v) = v$, $v \in V$; its image is the Nichols algebra of $V$, denoted $\mathscr{B}(V)$. In fact $\Omega$ is the quantum symmetrizer, see e.g. A, Section 3.3.

Another description: let $\mathcal{J}(V)$ be the largest element of the set $\mathfrak{S}$ of graded Hopf ideals of $T(V)$ trivially intersecting $\mathbb{k}\oplus V$. Then $\mathscr{B}(V) \simeq T(V)/\mathcal{J}(V)$.

2.4. Principal realizations

Theorems 1.1 and 1.3 are relevant for the classification of Hopf algebras with finite $\operatorname {GK-dim}$. Indeed a braided vector space arises (up to a mild condition) as a Yetter-Drinfeld module over a Hopf algebra; this is called a realization. Realizations are not unique and we single out a class of them for braidings of diagonal type. Let $H$ be a Hopf algebra. A YD-pair is a couple $(g, \chi ) \subset G(H) \times \operatorname {Hom}_{\text{Alg}} (H, \mathbb{k})$ satisfying

$$\begin{align*} \chi (h) g &= \chi (h_{(2)}) h_{(1)} g \mathcal{S}(h_{(3)}), && h \in H. \end{align*}$$

Compare with AS1, p. 671. This compatibility guarantees that $\mathbb{k}^{\chi }_g$ (i. e. $H$ acting and coacting on $\mathbb{k}$ by $\chi$ and $g$, respectively) is a Yetter-Drinfeld module over $H$. Let $(V,c^{\mathfrak{q}})$ be a braided vector space of diagonal type. Following AS1, p. 673, a principal realization of $(V,c^{\mathfrak{q}})$ over $H$ is a family $(g_i, \chi _i )_{ i \in \mathbb{I}}$ of YD-pairs such that $q_{ij} = \chi _j(g_i)$ for all $i, j$. In this case $V = \bigoplus _i \mathbb{k}^{\chi _i}_{g_i} \in {}^{H}_{H}\mathcal{YD}$.

2.5. Pre-Nichols and post-Nichols algebras

We present in detail the objects of interest in this paper.

•

Let $\mathscr{B} = \bigoplus _{n \in \mathbb{N}_0} \mathscr{B}^n$ be a graded connected braided Hopf algebra with $\mathscr{B}^1 \simeq V$. Then $\mathscr{B}$ is a pre-Nichols algebra of $V$ if it is generated by $\mathscr{B}^1$. In this case there are epimorphisms of (graded) braided Hopf algebras$$\begin{align*} T(V) \twoheadrightarrow \mathscr{B} \twoheadrightarrow \mathscr{B}(V). \end{align*}$$

Hence the set $\mathfrak{Pre}(V)$ of isomorphism classes of pre-Nichols algebras of $V$ is partially ordered with $T(V)$ minimal and $\mathscr{B}(V)$ maximal:

$$\begin{align*} \vcenter{\img[][158pt][86pt][{$\xymatrix@C-15pt{ &&& T(V) \ar@{->>}[1,-3] \ar@{->>}[1,0] \ar@{->>}[1,3] &&& \\ \mathscr{B} & \dots& \dots&\mathscr{B}' & \dots& \dots& \mathscr{B}'' \\ & & & \mathscr{B}(V) \ar@{<<-}[-1,-3] \ar@{<<-}[-1,0] \ar@{<<-}[-1,3] & & &}$}]{Images/imgae4fd1685ca9d68b74fc27b840d82894.svg}} \end{align*}$$

•

Dually, a graded connected braided Hopf algebra $\mathcal{E} = \bigoplus _{n \in \mathbb{N}_0} \mathcal{E}^n$ with $\mathcal{E}^1 \simeq V$ is a post-Nichols algebra of $V$ if it is coradically graded. Thus we have monomorphisms of (graded) braided Hopf algebras$$\begin{align*} \mathscr{B}(V) \hookrightarrow \mathcal{E} \hookrightarrow T^c(V). \end{align*}$$

Hence the set $\mathfrak{Post}(V)$ of isomorphism classes of post-Nichols algebras of $V$ is partially ordered with $T^c(V)$ maximal and $\mathscr{B}(V)$ minimal:

$$\begin{align*} \vcenter{\img[][158pt][86pt][{$\xymatrix@C-15pt{ &&& \mathscr{B}(V) \ar@{_{(}->}[1,-3] \ar@{^{(}->}[1,0] \ar@{^{(}->}[1,3] &&& \\ \mathscr{B} & \dots& \dots&\mathscr{B}' & \dots& \dots& \mathscr{B}'' \\ & & & T(V) \ar@{<-_{)}}[-1,-3] \ar@{<-_{)}}[-1,0] \ar@{<-^{)}}[-1,3] & & &}$}]{Images/imga0ba24c092134e75ee644a286c80b51a.svg}} \end{align*}$$

The only pre-Nichols which is also a post-Nichols algebra of $V$ is $\mathscr{B}(V)$ itself.

2.6. Eminent pre- and post-Nichols algebras

For the purposes of classifying Hopf algebras with finite $\operatorname {GK-dim}$, it is important to describe the (partially ordered) subset $\mathfrak{Post}_{\mathrm{fGK}}(V)$ of $\mathfrak{Post}(V)$ consisting of post-Nichols algebras with finite $\operatorname {GK-dim}$. In this paper we are mainly interested in the (partially ordered) subset $\mathfrak{Pre}_{\mathrm{fGK}}(V)$ of $\mathfrak{Pre}(V)$ consisting of pre-Nichols algebras with finite $\operatorname {GK-dim}$. The reason to start with this is given by the following result:

Lemma 2.2 (AAH3).

Let $\mathscr{B}$ be a pre-Nichols algebra of $V$ and let $\mathcal{E} = \mathscr{B}^d$ be the graded dual of $\mathscr{B}$. Then $\operatorname {GK-dim}\mathcal{E} \leq \operatorname {GK-dim}\mathscr{B}$. If $\mathcal{E}$ is finitely generated, then the equality holds.■

A first approximation to the determination of $\mathfrak{Post}_{\mathrm{fGK}}(V)$ and $\mathfrak{Pre}_{\mathrm{fGK}}(V)$ is through the following notion.

Definition 2.3.

(a)

A pre-Nichols algebra $\widehat{\mathscr{B}}$ is eminent if it is the minimum of $\mathfrak{Pre}_{\mathrm{fGK}}(V)$; i. e. there is an epimorphism of braided Hopf algebras $\widehat{\mathscr{B}} \twoheadrightarrow \mathscr{B}$ that is the identity on $V$ for any $\mathscr{B} \in \mathfrak{Pre}_{\mathrm{fGK}}(V)$.

(b)

A post-Nichols algebra $\widehat{\mathcal{E}}$ is eminent if it is the maximum of $\mathfrak{Post}_{\mathrm{fGK}}(V)$; that is for any $\mathcal{E} \in \mathfrak{Post}_{\mathrm{fGK}}(V)$, there is a monomorphism of braided Hopf algebras $\mathcal{E} \hookrightarrow \widehat{\mathcal{E}}$ that is the identity on $V$.

Beware that there are braided vector spaces without eminent pre-Nichols algebras; e. g., if $\dim V > 1$ and the braiding is the usual flip, then $\mathfrak{Pre}_{\mathrm{fGK}}(V)$ has infinite chains. An intermediate situation could be described as follows.

Definition 2.4.

A family $(\widehat{\mathscr{B}}_i)_{i\in I} \subset \mathfrak{Pre}_{\mathrm{fGK}}(V)$ is eminent if

(a)

for any $\mathscr{B} \in \mathfrak{Pre}_{\mathrm{fGK}}(V)$, there exists $i\in I$ and an epimorphism of braided Hopf algebras $\widehat{\mathscr{B}}_i \twoheadrightarrow \mathscr{B}$ that is the identity on $V$, and

(b)

$(\widehat{\mathscr{B}}_i)_{i\in I}$ is minimal among the families in $\mathfrak{Pre}_{\mathrm{fGK}}(V)$ satisfying (a).

Eminent families of post-Nichols algebras are defined similarly.

All the notions above about braided Hopf algebras related to braided vector spaces have a counterpart for Yetter-Drinfeld modules. Namely, suppose that $(V,c)$ is realized in ${}^{H}_{H}\mathcal{YD}$ for some Hopf algebra $H$. Then $\mathfrak{Pre}^H(V)$ is the subset of $\mathfrak{Pre}(V)$ of pre-Nichols algebras that belong to ${}^{H}_{H}\mathcal{YD}$; similarly we have $\mathfrak{Pre}_{\mathrm{fGK}}^H(V)$, $\mathfrak{Post}^H(V)$, $\mathfrak{Post}_{\mathrm{fGK}}^H(V)$, and also $H$-eminent pre-Nichols or post-Nichols algebras.

2.7. The adjoint representation and $q$-brackets

Any Hopf algebra $R$ in ${}^{H}_{H}\mathcal{YD}$ comes equipped with the (left) adjoint representation $\operatorname {ad}_c \colon R \to \operatorname {End}R$, given by

$$\begin{align*} (\operatorname {ad}_c x) y&= \mu (\mu \otimes \mathcal{S}) (\operatorname {id}\otimes c) (\Delta \otimes \operatorname {id}) (x \otimes y) , && x, y \in R, \end{align*}$$

where $\mu$, $\Delta$ and $\mathcal{S}$ denote the multiplication, comultiplication and antipode of $R$, respectively. The adjoint action of a primitive element $x \in R$ is

$$\begin{align*} (\operatorname {ad}_cx)y&=xy - (x_{(-1)}\cdot y)x_{(0)}, & y \in R. \end{align*}$$

Given $x_{i_1}, x_{i_2} \dots , x_{i_k} \in R$, put

$$\begin{align} x_{i_1 i_2 \dots i_k} = (\operatorname {ad}_c x_{i_1}) \dots (\operatorname {ad}_c x_{i_{k-1}}) x_{i_k}. {\tag{2.1}} \end{align}$$

We also set $x_{(k \, h)} = x_{k\,(k+1)\, (k+2) \dots h}$ for $k < h$.

On the other hand, the braided commutator is defined by

$$\begin{align*} [x, y]_c&=xy - (x_{(-1)}\cdot y)x_{(0)}, & x,y \in R. \end{align*}$$

We refer to AA2, Introduction for a more detailed treatment.

2.8. Nichols algebras of diagonal type

Fix a natural number $\theta$ and let $\mathbb{I} = \mathbb{I}_{\theta }$. Any matrix $\mathfrak{q} = (q_{ij})_{i, j \in \mathbb{I}}$ with coefficients in $\mathbb{k}^{\times }$ determines a braided vector space of diagonal type $(V, c^{\mathfrak{q}})$, where

$$\begin{align} V \text{ has a basis } (x_i) _{i \in \mathbb{I}}, && c^{\mathfrak{q}}(x_i \otimes x_j) = q_{ij} x_j \otimes x_i, && i, j \in \mathbb{I}. {\tag{2.2}} \end{align}$$

The Dynkin diagram associated to $\mathfrak{q}$ is a non-oriented graph with $\theta$ vertices. The vertex $i$ is labelled by $q_{ii}$, and there is an edge between $i$ and $j$ if and only if $\widetilde{q}_{ij}\coloneq q_{ij}q_{ji} \neq 1$; in this case, the edge is labeled by $\widetilde{q}_{ij}$. Thus we may speak of the connected components of this diagram and by abuse of notation of $\mathfrak{q}$. The following useful result says that a connected component with at least 2 vertices one of them labelled by 1 gives rise to an infinite GK-dimensional Nichols algebra.

Lemma 2.5 (AAH1, Lemma 2.8).

Let $U$ be a braided vector space of diagonal type with Dynkin diagram

$$\begin{align*} \vcenter{\img[][68pt][12pt][{$\xymatrix{ \underset{ \ }{\overset{q}{\circ}} \ar@{-}[rr]^{r} & & \underset{ \ }{\overset{ 1 }{\circ} }}$}]{Images/img4f1078639026e3b879e0cecd09b0281e.svg}} , \quad r\neq 1. \end{align*}$$

Then $\operatorname {GK-dim}\mathscr{B}(U) = \infty$.■

Let $\alpha _1, \dots , \alpha _{\theta }$ be the canonical basis of $\mathbb{Z}^{\theta }$. From the braiding matrix $\mathfrak{q}$ we obtain a $\mathbb{k}^{\times }$-valued bilinear form on $\mathbb{Z}^{\theta }$, still denoted $\mathfrak{q}$ and determined by $\mathfrak{q} (\alpha _i, \alpha _j) = q_{ij}$, $i, j \in \mathbb{I}$. Put also

$$\begin{align} \widetilde{\mathfrak{q}} (\alpha , \beta )& \coloneq \mathfrak{q}(\alpha , \beta ) \mathfrak{q}(\beta , \alpha ), & \alpha , \, \beta \in \mathbb{Z}^{\theta }. {\tag{2.3}} \end{align}$$

For sake of brevity, we use $\mathfrak{q}_{\alpha \beta }=\mathfrak{q}(\alpha , \beta )$ and $\widetilde{\mathfrak{q}}_{\alpha \beta }=\widetilde{\mathfrak{q}}(\alpha , \beta )$ as well.

The braided vector space $(V,c^{\mathfrak{q}})$ as in 2.2 is realized in ${}^{\mathbb{Z}^{\theta }}_{\mathbb{Z}^{\theta }}\mathcal{YD}$ by declaring

$$\begin{align} \deg (x_i) &= \alpha _i,& \alpha _i \cdot x_j &= q_{ij}x_j, & i,j &\in \mathbb{I}. {\tag{2.4}} \end{align}$$

The algebra $T(V)$ becomes $\mathbb{Z}^{\theta }$-graded. Thus any quotient algebra $R$ of $T(V)$ by a graded ideal inherits the grading: $R = \bigoplus _{\alpha \in \mathbb{Z}^{\theta }} R^{\alpha }$. We keep the notation $\deg$ for this degree. Furthermore, if $R$ is an algebra obtained as a quotient of $T(V)$ by a graded ideal $I$ (thus a subobject in ${}^{\mathbb{Z}^{\theta }}_{\mathbb{Z}^{\theta }}\mathcal{YD}$), then the braiding on the homogeneous subspaces is given by

$$\begin{align} c(u \otimes v) &= \mathfrak{q}_{\alpha , \beta } \, v \otimes u, && u\in R^{\alpha }, \, v \in R^{\beta }. {\tag{2.5}} \end{align}$$

We shall use 2.5 many times. The braided commutators satisfy

$$\begin{align} [u,vw]_c &= [u,v]_c w + \mathfrak{q}_{\alpha \beta } v [u,w]_c, {\tag{2.6}}\\ [uv,w]_c &= \mathfrak{q}_{ \beta \gamma }[u,w]_c v + u[v,w]_c, {\tag{2.7}}\\ \big [ [u,v]_c, w \big ]_c &= \big [ u, [v, w]_c \big ]_c - \mathfrak{q}_{\alpha \beta } v [u,w]_c + \mathfrak{q}_{ \beta \gamma }[u,w]_c v, {\tag{2.8}} \end{align}$$

for homogeneous elements $u\in R^{\alpha }$, $v \in R^{\beta }$, $w \in R^{\gamma }$.

In the diagonal setting 2.2 we set as usual $\mathcal{J}_{\mathfrak{q}} = \mathcal{J}(V)$, $\mathscr{B}_{\mathfrak{q}} =\mathscr{B}(V)$, $\widetilde{\mathscr{B}}_{\mathfrak{q}} =\widetilde{\mathscr{B}}(V)$, etc. Nichols algebras of diagonal type (i. e. those arising from braided vector spaces of diagonal type) have been intensively studied. The classification of all matrices $\mathfrak{q}$ such that $\mathscr{B}_{\mathfrak{q}}$ has finite root system was provided in H1; the defining relations of these Nichols algebras are given in An1An2. Clearly, finite dimensional Nichols algebras of diagonal type have finite root system. It was conjectured that those of finite $\operatorname {GK-dim}$ share the same property.

Conjecture 2.6 (AAH1, Conjecture 1.5).

The root system of a Nichols algebra of diagonal type with finite GK-dimension is finite.

The validity of Conjecture 2.6 would imply the classification of finite GK-dimensional Nichols algebras of diagonal type. There is strong evidence supporting it. The conjecture holds when $\theta = 2$ AAH2, Thm. 4.1, when the braiding is of affine Cartan type AAH2, Thm. 1.2, or when $\mathfrak{q}$ is generic, that is $q_{ii} \notin \mathbb{G}_{\infty }$, and $q_{ij}q_{ji}=1$ or $q_{ij}q_{ji}\notin \mathbb{G}_{\infty }$, for all $i\ne j\in \mathbb{I}$ RAA1.

We include for completeness proofs of the following well-known results.

Lemma 2.7.

Let $0\neq v, w \in T(V)$ be homogeneous primitive elements with $\deg v = \alpha$ and $\deg w = \beta$. Then $(\operatorname {ad}_cv)w$ is primitive if and only if $\widetilde{\mathfrak{q}}_{\alpha \beta } = 1$.

Proof.

Using 2.5, compute $\Delta ((\operatorname {ad}_cv)w)= \Delta (vw - \mathfrak{q}_{\alpha \beta } \, wv)=$

$$\begin{align*} &= (v \otimes 1 + 1 \otimes v)(w \otimes 1 + 1 \otimes w) - \mathfrak{q}_{\alpha \beta } \, (w \otimes 1 + 1 \otimes w)(v \otimes 1 + 1 \otimes v)\\ &= vw \otimes 1 + v\otimes w + \mathfrak{q}_{\alpha \beta } \, w\otimes v + 1 \otimes vw\\ &\quad - \mathfrak{q}_{\alpha \beta } \, (wv \otimes 1 + w\otimes v + \mathfrak{q}_{\beta \alpha } \, v\otimes w + 1 \otimes wv)\\ &=(\operatorname {ad}_cv)w \otimes 1 + 1 \otimes (\operatorname {ad}_cv)w + \big (1-\widetilde{\mathfrak{q}}_{\alpha \beta } \big ) v\otimes w. \end{align*}$$ ■

Lemma 2.8.

Let $R$ be a graded braided Hopf algebra. If $W$ is any braided subspace of $R$ contained in $\mathcal{P}(R)$ then $\operatorname {GK-dim}\mathscr{B}(W) \leq \operatorname {GK-dim}R$.

Proof.

We follow AS4, Lemma 5.4. Since the elements of $W$ are primitive, the subalgebra $\mathbb{k}\langle W \rangle$ is a braided Hopf subalgebra of $R$; by definition of the Nichols algebra it follows that $\operatorname {gr}\mathbb{k}\langle W \rangle$ projects onto $\mathscr{B}(W)$, so $\operatorname {GK-dim}\mathscr{B}(W) \leq \operatorname {GK-dim}\operatorname {gr}\mathbb{k}\langle W \rangle$. But $\operatorname {GK-dim}\operatorname {gr}\mathbb{k}\langle W \rangle \leq \operatorname {GK-dim}\mathbb{k}\langle W \rangle$ by KL, Lemma 6.5, and this proves the desired inequality.

■

2.9. Pre-Nichols algebras of diagonal type

Let $(V, c^{\mathfrak{q}})$ be a braided vector space of diagonal type associated to the matrix $\mathfrak{q} = (q_{ij})_{i, j \in \mathbb{I}}$. Recall that $\widetilde{q}_{ij} = q_{ij}q_{ji}$, $i\neq j$. We write $\mathfrak{Pre}_{\mathrm{fGK}}^{\mathbb{Z}^{\theta }} (V)$ for $\mathfrak{Pre}_{\mathrm{fGK}}^{\mathbb{k}\mathbb{Z}^{\theta }} (V)$, cf. 2.4.

2.9.1. Pre-Nichols algebras under twist-equivalence

Let $\mathfrak{p} = (p_{ij})_{i, j \in \mathbb{I}}$ be another braiding matrix such that

$$\begin{align*} q_{ii} &= p_{ii},& \widetilde{q}_{ij} &= \widetilde{p}_{ij},& i, j &\in \mathbb{I}. \end{align*}$$

In this case, $(V, c^{\mathfrak{q}})$ and the braided vector space $(W, c^{\mathfrak{p}})$ with basis $(y_i)_{i\in \mathbb{I}}$ are said to be twist-equivalent.

Lemma 2.9.

There is an isomorphism of posets $\mathfrak{Pre}_{\mathrm{fGK}}^{\mathbb{Z}^{\theta }} (W) \simeq \mathfrak{Pre}_{\mathrm{fGK}}^{\mathbb{Z}^{\theta }} (V)$.

Proof.

Let $\sigma : \mathbb{Z}^{\theta } \times \mathbb{Z}^{\theta } \to \mathbb{k}^{\times }$ be the bilinear form, hence a 2-cocycle, given by $\sigma (\alpha _{i}, \alpha _{j}) = \begin{cases} p_{ij} q^{-1}_{ij}, & i\le j, \\ 1, & i > j\end{cases}$. Let $T(V)_{\sigma }$ be the corresponding cocycle deformation of $T(V)$, i. e. with multiplication

$$\begin{align} u._{\sigma }v &= \sigma (\alpha , \beta ) u v,& u&\in T(V)^{\alpha }, \, v \in T(V)^{\beta }, \, \alpha , \beta \in \mathbb{Z}^{\theta }. {\tag{2.9}} \end{align}$$

By the proof of AS3, Prop. 3.9 the linear map $\varphi : W \to V$, $\varphi (y_{i}) = x_{i}$, $i\in \mathbb{I}$, induces an isomorphism $\varphi : T(W) \to T(V)_{\sigma }$ of Hopf algebras in ${}^{\mathbb{Z}^{\theta }}_{\mathbb{Z}^{\theta }}\mathcal{YD}$. Let $I$ be a Hopf ideal of $T(V)$ that belongs to ${}^{\mathbb{Z}^{\theta }}_{\mathbb{Z}^{\theta }}\mathcal{YD}$; then it is also a Hopf ideal of $T(V)_{\sigma }$ and $\operatorname {GK-dim}T(V)/ I = \operatorname {GK-dim}T(V)_{\sigma }/ I$ by Lemma 2.1.

■

2.9.2. Pre-Nichols algebras of super symmetric algebras

Assume that $\widetilde{q}_{ij} = 1 = q_{ii}^2$, for all $j\neq i \in \mathbb{I}$. Then $V = V_0 \oplus V_1$ is a super vector space where $V_{j}$ is spanned by those $x_i$’s such that $q_{ii} = (-1)^j$, $j =0, 1$. Let $\mathfrak{p} = (p_{ij})_{i, j \in \mathbb{I}}$ be the matrix corresponding to the associated super symmetry. Then

•

The pre-Nichols algebras of $(V, c^{\mathfrak{p}})$ are the enveloping superalgebras $U(\mathfrak{n})$, where $\mathfrak{n} = \oplus _{j\in \mathbb{N}} \mathfrak{n}^j$ is a graded Lie superalgebra generated by $\mathfrak{n}^1 \simeq V$.

•

$\mathfrak{Pre}_{\mathrm{fGK}} (V, c^{\mathfrak{p}})$ consists of the enveloping superalgebras $U(\mathfrak{n})$, where $\mathfrak{n} = \oplus _{j\in \mathbb{N}} \mathfrak{n}^j$ is a graded Lie superalgebra generated by $\mathfrak{n}^1 \simeq V$ with $\dim \mathfrak{n} < \infty$.

•

Hence $\mathfrak{Pre}_{\mathrm{fGK}}^{\mathbb{Z}^{\theta }} (V, c^{\mathfrak{p}})$ consists of the enveloping superalgebras $U(\mathfrak{n})$, where $\mathfrak{n} = \oplus _{\beta \in \mathbb{Z}^{\theta }} \mathfrak{n}^\beta$ is a finite-dimensional $\mathbb{Z}^{\theta }$-graded Lie superalgebra generated by $\mathfrak{n}^1 = \oplus _{i\in \mathbb{I}} \mathfrak{n}^{\alpha _{i}} \simeq V$. In particular $\mathfrak{Pre}_{\mathrm{fGK}}^{\mathbb{Z}^{\theta }} (V, c^{\mathfrak{p}}) \subsetneq \mathfrak{Pre}_{\mathrm{fGK}} (V, c^{\mathfrak{p}})$.

•

By Lemma 2.9, $\mathfrak{Pre}_{\mathrm{fGK}}^{\mathbb{Z}^{\theta }} (V, c^{\mathfrak{q}})$ is isomorphic as a poset to the set of isomorphism classes of finite-dimensional $\mathbb{Z}^{\theta }$-graded Lie superalgebras as in the previous point.

3. Quantum linear spaces

In this section we investigate finite GK-dimensional pre-Nichols algebras of quantum linear spaces. These are Nichols algebras of braided vector spaces of diagonal type with totally disconnected Dynkin diagram. More precisely, fix a matrix $\mathfrak{q} = (q_{ij})_{i,j \in \mathbb{I}}$ and a vector space $V$ with basis $(x_i)_{i \in \mathbb{I}}$ and braiding given by $c^{\mathfrak{q}}(x_i \otimes x_j) = q_{ij} x_j \otimes x_i$, $i, j \in \mathbb{I}$. In this section we assume that

$$\begin{align} q_{ij} q_{ji} = 1, && i \neq j \in \mathbb{I}. {\tag{3.1}} \end{align}$$

Then $\mathscr{B}_{\mathfrak{q}}$ is presented by generators $(x_i)_{i \in \mathbb{I}}$ and relations

$$\begin{align} x_{ij} &= 0, \, \, \text{ if } i<j, & & {\tag{3.2}}\\ x_i^{N_i} &= 0, \, \, \text{ if } q_{ii} \in \mathbb{G}'_{\infty },& \text{where } N_i &\coloneq \operatorname {ord}q_{ii} \in \mathbb{N} \cup \infty ; \tag{3.3} \end{align}$$

here we are using the notation 2.1. It has a PBW-basis:

$$\begin{align} \{ x_1^{ a_1} x_2^{ a_2} \cdots x_\theta ^{ a_{\theta }} \colon 0 \leq a_i < N_i \text{ if } q_{ii} \in \mathbb{G}'_{\infty }; \, \, 0\leq a_i \text{ otherwise} \}. {\tag{3.4}} \end{align}$$

As defined in the Introduction, the distinguished pre-Nichols algebra $\widetilde{\mathscr{B}}_{\mathfrak{q}}$ of $V$ is presented by generators $(x_i)_{i \in \mathbb{I}}$ and relations 3.2; it is a domain of $\operatorname {GK-dim}$ $\theta$. Recall the partition $\mathbb{I} = \mathbb{I}^{\pm } \sqcup \mathbb{I}^3 \sqcup \mathbb{I}^t \sqcup \mathbb{I}^{\infty }$ where as in 1.1 we set

$$\begin{align*} \mathbb{I}^{\infty } &= \{i \in \mathbb{I} \colon q_{ii} \notin \mathbb{G}_{\infty } \},& \mathbb{I}^{\pm } &= \{i \in \mathbb{I} \colon q_{ii} = \pm 1 \}, \\ \mathbb{I}^N &= \{ i\in \mathbb{I} \colon q_{ii} \in \mathbb{G}'_N\},\ N \geq 3,& \mathbb{I}^t &= \bigcup _{N > 3} \mathbb{I}^N. \end{align*}$$

For $\star \in \{\pm , 3, t,\infty \}$, let $V^{\star }$ be the subspace of $V$ spanned by $(x_i)_{i \in \mathbb{I}^{\star }}$ and $\mathfrak{q}^{\star }$ the restriction of $\mathfrak{q}$ to $V^{\star }$. Then $V = V^{\pm } \oplus V^{3} \oplus V^{t} \oplus V^{\infty }$. As we have seen in §2.9.2 the $\mathbb{Z}^{\theta }$-graded pre-Nichols algebras of $V^{\pm }$ are twistings of enveloping algebras of nilpotent Lie superalgebras with suitable properties; in particular, there is no eminent pre-Nichols algebra of $V^{\pm }$.

3.1. Reduction to order $\leq 3$

Below we consider various braided vector spaces of diagonal type, see §2.8 for the recipe of the Dynkin diagram that encodes the matrix $\mathfrak{q}$ that determines the braiding.

Remark 3.1.

Let $i \neq j \in \mathbb{I}$. Recall that $x_{ij} \coloneq (\operatorname {ad}_c x_i) x_j= x_i x_j - q_{ij} x_j x_i$. The braiding of the 3-dimensional subspace $\mathbb{k}x_i+\mathbb{k}x_{ij} + \mathbb{k}x_j \subset T(V)$ is easily computed, and the corresponding Dynkin diagram is either

$$\begin{align} \vcenter{\img[][158pt][26pt][{$\xymatrix{ \underset{ i }{\overset{q_{ii}}{\circ}} \ar@{-}[rr]^{q_{ii}^2} & & \underset{ ij }{\overset{ q_{ii} q_{jj} }{\circ} } \ar@{-}[rr]^{q_{jj}^2} & & \underset{ j }{\overset{ q_{jj} }{\circ} }}$}]{Images/img7d8be42f11fa0fdbd1bdc822f91b005a.svg}}, {\tag{3.5}} \end{align}$$

or it is disconnected if the label of some edge is $1$. Indeed,

$$\begin{align*} q_{i, ij} &=q_{ii}q_{ij}, &q_{ij, i} &=q_{ii}q_{ji}, &q_{ij, ij} &=q_{ii} \widetilde{q}_{ij} q_{jj}, &q_{ij, j} &=q_{ij}q_{jj}, & q_{j, ij} &=q_{ji}q_{jj} \end{align*}$$

so $\widetilde{q}_{i, ij} = q_{ii}^2\widetilde{q}_{ij}$ and $\widetilde{q}_{ij, j} = q_{jj}^2\widetilde{q}_{ij}$. Since $\widetilde{q}_{ij} = 1$ because we are in the quantum linear space situation, 3.5 is the Dynkin diagram of $\mathbb{k}x_i+\mathbb{k}x_{ij} + \mathbb{k}x_j$.

Proposition 3.2.

Let $i, j \in \mathbb{I}$ such that $4< \operatorname {ord}q_{ii} + \operatorname {ord}q_{jj}$. Assume that Conjecture 2.6 is true. Then $x_{ij} = 0$ holds in any finite GK-dimensional pre-Nichols algebra of $\mathfrak{q}$.

We point out that Conjecture 2.6 is needed only to discard cases 1, 5 and 8 below, that require the conjecture only for dimension 3.

Proof.

Let $\mathscr{B}$ be a pre-Nichols algebra of $\mathfrak{q}$, so there is a braided Hopf algebra map $T(V) \to \mathscr{B}$. Let $y_1, y_2, y_3$ denote the image of $x_i, x_j, x_{ij}$, respectively, and consider $W \coloneq \mathbb{k}y_1 + \mathbb{k}y_2 + \mathbb{k}y_3$. By Lemma 2.7 we have $W\subset \mathcal{P} (\mathscr{B})$, hence Lemma 2.8 warranties $\operatorname {GK-dim}\mathscr{B}(W) \leq \operatorname {GK-dim}\mathscr{B}$.

Assume $y_3 \neq 0$, so $W$ is 3-dimensional by a degree argument and its Dynkin diagram $\mathfrak{D}$ is 3.5. We show that $\operatorname {GK-dim}\mathscr{B}(W) = \infty$.

Consider the subspaces $V_1 = \mathbb{k}y_1 \oplus \mathbb{k}y_3$, $V_2 = \mathbb{k}y_3 \oplus \mathbb{k}y_2 \subset W$; denote their corresponding Dynkin diagrams by $\mathfrak{D} _1$ and $\mathfrak{D} _2$, respectively. From $q_{ij} q_{ji} = 1$ it follows $x_{ij} = - q_{ij} x_{ji}$, so the image of $x_{ji}$ in $\mathscr{B}$ is not zero.

We split the proof in several cases according to the possibilities for $\operatorname {ord}q_{ii}$ and $\operatorname {ord}q_{jj}$.

Case 1 ($q_{ii} \notin \mathbb{G}_{\infty }$ or $q_{jj} \notin \mathbb{G}_{\infty }$).

This essentially goes back to R. Assume first $q_{ii} \notin \mathbb{G}_{\infty }$. If $\operatorname {GK-dim}\mathscr{B} < \infty$, it follows from AAH1, Lemmas 2.6 and 2.7 that there exists a natural number $k$ such that $(k)^! _{q_{ii}} \prod _{h=0}^{k-1} (1- q^h_{ii}) = 0$, which contradicts $q_{ii} \notin \mathbb{G}_{\infty }$. The case $q_{jj} \notin \mathbb{G}_{\infty }$ is similar: since the image of $x_{ji}$ is not zero, we may apply the same argument as with $q_{ii}$.

Case 2 ($q_{ii}=1$ or $q_{jj} = 1$).

We may suppose $q_{jj} = 1$; if $q_{ii}=1$, the same argument applies. By the previous case, we may assume $q_{ii}$ is a root of unity, and by hypothesis its order must be $N_i > 3$. The diagram $\mathfrak{D}_1$ is

$$\begin{align*} \vcenter{\img[][77pt][24pt][{$\xymatrix{ \underset{ i }{\overset{q_{ii}}{\circ}} \ar@{-}[rr]^{q_{ii}^2} & & \underset{ ij }{\overset{q_{ii} }{\circ} }}$}]{Images/img3d47fb588a7c9368d1517c93723395c9.svg}}, &&\text{Cartan type }\quad \begin{pmatrix} 2 & 2 - N_i \\ 2 - N_i & 2 \end{pmatrix}. \end{align*}$$

If $\operatorname {GK-dim}\mathscr{B}(V_1) < \infty$ then AAH2 implies that the Cartan matrix is of finite type. Thus we conclude $N_i=3$, a contradiction.

Case 3 ($q_{ii}=1$ or $q_{jj} = -1$).

Assume that $q_{jj}=1$. By Case 1, we may assume that $q_{ii}$ is a root of unity; by hypothesis, its order is $\geq 3$. By AAH2, $\operatorname {GK-dim}\mathscr{B} (V_1) = \infty$ since the Dynkin diagram of $V _1$ is

$$\begin{equation*} \vcenter{\img[][83pt][24pt][{$\xymatrix{ \underset{ i }{\overset{q_{ii}}{\circ}} \ar@{-}[rr]^{q_{ii}^2} & & \underset{ ij }{\overset{- q_{ii} }{\circ} }}$}]{Images/imgc68f13adf50ec5eed8230b348176686c.svg}} \end{equation*}$$

and this does not appear in H1, Table 1; indeed it is of Cartan, but not finite, type. The case $q_{ii} = -1$ is treated similarly.

Case 4 ($q_{ii},q_{jj} \in \mathbb{G}_{\infty } - \mathbb{G}_2$).

Now $W$ has connected Dynkin diagram

$$\begin{equation*} \mathfrak{D} = \vcenter{\img[][158pt][26pt][{$\xymatrix{ \underset{ i }{\overset{q_{ii}}{\circ}} \ar@{-}[rr]^{q_{ii}^2} & & \underset{ ij }{\overset{ q_{ii} q_{jj} }{\circ} } \ar@{-}[rr]^{q_{jj}^2} & & \underset{ j }{\overset{ q_{jj} }{\circ} }}$}]{Images/img7d8be42f11fa0fdbd1bdc822f91b005a.svg}} . \end{equation*}$$

If the Nichols algebra of $V_1$ is finite GK-dimensional, by exhaustion of H1, Table 1 we conclude that $q_{ii}$, $q_{jj}$ and $\mathfrak{D}_1$ satisfy one of the following:

(1)

$q_{ii} \in \mathbb{G}'_3, q_{ii}q_{jj} = -1$, $\vcenter{\img[][48pt][24pt][{$\xymatrix{ \underset{ i }{\overset{q_{ii}}{\circ}} \ar@{-}[r]^{q_{ii}^2} & \underset{ ij }{\overset{-1}{\circ} }}$}]{Images/imgb942b5649a985377bb3337c58e454aff.svg}}$

(2)

$q_{ii} \in \mathbb{G}'_4$, $q_{jj} = q_{ii}, \vcenter{\img[][48pt][21pt][{$\xymatrix{ \underset{ i }{\overset{q_{ii}}{\circ}} \ar@{-}[r]^{-1} & \underset{ij}{\overset{-1}{\circ}}}$}]{Images/imgd2fa72a71c27d16a911447c2dcb38bce.svg}}$

(3)

$q_{ii} \in \mathbb{G}'_3, q_{jj} = \pm q_{ii}$, $\vcenter{\img[][52pt][24pt][{$\xymatrix{ \underset{i}{\overset{q_{ii}}{\circ}} \ar@{-}[r]^{q_{ii} ^2} & \underset{ij}{\overset{\pm q_{ii}^2 }{\circ} }}$}]{Images/imgdf1966ddf18a9da327e6041c52fc9f4b.svg}}$

(4)

$q_{ii} \in \mathbb{G}'_3, q_{jj} = q_{ii}^2, \vcenter{\img[][45pt][24pt][{$\xymatrix{ \underset{i}{\overset{q_{ii}}{\circ}} \ar@{-}[r]^{q_{ii}^2} & \underset{ij}{\overset{ 1 }{\circ} }}$}]{Images/img7d17e22dcb65f597b661dd73b85621d1.svg}}$

(5)

$q_{ii} \in \mathbb{G}'_6, q_{ii} q_{jj} = -1$, $\vcenter{\img[][48pt][24pt][{$\xymatrix{ \underset{i}{\overset{q_{ii}}{\circ}} \ar@{-}[r]^{q_{ii}^2} & \underset{ij}{\overset{ -1 }{\circ} }}$}]{Images/img5ebba538a466381e4f2e22a3543dfd1d.svg}}$

(6)

$q_{ii} \in \mathbb{G}'_5, q_{jj} = q_{ii} ^2, \vcenter{\img[][47pt][24pt][{$\xymatrix{ \underset{i}{\overset{q_{ii}}{\circ}} \ar@{-}[r]^{q_{ii}^2} & \underset{ij}{\overset{ q_{ii} ^3 }{\circ} }}$}]{Images/img982d4bdc9aad1bccb9fa38811432a6ae.svg}}$

(7)

$q_{jj} \in \mathbb{G}'_9, q_{ii} = q_{jj} ^3, \vcenter{\img[][51pt][25pt][{$\xymatrix{ \underset{i}{\overset{q_{jj} ^3}{\circ}} \ar@{-}[r]^{q_{jj}^6} & \underset{ij}{\overset{ q_{jj} ^4 }{\circ} }}$}]{Images/img3b2f64712d78f72301cb0caf1111adb1.svg}}$

(8)

$q_{ii} \in \mathbb{G}'_5, q_{ii} q_{jj} = -1$, $\vcenter{\img[][48pt][24pt][{$\xymatrix{ \underset{i}{\overset{q_{ii}}{\circ}} \ar@{-}[r]^{q_{ii}^2} & \underset{ij}{\overset{ -1 }{\circ} }}$}]{Images/img5ebba538a466381e4f2e22a3543dfd1d.svg}}$

In the rest of the proof, we discard one by one all these possibilities.

2 Now $W$ is of Cartan type with Dynkin diagram and Cartan matrix:

$$\begin{align*} \mathfrak{D} = \vcenter{\img[][146pt][21pt][{$\xymatrix{ \underset{i}{\overset{q_{ii}}{\circ}} \ar@{-}[rr]^{-1} & & \underset{ij}{\overset{ -1 }{\circ} } \ar@{-}[rr]^{ -1} & & \underset{j}{\overset{ q_{ii} }{\circ} }}$}]{Images/img44ee55f047c60b4c90880edb8b13e912.svg}}, \, q_{ii} \in \mathbb{G}'_4; &&\begin{pmatrix} 2 & -2 & 0 \\ -1 & 2 & -1 \\ 0 & -2 & 2 \end{pmatrix}. \end{align*}$$

Since this matrix is of affine type, $\operatorname {GK-dim}\mathscr{B}(W)=\infty$ by AAH2.

3 Assume first $q_{jj} = - q_{ii}$. Then

$$\begin{equation*} \mathfrak{D}_2 = \vcenter{\img[][89pt][24pt][{$\xymatrix{ \underset{ij}{\overset{ - q_{ii} ^2}{\circ}} \ar@{-}[rr]^{q_{ii} ^2} & & \underset{j}{\overset{- q_{ii} }{\circ} }}$}]{Images/imgcc66807259549ade50b5e6cc54a1837b.svg}}, \quad q_{ii} \in \mathbb{G}'_3, \end{equation*}$$

which is not arithmetic. By AAH2 we see that $\operatorname {GK-dim}\mathscr{B}(V_2) = \infty$. Next, when $q_{jj} = q_{ii}$, $W$ is of Cartan type with Dynkin diagram and Cartan matrix:

$$\begin{align*} \mathfrak{D} = \vcenter{\img[][146pt][24pt][{$\xymatrix{ \underset{i}{\overset{q_{ii}}{\circ}} \ar@{-}[rr]^{q_{ii}^2} & & \underset{ij}{\overset{ q_{ii} ^2 }{\circ} } \ar@{-}[rr]^{ q_{ii} ^2} & & \underset{j}{\overset{ q_{ii} }{\circ} }}$}]{Images/img1de733d794b809784ba70cb3f8636275.svg}},\, q_{ii} \in \mathbb{G}'_3; &&\begin{pmatrix} 2 & -1 & 0 \\ -2 & 2 & -2 \\ 0 & -1 & 2\end{pmatrix}, \end{align*}$$

which is affine, so $\operatorname {GK-dim}\mathscr{B} (W) = \infty$ by AAH2. 4 Since $q_{ii} ^2 \neq 1$, we have $\operatorname {GK-dim}\mathscr{B}(V_1)=\infty$ by AAH1, Lemma 2.8. 6 In this case

$$\begin{equation*} \mathfrak{D}_2 = \vcenter{\img[][77pt][24pt][{$\xymatrix{ \underset{ij}{\overset{q_{ii}^3}{\circ}} \ar@{-}[rr]^{q_{ii}^4} & & \underset{j}{\overset{ q_{ii} ^2 }{\circ} }}$}]{Images/img3d777b2947e8775226d05e76bf60292b.svg}}, \quad q_{ii} \in \mathbb{G}'_5, \end{equation*}$$

is of indefinite Cartan type, so $\operatorname {GK-dim}\mathscr{B} (V_2) = \infty$ by AAH2. 7 Similarly,

$$\begin{equation*} \mathfrak{D}_2 = \vcenter{\img[][80pt][25pt][{$\xymatrix{ \underset{ij}{\overset{q_{jj} ^4}{\circ}} \ar@{-}[rr]^{q_{jj}^2} & & \underset{j}{\overset{ q_{jj} }{\circ} }}$}]{Images/img40efb99b35adf8defb0566ab894738b2.svg}}, \quad q_{jj} \in \mathbb{G}'_9, \end{equation*}$$

is indefinite Cartan, so $\operatorname {GK-dim}\mathscr{B} (V_2) = \infty$. In the remaining cases, $\mathfrak{D}$ is

1 $\mathfrak{D} = \vcenter{\img[][149pt][23pt][{$\xymatrix{ \underset{i}{\overset{\omega}{\circ}} \ar@{-}[rr]^{\omega^2} & & \underset{ij}{\overset{ - 1}{\circ} } \ar@{-}[rr]^{\omega} & & \underset{j}{\overset{ - \omega^2 }{\circ} }}$}]{Images/imgf550f1dc92348fff719c7d8635afe028.svg}}, \quad \omega \in \mathbb{G}'_3 .$

5 $\mathfrak{D} = \vcenter{\img[][149pt][23pt][{$\xymatrix{ \underset{i}{\overset{ - \omega^2 }{\circ}} \ar@{-}[rr]^{\omega} & & \underset{ij}{\overset{ - 1}{\circ} } \ar@{-}[rr]^{\omega^2} & & \underset{j}{\overset{ \omega}{\circ} }}$}]{Images/img1702c53e7732b33d2367f2a590645692.svg}}, \quad \omega \in \mathbb{G}'_3 .$

8 $\mathfrak{D} = \vcenter{\img[][146pt][23pt][{$\xymatrix{ \underset{i}{\overset{\zeta}{\circ}} \ar@{-}[rr]^{\zeta^2} & & \underset{ij}{\overset{ - 1}{\circ} } \ar@{-}[rr]^{\zeta^3 } & & \underset{j}{\overset{ - \zeta^4 }{\circ} }}$}]{Images/img2249d569d74ace00785c672b4914526f.svg}}, \quad \zeta \in \mathbb{G}'_5 .$

Now 1 and 5 are equal up to permutation of the indexes. Only here we need to assume the validity of Conjecture 2.6. Indeed, these diagrams do not appear in H1, Table 2, so $\operatorname {GK-dim}\mathscr{B} (W)= \infty$ in all cases.■

3.2. A pre-Nichols algebra of type $G_2$

We assume $(V,c^{\mathfrak{q}})$ has the following Dynkin diagram

$$\begin{align*} \vcenter{\img[][69pt][20pt][{$\xymatrix{ \underset{ 1 }{\overset{\omega}{\circ}} & & \underset{ 2 }{\overset{ 1 }{\circ} }}$}]{Images/imgf4d5ff78481f18dd0af44345f4a44576.svg}} , \qquad \omega \in \mathbb{G}'_3. \end{align*}$$

Proposition 3.3.

The algebra $\breve{\mathscr{B}}_{\mathfrak{q}} \coloneq T(V)/ \langle x_{11112}, \, x_{221} \rangle$ is an eminent pre-Nichols algebra of $(V, c^{\mathfrak{q}})$ and $\operatorname {GK-dim}\breve{\mathscr{B}}_{\mathfrak{q}} = 6$.

Proof.

We first claim that the elements $x_{11112}$ and $x_{221}$ are primitive in $T(V)$. This is verified by a direct computation, see S.

Second, we claim that the relations $x_{11112}=0$ and $x_{221}=0$ hold in any finite GK-dimensional pre-Nichols algebra $\mathscr{B}$ of $(V, c^{\mathfrak{q}})$.

Assume first $x_{11112}\neq 0$ in $\mathscr{B}$. Then also $x_{12}\neq 0$. From Lemma 2.7 and the previous claim, we have a braided subspace

$$\begin{gather*} W=\mathbb{k}x_1 +\mathbb{k}x_{12}+\mathbb{k}x_{11112} \subset \mathcal{P}(\mathscr{B}), \end{gather*}$$

so Lemma 2.8 gives $\operatorname {GK-dim}\mathscr{B}(W) \leq \operatorname {GK-dim}\mathscr{B}$. By a degree argument, $W$ has dimension three; from direct computation its Dynkin diagram is

$$\begin{align} \begin{aligned} \vcenter{\img[][93pt][69pt][{$\xymatrix{ &\underset{1}{\overset{\omega}{\circ}} \ar@{-}[rd]^{\omega^2} \ar@{-}[ld]_{\omega^2} & \\ \underset{12}{\overset{\omega}{\circ}} \ar@{-}[rr]^{\omega^2} & &\underset{11112}{\overset{\omega}{\circ}}, }$}]{Images/img576ad68603c137f2576f849f8289ddfc.svg}} \end{aligned} & &\text{ Cartan type } \begin{pmatrix} 2 & -1 & -1\\ -1&2&-1\\ -1&-1 & 2 \end{pmatrix}. {\tag{3.6}} \end{align}$$

Since the Cartan matrix is of affine type $A^1_2$, we have $\operatorname {GK-dim}\mathscr{B}(W) = \infty$ by AAH2, Theorem 1.2(a). Thus $\operatorname {GK-dim}\mathscr{B} = \infty$.

Assume now $x_{221}\neq 0$ in $\mathscr{B}$. Then $x_{21}\neq 0$, and since $q_{12}q_{21}=1$, we have $x_{12}=-q_{12} x_{21}\neq 0$. Consider $W'=\mathbb{k}x_1 +\mathbb{k}x_{12}+\mathbb{k}x_{221}$. We may now use the same argument as above. Indeed, $W' \subset \mathcal{P}(\mathscr{B})$ has Dynkin diagram 3.6 replacing $x_{11112}$ by $x_{221}$, so $\operatorname {GK-dim}\mathscr{B}(W') = \infty$ by the same reason as $\operatorname {GK-dim}\mathscr{B}(W) = \infty$. Hence $\operatorname {GK-dim}\mathscr{B} = \infty$. Thus $\breve{\mathscr{B}}_{\mathfrak{q}} \twoheadrightarrow \mathscr{B}$.

The verification of $\operatorname {GK-dim}\breve{\mathscr{B}}_{\mathfrak{q}} = 6$ is postponed to Proposition 4.5.

■

3.3. A further reduction

Let $\mathscr{B}$ be a finite GK-dimensional pre-Nichols algebra of $\mathscr{B}_{\mathfrak{q}}$. We are naturally led to consider

$$\begin{align} E \coloneq \{(i,j) \colon \, i \in \mathbb{I}^3, j \in \mathbb{I}^1, \, x_{ij} \neq 0 \text{ in } \mathscr{B} \}. \tag{3.7} \end{align}$$

That is, $(i,j) \in E$ means that $\operatorname {ord}q_{ii} = 3$, $q_{jj} = 1$ and $x_{ij} \neq 0$ in $\mathscr{B}$.

Remark 3.4.

If $(i,j) \in E$, the braided vector space $\mathbb{k}x_i \oplus \mathbb{k}x_{ij}\subset \mathscr{B}$ is of Cartan type $A_2$ by Remark 3.1.

Lemma 3.5.

If $(i, j_1), (i,j_2) \in E$ then $j_1=j_2$.

Proof.

Since $x_{ij_1}$ and $x_{ij_2}$ are $\mathbb{Z}^{\theta }$-homogeneous,

$$\begin{align*} c(x_{ij_1}\otimes x_{ij_2}) = \mathfrak{q}(\alpha _i + \alpha _{j_1}, \alpha _i + \alpha _{j_2}) \, x_{ij_2}\otimes x_{ij_1}= q_{ii}q_{ij_2}q_{j_1i}q_{j_1j_2} \, x_{ij_2}\otimes x_{ij_1}. \end{align*}$$

Assume $j_1 \neq j_2$. Then $x_i$, $x_{ij_1}$ and $x_{ij_2}$ have pairwise different $\mathbb{Z}^{\theta }$-degrees, so they span a 3-dimensional braided subspace $W = \mathbb{k}x_i + \mathbb{k}x_{ij_1} + \mathbb{k}x_{ij_2} \subset \mathcal{P}(\mathscr{B})$. Now the Dynkin diagram of $W$ is

$$\begin{align*} \begin{aligned} \vcenter{\img[][89pt][71pt][{$\xymatrix{ &\underset{i}{\overset{q_{ii}}{\circ}} \ar@{-}[rd]^{q_{ii}^2} \ar@{-}[ld]_{q_{ii}^2} & \\ \underset{ij_1}{\overset{q_{ii}}{\circ}} \ar@{-}[rr]^{q_{ii}^2} & &\underset{ij_2}{\overset{q_{ii}}{\circ}}, }$}]{Images/img9df0e1de03ee3ca0dd9224f53dbd99fb.svg}} \end{aligned} & q_{ii} \in \mathbb{G}'_3,&\text{Cartan type } \begin{pmatrix} 2 & -1 & -1\\ -1&2&-1\\ -1&-1 & 2 \end{pmatrix}. \end{align*}$$

Since the Cartan matrix is of affine type $A^{(1)}_2$, we have $\operatorname {GK-dim}\mathscr{B}(W) = \infty$ by AAH2, Theorem 1.2(a). Thus $\operatorname {GK-dim}\mathscr{B} = \infty$, a contradiction.

■

4. Cartan type

In this section we determine the finite GK-dimensional pre-Nichols algebras of braided vector spaces of finite Cartan type under some restrictions.

We fix a matrix $\mathfrak{q} = (q_{ij})_{i,j \in \mathbb{I}}$ of non-zero scalars such that $q_{ii} \neq 1$ for all $i\in \mathbb{I}$ and a braided vector space $(V, c^{\mathfrak{q}})$ with braiding given by $c^{\mathfrak{q}}(x_i \otimes x_j) = q_{ij} x_j \otimes x_i$, $i, j \in \mathbb{I}_{\theta }$, in a basis $\{x_1, \dots , x_{\theta }\}$. Let $N_i=\operatorname {ord}q_{ii} \in \mathbb{N} \cup \infty$.

Recall that $\mathfrak{q}$, or $(V,c^{\mathfrak{q}})$, is of Cartan type if there exists a Cartan matrix $\mathbf{a} = (a_{ij})_{i,j \in \mathbb{I}}$ such that $q_{ij}q_{ji}=q_{ii}^{a_{ij}}$ for all $i,j$. Let $i \in \mathbb{I}$. If $N_i=\infty$, then $a_{ij}$ are uniquely determined. Otherwise, we impose

$$\begin{align} -N_i &< a_{ij} \leq 0,& j & \in \mathbb{I}. {\tag{4.1}} \end{align}$$

In this way we say that $(V,c^{\mathfrak{q}})$, is of Cartan type $\mathbf{a}$.

We follow the terminology of K. Cartan matrices are arranged in three families, namely: finite, affine and indefinite. We say that $\mathfrak{q}$, or $(V,c^{\mathfrak{q}})$, belongs to one of these families if the corresponding $\mathbf{a}$ does.

In this section we assume that $\mathfrak{q}$ is of connected finite Cartan type and that $\dim \mathscr{B}_{\mathfrak{q}} < \infty$. Thus the possible Dynkin diagrams of $\mathfrak{q}$ have the following form, where $q$ is a root of unity in $\mathbb{k}$ of order $N > 1$:

$$\begin{align*} &A_{\theta }: \ \vcenter{\img[][107pt][14pt][{$\xymatrix{ \overset{q}{\underset{\ }{\circ}}\ar@{-}[r]^{q^{-1}} & \overset{q}{\underset{\ }{\circ}}\ar@{.}[r] & \overset{q}{\underset{\ }{\circ}} \ar@{-}[r]^{q^{-1}} & \overset{q}{\underset{\ }{\circ}}}$}]{Images/img913c6f3af584349723c3dc36527eee17.svg}}, & &B_{\theta }: \ \vcenter{\img[][122pt][14pt][{$\xymatrix{ \overset{\,\,q^2}{\underset{\ }{\circ}}\ar@{-}[r]^{q^{-2}} & \overset{\,\,q^2}{\underset{\ }{\circ}}\ar@{.}[r] & \overset{\,\,q^2}{\underset{\ }{\circ}} \ar@{-}[r]^{q^{-2}} & \overset{q}{\underset{\ }{\circ}}}$}]{Images/imgee7c102b058f4d66eb9778b9dafb220f.svg}} \\ &C_{\theta }: \vcenter{\img[][147pt][14pt][{$\xymatrix{ \overset{q}{\underset{\ }{\circ}}\ar@{-}[r]^{q^{-1}} & \overset{q}{\underset{\ }{\circ}}\ar@{-}[r]^{q^{-1}} & \overset{q}{\underset{\ }{\circ}}\ar@{.}[r] & \overset{q}{\underset{\ }{\circ}} \ar@{-}[r]^{q^{-2}} & \overset{\,\,q^2}{\underset{\ }{\circ}}}$}]{Images/img846ce33bd9da395f8768e5bac16c67fa.svg}}, && \\ &D_{\theta }: \begin{aligned} \vcenter{\img[][141pt][42pt][{$\xymatrix@R-15pt{ & & & \overset{q}{\circ} &\\ \overset{q}{\underset{\ }{\circ}}\ar@{-}[r]^{q^{-1}} & \overset{q}{\underset{\ }{\circ}}\ar@{.}[r] & \overset{q}{\underset{\ }{\circ}} \ar@{-}[r]^{q^{-1}} & \overset{q}{\underset{\ }{\circ}} \ar@<0.7ex> @{-}[u]_{q^{-1}}^{} \ar@{-}[r]^{q^{-1}} & \overset{q}{\underset{\ }{\circ}}}$}]{Images/img845b9286440e6e50b68f00034d4a1865.svg}} \end{aligned},& \\ & E_{\theta }: \begin{aligned} \vcenter{\img[][141pt][42pt][{$\xymatrix@R-15pt { & & \overset{q}{\circ} &\\ \overset{q}{\underset{\ }{\circ}}\ar@{-}[r]^{q^{-1}} & \overset{q}{\underset{\ }{\circ}}\ar@{.}[r] & \overset{q}{\underset{\ }{\circ}} \ar@<0.7ex> @{-}[u]_{q^{-1}}^{} \ar@{-}[r]^{q^{-1}} & \overset{q}{\underset{\ }{\circ}} \ar@{-}[r]^{q^{-1}} & \overset{q}{\underset{\ }{\circ}}}$}]{Images/imga163a4d4bc2b29c2b36a4f1800bb8796.svg}} \end{aligned}, && \theta \in \mathbb{I}_{6, 8}, \\ &F_4: \ \vcenter{\img[][117pt][14pt][{$\xymatrix{ \overset{\,\,q}{\underset{\ }{\circ}}\ar@{-}[r]^{q^{-1}} & \overset{\,\,q}{\underset{\ }{\circ}}\ar@{-}[r]^{q^{-2}} & \overset{q^2}{\underset{\ }{\circ}} \ar@{-}[r]^{q^{-2}} & \overset{q^2}{\underset{\ }{\circ}} }$}]{Images/imgafca3846378c0b8b513939305c16fad8.svg}}, & & G_2: \ \vcenter{\img[][43pt][14pt][{$\xymatrix{ \overset{\,\,q}{\underset{\ }{\circ}} \ar@{-}[r]^{q^{-3}}& \overset{q^3}{\underset{\ }{\circ}}}$}]{Images/imgb7c3c68bbfa1d769766dbde30a0739ec.svg}}. \end{align*}$$

We refer to the surveyAA2 for restrictions on $N$ and other features of $\mathscr{B}_{\mathfrak{q}}$ in each case. The quantum Serre relations are the following elements of $T(V)$:

$$\begin{align} (\operatorname {ad}_cx_i)^{1-a_{ij}}x_j, &&i, \,j \in \mathbb{I}, & \, \, i\neq j. {\tag{4.2}} \end{align}$$

By AS2, Lemma A.1 these are primitive in any pre-Nichols algebra. Let $\widetilde{\mathscr{B}}_{\mathfrak{q}} = T(V)/ \mathcal{I}_{\mathfrak{q}}$ be the distinguished pre-Nichols algebra of $(V,c^{\mathfrak{q}})$, see §1.1.3.

Remark 4.1.

From the detailed presentation in AA2, §4 we see that the quantum Serre relations 4.2 generate $\mathcal{I}_{\mathfrak{q}}$ in the following cases:

•

when $\mathbf{a}$ is of type $A_2$ or $B_2$ AA2, pp. 397, 399, 400,

•

when $\mathbf{a}$ is of type $G_2$ and $N\neq 4, 6$ AA2, pp. 410, 411,

•

when $\mathbf{a}$ is simply-laced and $N > 2$ AA2, pp. 397, 404, 407,

•

when $\mathbf{a}$ is of type B, C, or F and $N > 4$ AA2, pp. 399, 402, 409.

4.1. Quantum Serre relations

Let $\mathbf{a} = (a_{ij})_{i,j \in \mathbb{I}}$ be a symmetrizable indecomposable generalized Cartan matrix and $\mathbf{d}\in \operatorname {GL}_{\theta } (\mathbb{Z})$ diagonal such that $\mathbf{d} \mathbf{a}$ is symmetric. The datum $(\mathbf{a}, \mathbf{d})$ is equivalent to an irreducible Cartan datum as in Lu, 1.1.1 by setting

$$\begin{align*} \cdot : \mathbb{I} &\times \mathbb{I} \to \mathbb{Z}, & i\cdot j &= d_{i}a_{ij}, & i,j &\in \mathbb{I}. \end{align*}$$

Let $\mathfrak{g} = \mathfrak{g}(\mathbf{a})$ be the associated Kac-Moody algebra which has a triangular decomposition $\mathfrak{g}(\mathbf{a}) = \mathfrak{g}^{+} \oplus \mathfrak{h} \oplus \mathfrak{g}^{-}$.

Let $q \in \mathbb{k}^{\times }$ and consider the Dynkin diagram

$$\begin{align} \vcenter{\img[][138pt][25pt][{$\xymatrix{ & \overset{q^{d_i}}{\underset{i}{\circ}} \ar@{..}[l] \ar@{-}[rr]^{q^{d_{i}a_{ij}}} & & \overset{q^{d_j}}{\underset{j}{\circ}} \ar@{.}[r] & }$}]{Images/imga43cf0970d72d5d73270359aab4b9a65.svg}} {\tag{4.3}} \end{align}$$

Let $\mathfrak{q}$ be any matrix with Dynkin diagram 4.3 and $(V,c^{\mathfrak{q}})$ be the corresponding braided vector space with basis $(x_i)_{i\in \mathbb{I}}$. Notice that $\mathfrak{q}$ is of Cartan type but it is not necessarily of type $\mathbf{a}$ as 4.1 may not hold.

Let $\breve{\mathscr{B}}_{\mathfrak{q}} = T(V)$ modulo the ideal $\mathfrak{K}_{\mathfrak{q}}$ generated by the quantum Serre relations $(\operatorname {ad}_c x_i)^{1-a_{ij}} (x_j)$, $i\neq j \in \mathbb{I}$, which is a pre-Nichols algebra of $V$.

Proposition 4.2.

$\operatorname {GK-dim}\breve{\mathscr{B}}_{\mathfrak{q}} \geq \dim \mathfrak{g}^+$.

Proof.

If $\xi \in \mathbb{k}$, $\xi ^2 = q$, then $\mathfrak{p} = (\xi ^{d_ia_{ij}})$ has Dynkin diagram 4.3. Let $(W,c^{\mathfrak{q}})$ be the corresponding braided vector space with basis $(\widehat{x}_i)_{i\in \mathbb{I}}$.

Claim 1.

$\operatorname {GK-dim}\breve{\mathscr{B}}_{\mathfrak{q}} = \operatorname {GK-dim}\breve{\mathscr{B}}_{\mathfrak{p}}$.

Proof.

By the proof of AS3, Proposition 3.9 (or the proof of Lemma 2.9) there is a homogeneous linear isomorphism $\psi : T(V) \to T(W)$ determined by $\psi (x_i) = \widehat{x}_i$ for all $i\in \mathbb{I}$ and satisfying AS3, Remarks 3.10. Hence $\psi (\mathfrak{K}_{\mathfrak{q}}) = \mathfrak{K}_{\mathfrak{p}}$ and $\psi$ induces a homogeneous linear isomorphism $\psi : \breve{\mathscr{B}}_{\mathfrak{q}} \to \breve{\mathscr{B}}_{\mathfrak{p}}$. Then apply Lemma 2.1.

■

Let now $\mathbf{f}$ be the $\mathbb{Q}(v)$-algebra defined in Lu, 1.2.5, where $v$ is an indeterminate and let ${}_{\mathcal{A}}\mathbf{f}$ be the $\mathcal{A} \coloneq \mathbb{Z}[v, v^{-1}]$-subalgebra spanned by the quantum divided powers of the generators of $\mathbf{f}$ Lu, 1.4.7. By Lu, 14.4.3, ${}_{\mathcal{A}}\mathbf{f}$ is a free $\mathcal{A}$-module and

$$\begin{align} P_{{}_{\mathcal{A}}\mathbf{f}} &= P_{\mathbf{f}}. {\tag{4.4}} \end{align}$$

Consider $\mathbb{k}$ as $\mathcal{A}$-module via $v \mapsto \xi$. Then we have the algebras ${}_{\mathbb{k}}\mathbf{f} = \mathbb{k}\otimes _{\mathcal{A}} {}_{\mathcal{A}}\mathbf{f}$ and ${}_{\mathbb{k}}\widetilde{\mathbf{f}}$ defined in Lu, 33.1.1 (which is nothing else than $\breve{\mathscr{B}}_{\mathfrak{p}}$). By Lu, 1.4.3, the quantum Serre relations hold in ${}_{\mathbb{k}}\mathbf{f}$, hence we have a surjective algebra map $\breve{\mathscr{B}}_{\mathfrak{p}} = {}_{\mathbb{k}}\widetilde{\mathbf{f}} \twoheadrightarrow {}_{\mathbb{k}}\mathbf{f}$. Thus

$$\begin{align} \operatorname {GK-dim}\breve{\mathscr{B}}_{\mathfrak{p}} &\geq \operatorname {GK-dim}\mathbf{f}. {\tag{4.5}} \end{align}$$

On the other hand, let $\mathbb{k}_0$ be $\mathbb{k}$ as $\mathcal{A}$-module via $v \mapsto 1$. Then ${}_{\mathbb{k}_{0}}\widetilde{\mathbf{f}} \simeq U(\mathfrak{g}^{+})$ by Lu, 33.1.1 and $\dim _{\mathbb{Q}(v)} \mathbf{f}_{\nu } = \dim _{\mathbb{k}_0} ({}_{\mathbb{k}_{0}}\widetilde{\mathbf{f}}_{\nu })$ by Lu, 33.1.3; that is

$$\begin{align} \operatorname {GK-dim}\mathbf{f} &=\operatorname {GK-dim}( {}_{\mathbb{k}_{0}}\widetilde{\mathbf{f}})= \dim \mathfrak{g}^+, {\tag{4.6}} \end{align}$$

where the first equality holds by Lemma 2.1. The Proposition follows.

■

Example 4.3.

Let $\mathbf{a} = \begin{pmatrix} 2 & -5 \\ -1 & 2 \end{pmatrix}$. Then 4.3 takes the form $\vcenter{\img[][41pt][21pt][{$\xymatrix{ \overset{q}{\underset{1}{\circ}}\ar@{-}[r]^{q^{-5}} & \overset{q^5}{\underset{2 }{\circ}}}$}]{Images/img956a89447291babc59be156957aba472.svg}}$ with $q \in \mathbb{k}^{\times }$. If $q_{12} \in \mathbb{k}^{\times }$ and $q_{21} \coloneq q_{12}^{-1}q^{-5}$, then $\mathfrak{q} = \begin{pmatrix} q & q_{12} \\ q_{21} &q^{5} \end{pmatrix}$ has the Dynkin diagram above. Here $\breve{\mathscr{B}}_{\mathfrak{q}} = \mathbb{k}\langle x_1, x_2 \rangle$ modulo the relations

$$\begin{align*} x_2^2x_1 &-q_{21}(2)_q \, x_2x_1x_2+q q_{21}^2 \ x_1 x_2^2,\\ x_1^6x_2 &- 3q_{12}^2 \ x_1^4 x_2x_1^2 +3q_{12}^4 \ x_1^2x_2x_1^4 - q_{12}^6 \ x_2x_1^6 . \end{align*}$$

In this setting Proposition 4.2 gives $\operatorname {GK-dim}\breve{\mathscr{B}}_{\mathfrak{q}} = \infty$.

Example 4.4.

Let $\mathbf{a} = \begin{pmatrix} 2 & -3 \\ -1 & 2 \end{pmatrix}$. Then 4.3 takes the form $\vcenter{\img[][42pt][21pt][{$\xymatrix{ \overset{q}{\underset{1}{\circ}}\ar@{-}[r]^{q^{-3}} & \overset{q^3}{\underset{2 }{\circ}}}$}]{Images/imgaf7c33ec179afa24d8eedf88057ee2c8.svg}}$ with $q \in \mathbb{k}^{\times }$. If $q_{12} \in \mathbb{k}^{\times }$ and $q_{21} \coloneq q_{12}^{-1}q^{-3}$, then $\mathfrak{q} = \begin{pmatrix} q & q_{12} \\ q_{21} &q^{3} \end{pmatrix}$ has the Dynkin diagram above. Here $\breve{\mathscr{B}}_{\mathfrak{q}} = \mathbb{k}\langle x_1, x_2 \rangle$ modulo the relations

$$\begin{align*} &x_2^2x_1 -q_{21}(2)_{q^3} \, x_2x_1x_2+q_{21}^2 q^3 \ x_1 x_2^2,\\ &x_1^4 x_2 -q_{12} (4)_q \, x_1^3x_2x_1 + q_{12}^2 q \binom{4}{2}_{q} \, x_1^2x_2x_1^2 -q_{12} ^3 (4)_q \, x_1^3x_2x_1 + q_{12}^4 q^6 \, x_2 x_1^4. \end{align*}$$

In this situation Proposition 4.2 establishes $\operatorname {GK-dim}\breve{\mathscr{B}}_{\mathfrak{q}} \geq 6$.

This last example gains more relevance when the parameter $q\in \mathbb{k}^{\times }$ specializes to a root of unity with small order.

Proposition 4.5.

Let $\mathbf{a}$ and $q\in \mathbb{k}^{\times }$ as in Example 4.4.

(a)

If $q\in \mathbb{G}'_3$ then $\operatorname {GK-dim}\breve{\mathscr{B}}_{\mathfrak{q}} = 6$.

(b)

If $q\in \mathbb{G}'_2$ then $x_{112}^2=0$ in $\breve{\mathscr{B}}_{\mathfrak{q}}$.

Proof.

Let $q\in \mathbb{k}^{\times }$. Put $x_{1^32^2} = [x_{112},x_{12}]$. By direct computation, in $\breve{\mathscr{B}}_{\mathfrak{q}}$ the following relations hold:

$$\begin{align*} x_{12} x_{2} &= q_{12} q^3 x_{2} x_{12}, \\ x_{112}x_{2}&= q_{12}^2 q^3x_{2}x_{112}+ q_{12} q^2(q-1)(2)_q x_{12}^2,\\ x_{1112}x_{2}&= \text{$q_{12}^3 q ^3$} x_{2}x_{1112}+ \text{$q_{12} q(q^2-q-1)$} x_{1^32^2} + \text{$q_{12}^2 q^2(q-1)(3)_q$} x_{12}x_{112},\\ x_{1^32^2}x_{2}&=q_{12}^3 q^6 x_{2}x_{1^32^2} +q_{12}^2q^3(q-1)^2(2)_q x_{12}^3,\\ x_{1}x_{1^32^2}&= q_{12}^2 q^3 x_{1^32^2}x_{1} + x_{1112}x_{12} - q_{12}^2 q^3 x_{12}x_{1112},\\ (2)_qx_{1112}x_{12}&= q_{12}^2 q^3(2)_qx_{12}x_{1112}+ q_{12}q(q-1)(3)_qx_{112}^2. \end{align*}$$

(a) Here $q\in \mathbb{G}'_3$, so the last relation above becomes

$$\begin{align*} x_{1112}x_{12}= q_{12}^2 x_{12}x_{1112}. \end{align*}$$

Substituting this in the penultimate equation we get

$$\begin{align*} x_{1}x_{1^32^2}= q_{12}^2 x_{1^32^2}x_{1}. \end{align*}$$

These equalities imply more commutations:

$$\begin{align*} x_{1^32^2} x_{12} = q_{12} x_{12} x_{1^32^2},&& x_{112} x_{1^32^2} &= q_{12} x_{1^32^2} x_{112},\\ x_{1112} x_{112} = q_{12} x_{112} x_{1112}, && x_{1112}x_{1^32^2}&= q_{12}^3 x_{1^32^2}x_{1112}. \end{align*}$$

Now we claim that $\breve{\mathscr{B}}_{\mathfrak{q}}$ is linearly spanned by

$$\begin{align*} \mathbb{B}=\{x_2^{n_1} x_{12}^{n_2} x_{1^32^2}^{n_3}x_{112}^{n_4}x_{1112}^{n_5}x_1^{n_6}\colon 0\leq n_1, \dots , n_6 \}. \end{align*}$$

Denote by $\mathcal{I}$ the linear span of $\mathbb{B}$. Since $1\in \mathcal{I}$, it is enough to show that $\mathcal{I}$ is left ideal of $\breve{\mathscr{B}}_{\mathfrak{q}}$. If we multiply $x_2^{n_1} x_{12}^{n_2} x_{1^32^2}^{n_3}x_{112}^{n_4}x_{1112}^{n_5}x_1^{n_6}$ by $x_1$ on the left, we can use the previously deduced commutations between the (powers of the) $x_{\alpha }$’s to successively rearrange the terms until we get a linear combination of elements in $\mathbb{B}$. The claim follows. Let $K = \{2 , 12 , 1^32^2 , 112 , 1112 , 1\}$ be ordered by

$$\begin{align*} 2 > 12 > 1^32^2 > 112 > 1112 > 1. \end{align*}$$

This order is convex, that is for $\alpha , \beta \in K$ with $\alpha > \beta$, the braided commutator $[x_\alpha , x_\beta ]_c$ is a sum of monomials in the letters $\gamma$ such that $\beta < \gamma < \alpha$. Indeed this follows from the equalities above.

Consider next the lexicographical order on $\mathbb{B}$ induced by the order of $K$ and the corresponding $\mathbb{N}_0$-filtration $\mathcal{F}$ on $\breve{\mathscr{B}}_{\mathfrak{q}}$. Let $\operatorname {gr}_{\mathcal{F}} (\breve{\mathscr{B}}_{\mathfrak{q}})$ be the associated graded algebra. By the convexity of the order, there is a natural projection from a quantum polynomial algebra $\mathbb{k}_{\mathtt{q}} [y_1, \dots , y_6] \twoheadrightarrow \operatorname {gr}_{\mathcal{F}} (\breve{\mathscr{B}}_{\mathfrak{q}})$, hence $\operatorname {GK-dim}\operatorname {gr}_{\mathcal{F}} (\breve{\mathscr{B}}_{\mathfrak{q}})\leq 6$. By KL, Proposition 6.6 and Example 4.4 we also have $\operatorname {GK-dim}\operatorname {gr}_{\mathcal{F}} (\breve{\mathscr{B}}_{\mathfrak{q}}) = \operatorname {GK-dim}\breve{\mathscr{B}}_{\mathfrak{q}} \geq 6$, so the equality holds.

(b) This follows by specialization at $q=-1$ in the relation

$$\begin{equation*} (2)_qx_{1112}x_{12}= q_{12}^2 q^3(2)_qx_{12}x_{1112}+ q_{12}q(q-1)(3)_qx_{112}^2. \end{equation*}$$ ■

Remark 4.6.

Let us point out the relevance of (b). By Kharchenko’s theory Kh, $\breve{\mathscr{B}}_{\mathfrak{q}}$ has a PBW-basis. By Proposition 4.2 we know $\operatorname {GK-dim}\breve{\mathscr{B}}_{\mathfrak{q}} \geq 6$ but, when $q\in \mathbb{G}'_2$, the root $2\alpha _1+\alpha _2$ will not contribute to $\operatorname {GK-dim}\breve{\mathscr{B}}_{\mathfrak{q}}$ by (b) above. So even if $\mathbf{a}$ is of type $G_2$, one should not expect that the PBW generators are just those related to the six positive roots of $G_2$, as was the case in the proof of (a).

4.2. Type $A_2$

In this and the next subsections we seek eminent (families of) pre-Nichols algebras in order to determine finite $\operatorname {GK-dim}$ pre-Nichols algebras of braidings of finite Cartan type. The distinguished pre-Nichols algebra will serve as the principal guide in our exploration.

4.2.1. Type $A_2$ with $N>3$

Lemma 4.7.

Assume $\mathbf{a}$ is of Cartan type $A_2$ with $N>3$. If $\mathscr{B}$ is a finite GK-dimensional pre-Nichols algebra of $\mathfrak{q}$, then $x_{112}=0$ and $x_{221}=0$ in $\mathscr{B}$, i. e. the distinguished pre-Nichols algebra $\widetilde{\mathscr{B}}_{\mathfrak{q}}$ is eminent, cf. Definition 2.3.

Proof.

Assume $x_{iij}\neq 0$ for some $i\neq j \in \mathbb{I}_2$; the 3-dimensional braided subspace $W\coloneq \mathbb{k}x_j \oplus \mathbb{k}x_i \oplus \mathbb{k}x_{iij} \subset \mathcal{P}(\mathscr{B})$ has $\operatorname {GK-dim}\mathscr{B}(W) < \infty$.

Consider the braided subspace $W_1 = \mathbb{k}x_i \oplus \mathbb{k}x_{iij} \subset W$. By direct computation, the braiding on $W_1$ is of Cartan type with the following Dynkin diagram and Cartan matrix:

$$\begin{align*} \vcenter{\img[][77pt][24pt][{$\xymatrix{ \underset{ i }{\overset{q_{ii}}{\circ}} \ar@{-}[rr]^{q_{ii}^3} & & \underset{ iij }{\overset{q_{ii}^3 }{\circ} }}$}]{Images/img4ccda96e162801b4343252ba1f3186b6.svg}}, && A_1=\begin{pmatrix} 2 & 3-N\\ 1-M & 2 \end{pmatrix}, && M=\begin{cases} N/3, &\text{ if } 3 | N,\\ N, &\text{ otherwise}. \end{cases} \end{align*}$$

If either $N=5$ or $N>6$, it is evident that the Cartan matrix $A_1$ is not finite, so $\operatorname {GK-dim}\mathscr{B}(W_1)=\infty$ by AAH2, Theorem 1.2 (b). This contradicts $\operatorname {GK-dim}\mathscr{B} < \infty$.

For the remaining cases (i. e. $N=4$ and $N=6$), we consider the whole $W$. Since $\widetilde{\mathfrak{q}}(\alpha _j, 2\alpha _i+\alpha _j)=(q_{ji}q_{ij})^2q_{jj}^2=1$, the braiding on $W$ is of Cartan type with the following Dynkin diagram and Cartan matrix

$$\begin{align*} \vcenter{\img[][146pt][24pt][{$\xymatrix{ \underset{j}{\overset{q_{ii}}{\circ}} \ar@{-}[rr]^{q_{ii}^{-1}} & & \underset{i}{\overset{ q_{ii} }{\circ} } \ar@{-}[rr]^{q_{ii}^3} & & \underset{iij}{\overset{ q_{ii}^3 }{\circ} }}$}]{Images/img42284750686130b40e09ae66351005da.svg}}, &&A= \begin{pmatrix} 2 & -1 & 0\\ -1&2&3-N\\ 0&1-M & 2 \end{pmatrix}. \end{align*}$$

Now it is straightforward to verify that if $N=4$ or $6$, then $A$ is of affine type, which contradicts $\operatorname {GK-dim}\mathscr{B}(W) < \infty$ by AAH2, Theorem 1.2 (b).

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4.2.2. Type $A_2$ with $N=3$

Here is the first restriction.

Lemma 4.8.

Assume $\mathbf{a}$ is of Cartan type $A_2$ with $N=3$. Let $\mathscr{B}\in \mathfrak{Pre}_{\mathrm{fGK}}$. Then $x_{iiij}=0$ and $x_{jiij}=0$ in $\mathscr{B}$ for all $i\neq j \in \mathbb{I}_2$.

Proof.

Since $x_{iij}$ is primitive, using that $\widetilde{\mathfrak{q}}(\alpha _i, 2\alpha _i+\alpha _j)=q^4q^{-1}=1$ and $\widetilde{\mathfrak{q}}(\alpha _j, 2\alpha _i+\alpha _j)=q^2q^{-2}=1$, we get $x_{iiij}, x_{jiij} \in \mathcal{P}(\mathscr{B})$ by Lemma 2.7. Assume first $x_{iiij}\neq 0$ in $\mathscr{B}$. The braided subspace $\mathbb{k}x_i \oplus \mathbb{k}x_j \oplus \mathbb{k}x_{iiij} \subset \mathcal{P}(\mathscr{B})$ has finite $\operatorname {GK-dim}$ Nichols algebra. The Dynkin diagram is

$$\begin{align*} \begin{aligned} \vcenter{\img[][82pt][72pt][{$\xymatrix{ &\underset{iiij}{\overset{q}{\circ}} \ar@{-}[rd]^{q^{-1}} \ar@{-}[ld]_{q^{-1}} & \\ \underset{i}{\overset{q}{\circ}} \ar@{-}[rr]^{q^{-1}} & &\underset{j}{\overset{q}{\circ}}, }$}]{Images/imgbff1c7d06e63f44af260e80f722ce2ee.svg}} \end{aligned} & &\text{Cartan type } \begin{pmatrix} 2 & -1 & -1\\ -1&2&-1\\ -1&-1 & 2 \end{pmatrix}. \end{align*}$$

The Cartan matrix is of affine type $A^{(1)}_2$, and by AAH2, Theorem 1.2(a) this contradicts $\operatorname {GK-dim}\mathscr{B} < \infty$.

If $x_{jiij}\neq 0$, the same argument leads to a contradiction. Indeed, by direct computations, the Dynkin diagram of $U=\mathbb{k}x_i \oplus \mathbb{k}x_j \oplus \mathbb{k}x_{jiij}$ is

$$\begin{align*} \vcenter{\img[][83pt][72pt][{$\xymatrix{ &\underset{jiij}{\overset{q}{\circ}} \ar@{-}[rd]^{q^{-1}} \ar@{-}[ld]_{q^{-1}} & \\ \underset{i}{\overset{q}{\circ}} \ar@{-}[rr]^{q^{-1}} & &\underset{j}{\overset{q}{\circ}}. }$}]{Images/img3f2d563d367a29f0da00f5fc6a15836c.svg}} \end{align*}$$

so $\operatorname {GK-dim}\mathscr{B}(U)=\infty$ by AAH2, Theorem 1.2(a).

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Remark 4.9.

Denote $\widehat{\mathscr{B}}=T(V)/\langle x_{1112}, x_{2221}, x_{2112}, x_{1221}\rangle$. The defining ideal of $\widehat{\mathscr{B}}$ is a Hopf ideal by the proof of Lemma 4.8. Let $\pi \colon \widehat{\mathscr{B}} \to \mathscr{B}(V)$ denote the natural projection. Let $\widehat{\mathcal{Z}}$ be the subalgebra of $\widehat{\mathscr{B}}$ generated by

$$\begin{align*} z_1 \coloneq x_2^3, \qquad z_2 \coloneq x_{221}, \qquad z_3 \coloneq x_{112}, \qquad z_4 \coloneq x_1^3, \qquad z_5 \coloneq x_{12}^3. \end{align*}$$

The next results are devoted to prove that $\widehat{\mathscr{B}}$ is eminent.

Lemma 4.10.

(a)

Given $i\neq j \in \mathbb{I}_2$, the following relations hold in $\widehat{\mathscr{B}}$:$$\begin{align*} [ x_{ij}, x_{iij}]_c = 0 = [ x_{ij}, x_{jji}]_c; \qquad [ x_{iij}, x_{jji}]_c=0. \end{align*}$$

(b)

$\widehat{\mathcal{Z}}$ is a normal braided Hopf subalgebra of $\widehat{\mathscr{B}}$.

(c)

The $z_i$’s $q$-commute; $B= \left\{ z_1^{n_1} z_2^{n_2} z_3^{n_3} z_4^{n_4} z_5^{n_5} \colon n_i \in \mathbb{N}_0 \right\}$ is a basis of $\widehat{\mathcal{Z}}$.

(d)

$\widehat{\mathcal{Z}} = {}^{\operatorname {co}\pi }\widehat{\mathscr{B}}$.

Proof.

(a) Just compute using 2.8:

$$\begin{align*} \big [ x_{ij}, x_{iij} \big ]_c &= \big [ x_i, [x_{j}, x_{iij}]_c \big ]_c - q_{ij} x_{j} \big [ x_i , x_{iij} \big ]_c + q_{jj}q_{ji}^2\big [ x_i , x_{iij} \big ]_c x_{j} = 0; \\ \big [ x_{ij}, x_{jji} \big ]_c &= \big [ x_i, [x_{j}, x_{jji}]_c \big ]_c - q_{ij} x_{j} \big [ x_i , x_{jji} \big ]_c + q_{ii}q_{ij}^2\big [ x_i , x_{jji} \big ]_c x_{j} = 0; \\ \big [ x_{iij}, x_{jji} \big ]_c &= \big [ [x_i, x_{ij}]_c, x_{jji} \big ]_c \\ &= \big [ x_i, [x_{ij}, x_{jji}]_c \big ]_c - q_{ii}q_{ij} x_{ij} \big [ x_i , x_{jji} \big ]_c +q_{ii}^2 q_{ij}\big [ x_i , x_{jji} \big ]_c x_{ij} \\ & = \big [ x_i, [x_{ij}, x_{jji}]_c \big ]_c = 0. \end{align*}$$

(b) We claim that the generators of $\widehat{\mathcal{Z}}$ are annihilated by the braided adjoint action of $\widehat{\mathscr{B}}$. Fix $i \in \mathbb{I}_2$. By definition $(\operatorname {ad}_c x_i)z_2=0=(\operatorname {ad}_c x_i)z_3$. In $T(V)$ we have $(\operatorname {ad}_cx_i) x_i^3= x_i^4 - q_{ii}^3 x_i^4 = 0$, and if $j\neq i$ then

$$\begin{align} \begin{aligned} x_{jjji} = (\operatorname {ad}_c x_j)^3 x_i &= \sum _{k=0}^3 (-1)^k q_{ji}^k q_{jj}^{k(k-1)/2} \binom{3}{k}_{q_{jj}} x_j^{3-k} x_i x_j^k \\ &= x_j^3 x_i - q_{ji}^3 x_i x_j^3 = - q_{ji}^3 (\operatorname {ad}_c x_i) x_j^3. \end{aligned} {\tag{4.7}} \end{align}$$

Thus $\operatorname {ad}_c x_i$ annihilates $z_1$ and $z_4$. Finally, we proceed with $z_5$. From (a) we get the commutation $x_{112}x_{12}=q_{11}^2q_{12} x_{12} x_{112}$ in $\widehat{\mathscr{B}}$. Then using 2.6

$$\begin{align} \begin{aligned} (\operatorname {ad}_c x_1)z_5 & =x_{112}x_{12}^2 +q_{11}q_{12}x_{12}x_{112}x_{12}+q_{11}^2q_{12}^2 x_{12}^2 x_{112} \\ &=q_{12}^2(q_{11}^4+q_{11}^3+q_{11}^2) x_{12}^2 x_{112} = 0. \end{aligned} {\tag{4.8}} \end{align}$$

For $(\operatorname {ad}_c x_2)z_5$, notice that on the one hand

$$\begin{align} \begin{aligned} \big [x_{12}, - \big ]_c^3 x_2 &= \sum _{k=0}^3 (-1)^k (q_{12}q_{22})^k q_{22}^{k(k-1)/2} \binom{3}{k}_{q_{22}} x_{12}^{3-k} x_2 x_{12}^k \\ &= x_{12}^3 x_i - q_{12}^3 x_i x_{12}^3 = - q_{12}^3 (\operatorname {ad}_c x_2) x_{12}^3. \end{aligned} {\tag{4.9}} \end{align}$$

On the other hand, using $[x_{12}, x_{2}]_c= q_{12}^2q_{22} x_{221}$ and (a) we get

$$\begin{align} \begin{aligned} \big [x_{12}, - \big ]_c^3 x_2 &= \Big [ x_{12},\big [ x_{12}, [x_{12}, x_{2}]_c \big ]_c \Big ]_c \\ &= q_{12}^2q_{22} \Big [ x_{12}, [x_{12}, x_{221}]_c \Big ]_c = \Big [ x_{12}, 0 \Big ]_c = 0, \end{aligned} {\tag{4.10}} \end{align}$$

so $(\operatorname {ad}_c x_2)z_5 = 0$. This shows that $\widehat{\mathcal{Z}}$ is a normal subalgebra.

Next we verify that $\Delta (z_i) \in \widehat{\mathcal{Z}} \otimes \widehat{\mathcal{Z}}$ for $i\in \mathbb{I}_5$. This is clear for $i \in \mathbb{I}_4$, because those elements are primitive in $T(V)$; for $i=5$ we compute in $T(V)$:

$$\begin{align} \begin{aligned} \Delta (x_{12}^3) =& x_{12}^3 \otimes 1 + 1 \otimes x_{12}^3 \\ &+(q^{-1}-q^{-2}) x_{112}\otimes x_{221} + (1-q^{-1})^3 q_{21}^{3} x_{1}^3\otimes x_2^3\\ &+ (1-q)^2 q_{21}^{3} x_{1112}\otimes x_2^2 - (1-q^{-1})^2q^{-1} x_1^2 \otimes x_{2221} \\ &+(q-1) x_1 \otimes [x_{12}, x_{221}]_c - (1-q^{-1}) q_{21} [x_{12}, x_{112}]_c \otimes x_2. \end{aligned} {\tag{4.11}} \end{align}$$

Using (a) and the defining relations of $\widehat{\mathscr{B}}$ we see that $\widehat{\mathcal{Z}}$ is a Hopf subalgebra.

(c) We show that any pair of generators of $\widehat{\mathcal{Z}}$ $q$-commute. By definition of $\widehat{\mathscr{B}}$, both $x_1$ and $x_2$ $q$-commute with $z_2$ and $z_3$, so $z_4$ and $z_1$ $q$-commute with $z_2$ and $z_3$. Secondly, 4.7 implies that $z_1$ and $z_4$ $q$-commute. Thirdly, (a) shows that $z_5$ $q$-commutes with $z_3$ and $z_2$, and also that $z_2$ and $z_3$ $q$-commute. Lastly, $z_5$ $q$-commutes with $z_4$ by 4.8, and with $z_1$ by 4.9 and 4.10. Hence $B$ linearly generates $\widehat{\mathcal{Z}}$.

The linear independence is proven by steps.

Step 1.

The set $\left\{ z_1^{n_1} z_2^{n_2} z_3^{n_3} z_4^{n_4} \colon n_i \in \mathbb{N}_0 \right\}$ is linearly independent.

Proof.

Consider the Hopf algebra $\widehat{\mathscr{B}} \# \mathbb{k}\mathbb{Z}^2$; let $A$ denote the subalgebra generated by $z_1, \dots , z_4$ and $\mathbb{Z}^2$. Since all the generators of $A$ are either skew-primitives or group-likes, it follows that $A$ itself is a pointed Hopf algebra. Notice that $z_1, \dots , z_4 \in \mathcal{P}(\widehat{\mathcal{Z}})$ are linearly independent. Indeed, they are non-zero because their $\mathbb{Z}$-degree is $3$, so they are linearly independent since their $\mathbb{Z}^2$-degrees are pairwise different (here we are using that the defining ideal of $\widehat{\mathscr{B}}$ is a Hopf ideal generated by $\mathbb{Z}^2$-homogeneous elements of $\mathbb{Z}$-degree $4$). Hence the infinitesimal braiding of $A$ contains the braided vector space $\mathbb{k}z_1 \oplus \dots \oplus \mathbb{k}z_4$, which is quantum linear space with all points labeled by $1$. Thus $\left\{ z_1^{n_1} z_2^{n_2} z_3^{n_3} z_4^{n_4} g \colon n_i \in \mathbb{N}_0, g \in \mathbb{Z}^2 \right\} \subset A$ is linearly independent.

■

Step 2.

The element $z_5$ does not belong to the left ideal $\widehat{\mathscr{B}}\langle z_1, z_2, z_3, z_4\rangle$.

Proof.

We verify this using GAP.

■

The ideal $\widehat{\mathscr{B}}\langle z_1, z_2, z_3, z_4\rangle$ is a Hopf ideal because the generators are primitive. Denote the quotient by $R$ and consider the projection $\pi _R \colon \widehat{\mathscr{B}} \twoheadrightarrow R$.

Step 3.

The set $\left\{ \pi _R(z_5)^{n} \colon n \in \mathbb{N}_0 \right\}$ is linearly independent.

Proof.

Consider the Hopf algebra $R \# \mathbb{Z}^2$. The subalgebra generated by $\pi _R(z_5)$ and $\mathbb{Z}^2$ is a pointed Hopf algebra. Moreover, its infinitesimal braiding contains $\pi _R(z_5)$, which is a non-zero point by Step 2 and is labeled by $1$. Now proceed as in the proof of Step 1.

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Step 4.

We have $(\operatorname {id}\otimes \pi _R) \Delta (z_5^n) = \sum _{k=0}^n \binom{n}{k} z_5^k \otimes \pi _R(z_5)^{n-k}$ for all $n\in \mathbb{N}_0$.

Proof.

The case $n=0$ is obvious, and $n=1$ follows from 4.11. An standard inductive argument for braided comultiplication yields the desired result.

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Step 5.

The set $B$ is linearly independent.

Proof.

Let $\sum _{n_1, \dots ,n_5 \in \mathbb{N}_0} \lambda _{n_1, \dots ,n_5} z_1^{n_1} z_2^{n_2} z_3^{n_3} z_4^{n_4} z_5^{n_5} = 0$. Assume there exists $n_5$ such that $\lambda _{n_1, \dots ,n_5}\ne 0$ for some $n_1, \dots , n_4 \in \mathbb{N}_0$; take $N$ as the maximal one. By Step 3 there is a linear map $f\colon R \to \mathbb{k}$ such that $f(\pi _R(z_5)^n) = \delta _{n,N}$ for all $n\in \mathbb{N}_0$. Now using Step 4 we compute

$$\begin{align*} 0 &=(\operatorname {id}\otimes f) (\operatorname {id}\otimes \pi _R) \Delta \left(\sum _{n_1, \dots ,n_5 \in \mathbb{N}_0} \lambda _{n_1, \dots ,n_5} z_1^{n_1} z_2^{n_2} z_3^{n_3} z_4^{n_4} z_5^{n_5} \right) \\ &=\sum _{n_1, \dots ,n_5 \in \mathbb{N}_0} \lambda _{n_1, \dots ,n_5} (\operatorname {id}\otimes f) (\operatorname {id}\otimes \pi _R) \left( \left( \prod _{i=1}^{4} \sum _{j=0}^{n_i} \binom{n_i}{j}z_i^{j} \otimes z_i^{n_i-j} \right)\Delta z_5^{n_5} \right) \\ &=\sum _{n_1, \dots ,n_4 \in \mathbb{N}_0, \ n_5\leq N} \lambda _{n_1, \dots ,n_5} (\operatorname {id}\otimes f) \left( \sum _{j=0}^{n_5} z_1^{n_1} z_2^{n_2} z_3^{n_3} z_4^{n_4} z_5^{j} \otimes \pi _R(z_5^{n_5-j}) \right) \\ &=\sum _{n_1, \dots ,n_4 \in \mathbb{N}_0} \lambda _{n_1, \dots ,n_4, N} z_1^{n_1} z_2^{n_2} z_3^{n_3} z_4^{n_4} \otimes 1. \end{align*}$$

This contradicts Step 1.

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(d) Since $\Delta (z_i) \in \widehat{\mathcal{Z}}\otimes \widehat{\mathcal{Z}}$ and $\widehat{\mathcal{Z}}$ is normal, the right ideal $\widehat{\mathscr{B}} \widehat{\mathcal{Z}} ^+$ is a Hopf ideal. By A+, Proposition 3.6 (c) we get that the equality $\widehat{\mathcal{Z}} = {}^{\operatorname {co}\pi }\widehat{\mathscr{B}}$ is equivalent to $\mathscr{B}_{\mathfrak{q}} \simeq \widehat{\mathscr{B}}/\widehat{\mathscr{B}} \widehat{\mathcal{Z}} ^+$. This last isomorphism holds because the diagram

$$\begin{align*} \begin{aligned} \vcenter{\img[][129pt][55pt][{$\xymatrix@C+8pt { \mathcal{J}_{\mathfrak{q}} \ar@{^{(}->}[r] \ar@{->>}[d] & T(V) \ar@{->>}[d] \ar@{->>}[dr] & \\ \widehat{\mathscr{B}} \widehat{\mathcal{Z}} ^+ \ar@{^{(}->}[r] & \widehat{\mathscr{B}} \ar@{->>}[r]^{\pi}& \mathscr{B}_{\mathfrak{q}} }$}]{Images/img4a4892e9596d2cfa11e734414b74cbb5.svg}} \end{aligned} \end{align*}$$

commutes.

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Proposition 4.11.

(a)

There is an extension of braided Hopf algebras$$\begin{align*} \mathbb{k}\to \widehat{\mathcal{Z}} \hookrightarrow \widehat{\mathscr{B}} \twoheadrightarrow \mathscr{B}_{\mathfrak{q}} \to \mathbb{k}. \end{align*}$$

(b)

The pre-Nichols algebra $\widehat{\mathscr{B}}$ is eminent and $\operatorname {GK-dim}\widehat{\mathscr{B}} = 5$.

Proof.

(a) Follows from Lemma 4.10 (d).

(b) We know that $\widehat{\mathscr{B}}$ covers all elements of $\mathfrak{Pre}_{\mathrm{fGK}}$ by Lemma 4.8; it remains to show that $\widehat{\mathscr{B}}$ itself belongs to $\mathfrak{Pre}_{\mathrm{fGK}}$. By A+, Proposition 3.6 (d) there is a right $\widehat{\mathcal{Z}}$-linear isomorphism $\mathscr{B}_{\mathfrak{q}} \otimes \widehat{\mathcal{Z}} \simeq \widehat{\mathscr{B}}$. Since $\mathscr{B}_{\mathfrak{q}}$ is finite dimensional, this implies that $\widehat{\mathscr{B}}$ is finitely generated as a $\widehat{\mathcal{Z}}$-module. Now KL, Proposition 5.5 provides $\operatorname {GK-dim}\widehat{\mathscr{B}} = \operatorname {GK-dim}\widehat{\mathcal{Z}} = 5$.

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4.3. Type $B_2$

Lemma 4.12.

Assume that $\mathbf{a}$ is of Cartan type $B_2$. Then the distinguished pre-Nichols algebra $\widetilde{\mathscr{B}}_{\mathfrak{q}}$ is eminent.

Proof.

Here $N>2$. We may fix a braiding matrix $\mathfrak{q}$ such that $q_{11}=q_{22}^2$, so $q= q_{22}$ and $\widetilde{q_{12}}=q^{-2}$. Let $\mathscr{B}$ be a finite GK-dimensional pre-Nichols algebra of $V$. It is enough to prove that $x_{112}=0=x_{2221}$ in $\mathscr{B}$.

Assume first $x_{112}\neq 0$, and consider the 3-dimensional braided subspace $W\coloneq \mathbb{k}x_1 \oplus \mathbb{k}x_2\oplus \mathbb{k}x_{112} \subset \mathcal{P}(\mathscr{B})$. Then $\operatorname {GK-dim}\mathscr{B}(W)<\infty$ from Lemma 2.8. We split the proof according to the several possibilities for $N$.

$\heartsuit$ $N=3$. Now the braiding on $W$ is of Cartan type

$$\begin{align*} \vcenter{\img[][143pt][21pt][{$\xymatrix{ \underset{1}{\overset{q^2}{\circ}} \ar@{-}[rr]^{q^{-2}} & & \underset{2}{\overset{q}{\circ} } \ar@{-}[rr]^{q^{-2}} & & \underset{112}{\overset{q^2}{\circ} }}$}]{Images/imgabd664061a1244e195956976e28023f3.svg}}, &&A= \begin{pmatrix} 2 & -1 & 0\\ -2&2&-2\\ 0&-1 & 2 \end{pmatrix}. \end{align*}$$

Since $A$ is of affine type $C_2^{(1)}$, this contradicts AAH2, Theorem 1.2(a).

$\heartsuit$ $N=6$. In this case the braiding on $W_2\coloneq \mathbb{k}x_2\oplus \mathbb{k}x_{112} \subset W$ is

$$\begin{align*} \vcenter{\img[][75pt][21pt][{$\xymatrix{ \underset{ 2 }{\overset{q}{\circ}} \ar@{-}[rr]^{q^{-2}} & & \underset{ 112 }{\overset{q^5}{\circ} }}$}]{Images/img46431153a8559bdbb922c704241afa77.svg}}, && \text{Cartan type } \begin{pmatrix} 2 & -2 \\ -4 & 2 \end{pmatrix}. \end{align*}$$

The Cartan matrix is of indefinite type, and by AAH2, Theorem 1.2(b) this contradicts $\operatorname {GK-dim}\mathscr{B}(W)<\infty$.

$\heartsuit$ $N\neq 3, 6$. The Dynkin diagram of $W_1\coloneq \mathbb{k}x_1\oplus \mathbb{k}x_{112} \subset W$ is

$$\begin{equation*} \mathfrak{D}_1= \vcenter{\img[][78pt][21pt][{$\xymatrix{ \underset{1}{\overset{q^2}{\circ}} \ar@{-}[rr]^{q^6} & & \underset{112}{\overset{q^5 }{\circ}}}$}]{Images/imge65067eb97ec118584a28dc4fa7f73d1.svg}}. \end{equation*}$$

Since $\operatorname {GK-dim}\mathscr{B}(W_1)<\infty$, it follows from AAH2, Theorem 1.2(b) that the associated root system is finite. Now $\mathfrak{D}_1$ is connected; by exhaustion on H1, Table 1, we deduce that we must have $N=4$ or $N=8$. We turn again to $W_2$, whose Dynkin diagram is easily computed in each case:

$$\begin{align*} \heartsuit \heartsuit N=4: && \vcenter{\img[][75pt][22pt][{$\xymatrix{ \underset{ 2 }{\overset{q}{\circ}} \ar@{-}[rr]^{q^{-2}} & & \underset{ 112 }{\overset{q}{\circ} }}$}]{Images/imgd58ca36aeab3453ed400814db737d940.svg}}, && \text{Cartan type } \begin{pmatrix} 2 & -2 \\ -2 & 2 \end{pmatrix}, \\ \heartsuit \heartsuit N=8: && \vcenter{\img[][75pt][21pt][{$\xymatrix{ \underset{ 2 }{\overset{q}{\circ}} \ar@{-}[rr]^{q^{-2}} & & \underset{ 112 }{\overset{q^5 }{\circ} }}$}]{Images/img0d9a1ce7cc19865f11a9df4a6adb783c.svg}}, && \text{Cartan type } \begin{pmatrix} 2 & -2\\ -2 & 2 \end{pmatrix}. \end{align*}$$

In any case the Cartan matrix is of affine type $A_1^{(1)}$, so $\operatorname {GK-dim}\mathscr{B}(W_2) = \infty$ by AAH2, Theorem 1.2(b).

Assume $x_{2221}\neq 0$ in $\mathscr{B}$. The subspace $U \coloneq \mathbb{k}x_1 \oplus \mathbb{k}x_2\oplus \mathbb{k}x_{2221} \subset \mathcal{P}(\mathscr{B})$ has dimension 3 and $\operatorname {GK-dim}\mathscr{B}(U)<\infty$. Now $U_1\coloneq \mathbb{k}x_1\oplus \mathbb{k}x_{2221} \subset U$ has connected Dynkin diagram

$$\begin{equation*} \vcenter{\img[][82pt][21pt][{$\xymatrix{ \underset{1}{\overset{q^2}{\circ}} \ar@{-}[rr]^{q^2} & & \underset{2221}{\overset{q^5 }{\circ}}}$}]{Images/imge03055a8b76eb4ec61f2a569ed4885b0.svg}}, \end{equation*}$$

and it is finite by AAH2, Theorem 1.2(b). By exhaustion on H1, Table 1 we deduce that $N=4$. Then the Dynkin diagram of $U$ is of Cartan type

$$\begin{align*} \vcenter{\img[][148pt][20pt][{$\xymatrix{ \underset{2}{\overset{q}{\circ}} \ar@{-}[rr]^{-1} & & \underset{1}{\overset{-1}{\circ} } \ar@{-}[rr]^{-1} & & \underset{2221}{\overset{q}{\circ} }}$}]{Images/imgf3440a60326aefb186420755eff231eb.svg}}, &&A= \begin{pmatrix} 2 & -2 & 0\\ -1&2&-1\\ 0&-2 & 2 \end{pmatrix}. \end{align*}$$

Since $A$ is of affine type $C_2^{(1)}$, this contradicts AAH2, Theorem 1.2(a).

■

4.4. Type $G_2$

Lemma 4.13.

Assume that $\mathbf{a}$ is of Cartan type $G_2$. Then the quantum Serre relations hold in any $\mathscr{B} \in \mathfrak{Pre}_{\mathrm{fGK}}$. In particular, the distinguished pre-Nichols algebra $\widetilde{\mathscr{B}}_{\mathfrak{q}}$ is eminent if $N\neq 4, 6$.

Proof.

Here $N>3$. Let $\mathscr{B} \in \mathfrak{Pre}_{\mathrm{fGK}}(V)$; we show first that the quantum Serre relations $x_{11112}=0=x_{221}$ hold in $\mathscr{B}$.

Start assuming $x_{11112}\neq 0$. Then the 3-dimensional subspace $W\coloneq \mathbb{k}x_1 \oplus \mathbb{k}x_2\oplus \mathbb{k}x_{11112} \subset \mathcal{P}(\mathscr{B})$ satisfies $\operatorname {GK-dim}\mathscr{B}(W)\leq \operatorname {GK-dim}\mathscr{B}$ by Lemma 2.8. The Dynkin diagram of $W_1\coloneq \mathbb{k}x_1\oplus \mathbb{k}x_{11112} \subset W$ is

$$\begin{equation*} \mathfrak{D}_1= \vcenter{\img[][82pt][21pt][{$\xymatrix{ \underset{1}{\overset{q}{\circ}} \ar@{-}[rr]^{q^5} & & \underset{11112}{\overset{q^7 }{\circ}}}$}]{Images/imgb9da8c7a507c5484dd82c7139ec3be60.svg}}. \end{equation*}$$

Since $\operatorname {GK-dim}\mathscr{B}(W_1)<\infty$, it follows from AAH2, Theorem 1.2(b) that the root system of $\mathfrak{D}_1$ is finite. We split the proof according to the several possibilities for $N$.

$\heartsuit$ $N=5$. The diagram $\mathfrak{D}_1$ is disconnected, but we might consider instead $W_2\coloneq \mathbb{k}x_{11112}\oplus \mathbb{k}x_2\subset W$, that satisfies $\operatorname {GK-dim}\mathscr{B}(W_2)<\infty$ as well. By direct computation $W_2$ is of indefinite Cartan type:

$$\begin{align*} \vcenter{\img[][85pt][21pt][{$\xymatrix{ \underset{11112}{\overset{q^2}{\circ}} \ar@{-}[rr]^{q^{-6}} & & \underset{2}{\overset{q^3}{\circ} }}$}]{Images/img9dcbbc77f93e124a898af8608c6a53aa.svg}}, && \begin{pmatrix} 2 & -3 \\ -2 & 2 \end{pmatrix}, \end{align*}$$

which is in contradiction with AAH2, Theorem 1.2(a).

$\heartsuit$ $N\neq 5$. Now $\mathfrak{D}_1$ is connected and finite; by inspection on H1, Table 1, we must have $N=4$ or $N=6$.

$\heartsuit \heartsuit N=4$. In this case $W_2$ is of Cartan type

$$\begin{align*} \vcenter{\img[][85pt][21pt][{$\xymatrix{ \underset{11112}{\overset{q^3}{\circ}} \ar@{-}[rr]^{q^{-6}} & & \underset{2}{\overset{q^3}{\circ} }}$}]{Images/imga9e0ad64644b4395f9af64a507782197.svg}}, && \begin{pmatrix} 2 & -2 \\ -2 & 2 \end{pmatrix}, \end{align*}$$

which is of affine type $A_1^{(1)}$, now contradicting AAH2, Theorem 1.2(b).

$\heartsuit \heartsuit N=6$. In this case the Dynkin diagram of $W$ is of Cartan type

$$\begin{align*} \vcenter{\img[][149pt][21pt][{$\xymatrix{ \underset{11112}{\overset{q}{\circ}} \ar@{-}[rr]^{q^{-1}} & & \underset{1}{\overset{q}{\circ} } \ar@{-}[rr]^{q^{-3}} & & \underset{2}{\overset{q^3}{\circ} }}$}]{Images/img205fdb694158a26bb90827c5ea150eeb.svg}}, &&A=\begin{pmatrix} 2 & -1 & 0\\ -1&2&-3\\ 0&-1 & 2 \end{pmatrix}. \end{align*}$$

By AAH2, Theorem 1.2(b) this contradicts $\operatorname {GK-dim}\mathscr{B}(W) < \infty$, since $A$ is of affine type $G_2^{(1)}$.

\kern2.22198ptAssume now $x_{221}\neq 0$ in $\mathscr{B}$. The subspace $U \coloneq \mathbb{k}x_1 \oplus \mathbb{k}x_2\oplus \mathbb{k}x_{221} \subset \mathcal{P}(\mathscr{B})$ has dimension 3 and $\operatorname {GK-dim}\mathscr{B}(U)<\infty$. Consider two possibilities for $N$.

$\heartsuit N\neq 4$. Now $U_1\coloneq \mathbb{k}x_1\oplus \mathbb{k}x_{221} \subset U$ has connected Dynkin diagram

$$\begin{equation*} \vcenter{\img[][75pt][21pt][{$\xymatrix{ \underset{1}{\overset{q}{\circ}} \ar@{-}[rr]^{q^{-4}} & & \underset{221}{\overset{q^7 }{\circ}}}$}]{Images/img634ea12aa0b479549fcbf909bcda5e07.svg}}. \end{equation*}$$

By exhaustion on H1, Table 1 we conclude that this diagram is never finite, which contradicts AAH2, Theorem 1.2(b), as $\operatorname {GK-dim}\mathscr{B}(U_1)<\infty$.

$\heartsuit N=4$. In this case the braiding on $U$ is of Cartan type

$$\begin{align*} \vcenter{\img[][142pt][21pt][{$\xymatrix{ \underset{1}{\overset{q}{\circ}} \ar@{-}[rr]^{q^{-3}} & & \underset{2}{\overset{q^{3}}{\circ} } \ar@{-}[rr]^{q^{-3}} & & \underset{221}{\overset{q^{3}}{\circ} }}$}]{Images/imgae28ac8115a74f16aa33c34b8dc6f51f.svg}}, &&A= \begin{pmatrix} 2 & -3 & 0\\ -1&2&-1\\ 0&-1 & 2 \end{pmatrix}. \end{align*}$$

Since $A$ is of affine type $G^{(1)}_2$, this contradicts AAH2, Theorem 1.2(a).

Thus the quantum Serre relations hold in $\mathscr{B}$. By Remark 4.1 this proves the assertion regarding $N\neq 4, 6$.

■

4.5. Type $A_3$

Lemma 4.14.

If $\mathbf{a}$ is of Cartan type $A_3$ with $N>2$, then $\widetilde{\mathscr{B}}_{\mathfrak{q}}$ is eminent.

Proof.

As $N>2$, the ideal $\mathcal{I}_{\mathfrak{q}}$ is generated by the quantum Serre relations $x_{13}=0$ and $x_{iij}=0$ for $|j-i|=1$, cf. AA2, p. 397. Let $\mathscr{B} \in \mathfrak{Pre}_{\mathrm{fGK}}(\mathfrak{q})$. Then $x_{13}=0$ holds in $\mathscr{B}$ since the braided vector space $\mathbb{k}x_1\oplus \mathbb{k}x_3$ satisfies the hypothesis in Proposition 3.2.

Turn to $x_{iij}$ for some fix $i,j \in \mathbb{I}_3$ with $|j-i|=1$; in this case $\mathbb{k}x_i\oplus \mathbb{k}x_j$ is of Cartan type $A_2$. If $N>3$, then $x_{iij}=0$ in $\mathscr{B}$ by Lemma 4.7. Only the case $N=3$ remains. Now we have $\mathfrak{q}(2\alpha _i+\alpha _j, 2\alpha _i+\alpha _j )=q_{ii}^4\widetilde{q_{ij}}^2q_{jj}=q^5q^{-2}=1$. Using AAH1, Lemma 2.8, in order to guarantee $x_{iij}=0$ in $\mathscr{B}$ it is enough to find $k \in \mathbb{I}_3$ such that $\widetilde{\mathfrak{q}} (\alpha _k, 2\alpha _i+\alpha _j) \neq 1$. It is straightforward to verify that the unique $k \in \mathbb{I}_3$ different from $i$ and $j$ does the trick.

■

4.6. Types $B_3$ and $C_3$

Lemma 4.15.

The distinguished pre-Nichols algebra $\widetilde{\mathscr{B}}_{\mathfrak{q}}$ is eminent if either

(i)

$\mathbf{a}$ is of type $B_3$, or

(ii)

$\mathbf{a}$ is of type $C_3$.

Proof.

Let $\mathscr{B} \in \mathfrak{Pre}_{\mathrm{fGK}}(\mathfrak{q})$. Then $x_{13}=0$ holds in $\mathscr{B}$. Indeed, the braided vector space $\mathbb{k}x_1\oplus \mathbb{k}x_3$ satisfies the hypothesis in Proposition 3.2. Similarly, since $\mathbb{k}x_2 \oplus \mathbb{k}x_3$ is of type $B_2$, it follows from Lemma 4.12 that the quantum Serre relations involving $x_2$ and $x_3$ hold in $\mathscr{B}$.

Step 1.

If $\mathbf{a}$ is of Cartan type $B_3$, then the quantum Serre relations hold in any finite $\operatorname {GK-dim}$ pre-Nichols algebra.

Proof.

Here $\mathbb{k}x_1 \oplus \mathbb{k}x_2$ has Dynkin diagram $\vcenter{\img[][45pt][14pt][{$\xymatrix{\overset{q^2}{\circ}\ar@{-}[r]^{q^{-2}} &\overset{q^2}{\circ}}$}]{Images/imgbe77558e3464ce0c439179e238ecaba1.svg}}$, type $A_2$. Hence, if $\operatorname {ord}q^2 > 3$, we know from Lemma 4.7 that the quantum Serre relations between $x_1$ and $x_2$ hold in $\mathscr{B}$. Let us show that in the cases $\operatorname {ord}q^2 =2, 3$ the same happens.

$\heartsuit$ $\operatorname {ord}q^2 =2$. If $x_{112} \neq 0$ in $\mathscr{B}$, we get a subspace $\mathbb{k}x_2 \oplus \mathbb{k}x_3 \oplus \mathbb{k}x_{112} \subset \mathcal{P}(\mathscr{B})$ of dimension $3$ with the following Dynkin diagram

$$\begin{align*} \vcenter{\img[][144pt][20pt][{$\xymatrix{ \underset{2}{\overset{-1}{\circ}} \ar@{-}[rr]^{-1} & & \underset{3}{\overset{q}{\circ} } \ar@{-}[rr]^{-1} & & \underset{112}{\overset{-1}{\circ}} }$}]{Images/img92e36667ebcb9ff8fa5c0fada51674e3.svg}}, && \text{Cartan type } \begin{pmatrix} 2 & -1 & 0\\ -2&2&-2\\ 0&-1 & 2 \end{pmatrix}. \end{align*}$$

This matrix is of affine type $C_2^{(1)}$, hence $\operatorname {GK-dim}\mathscr{B} = \infty$, a contradiction.

Similarly, the assumption $x_{221} \neq 0$ yields a subspace of $\mathcal{P}(\mathscr{B})$ with braiding

$$\begin{align*} \begin{aligned} \vcenter{\img[][135pt][71pt][{$\xymatrix@C+24pt { & \overset{-1}{\underset{221}{\circ}} \\ \overset{-1}{\underset{1}{\circ}} \ar@{-}[r]^{-1} & \overset{-1}{\underset{2}{\circ}} \ar@{-}[u]^{-1}_{} \ar@{-}[r]^{-1} & \overset{q}{\underset{3}{\circ}},}$}]{Images/imgc74ed6dc2a39652fda6af1bd5a1a0871.svg}} \end{aligned} & &\text{Cartan type } \begin{pmatrix} 2&-1&0&0\\ -1&2&-1&-1 \\ 0&-2&2&0 \\ 0&-1&0&2 \end{pmatrix}. \end{align*}$$

The Cartan matrix is of affine type $B_3^{(1)}$, and again $\operatorname {GK-dim}(\mathscr{B}) = \infty$.

$\heartsuit$ $\operatorname {ord}q^2 =3$. Notice that

$$\begin{align*} \mathfrak{q}(2\alpha _1+\alpha _2, 2\alpha _1+\alpha _2)&=q^6=1, & \widetilde{\mathfrak{q}}(2\alpha _1+\alpha _2, \alpha _3)&=q^{-2}\neq 1; \\ \mathfrak{q}(\alpha _1+2\alpha _2, \alpha _1+2\alpha _2)&=q^6=1, & \widetilde{\mathfrak{q}}(\alpha _1+2\alpha _2, \alpha _3)&=q^{-4}\neq 1. \end{align*}$$

Assuming $x_{112}\neq 0$ in $\mathscr{B}$ we get $\mathbb{k}x_3 \oplus \mathbb{k}x_{112} \subset \mathcal{P}(\mathscr{B})$ with Dynkin diagram $\vcenter{\img[][39pt][14pt][{$\xymatrix{\overset{q}{\circ}\ar@{-}[r]^{q^{-2}} &\overset{1}{\circ}}$}]{Images/img35ecdfbc60d1bea59e739b687ed2cc97.svg}}$. Then by AAH1, Lemma 2.8 it follows that $\operatorname {GK-dim}\mathscr{B} =\infty$, a contradiction. By the same argument, we can not have $x_{221}\neq 0$ in $\mathscr{B}$.

■

The assertion (i) for $N>4$ follows since, in that case, $\widetilde{\mathscr{B}}_{\mathfrak{q}}$ is presented by the quantum Serre relations, cf. Remark 4.1.

Step 2.

If $\mathbf{a}$ is of Cartan type $B_3$ with $N=3$, then $\widetilde{\mathscr{B}}_{\mathfrak{q}}$ is eminent.

Proof.

By AA2, pp. 399, 400, $\widetilde{\mathscr{B}}_{\mathfrak{q}}$ is presented by the quantum Serre relations and $[x_{3321}, x_{32}]_c=0$. Given $\mathscr{B} \in \mathfrak{Pre}_{\mathrm{fGK}}$, let us show that $[x_{3321}, x_{32}]_c \in \mathcal{P}(\mathscr{B})$. Using $x_{13}=0$ a straightforward computation gives

$$\begin{align*} \Delta (x_{3321}) =& x_{3321}\otimes 1 + 1\otimes x_{3321} + (1-q_{33})x_{332}\otimes x_1 \\ &+ (1-q_{33})q_{33} x_3 \otimes x_{321} + (1-q_{33}) (1-q_{22}) x_3^2 \otimes x_{21}. \end{align*}$$

With this we compute

$$\begin{align*} \Delta ([x_{3321}, x_{32}]_c) =& [x_{3321}, x_{32}]_c\otimes 1 + 1\otimes [x_{3321}, x_{32}]_c \\ &- (1-q_{33})^2(1-q_{22})q_{12} \, x_3^2\otimes x_{221}\\ &- (1-q_{33})q_{12}q_{13}q_{23} q_{33} \, x_{3332} \otimes x_{21}\\ &+ (1-q_{33})q_{33}^2q_{12}q_{13} \, x_{332} \otimes (x_{321}-[x_{32},x_1]_c)\\ &- (1-q_{33})q_{23}q_{13} \, x_{33321} \otimes x_2\\ &+ (1-q_{33})q_{13}q_{12}\, [x_{332},x_{32}]_c \otimes x_1 \\ &+ (1-q_{33})q_{33} \, x_3 \otimes (q_{13}q_{23}q_{33} [x_{3321},x_2]_c +[x_{321},x_{32}]_c) \end{align*}$$

The third and fourth terms vanish in $\mathscr{B}$ by Step 1. For the fifth term, a straightforward computation involving $x_{13}=0$ shows that $x_{321}=[x_{32},x_1]_c$. The last three terms also vanish, but they require a more detailed analysis.

$\heartsuit$ $x_{33321}=0$ in $\mathscr{B}$. Notice that

$$\begin{align*} \Delta (x_{33321}) =& x_{33321}\otimes 1 + 1\otimes x_{33321} \\ &+ (1-q_{33})x_{3332}\otimes x_1 - (1-q_{33}^2)q_{32} x_{332} \otimes x_{31}, \end{align*}$$

so this element is primitive in $\mathscr{B}$ by Step 1. Assuming $x_{33321}\neq 0$ we get a subspace $\mathbb{k}x_1 \oplus \mathbb{k}x_{33321} \subset \mathcal{P}(\mathscr{B})$ where the braiding is given by

$$\begin{align*} \vcenter{\img[][86pt][21pt][{$\xymatrix{ \underset{1}{\overset{q^2}{\circ}} \ar@{-}[rr]^{q^{-1}} & & \underset{33321}{\overset{q^2}{\circ} }}$}]{Images/img572aa66c21bf1665ca777180032d06cf.svg}}, && \text{ Cartan type } \begin{pmatrix} 2 & -2 \\ -2 & 2 \end{pmatrix}, && \text{ affine type } A_1^{(1)}. \end{align*}$$

this contradicts $\operatorname {GK-dim}\mathscr{B} <\infty$ by AAH2, Theorem 1.2.

$\heartsuit$ $[x_{332},x_{32}]_c=0$ in $\mathscr{B}$. Now we have

$$\begin{align*} \Delta ([x_{332}, x_{32}]_c) =& [x_{332}, x_{32}]_c\otimes 1 + 1\otimes [x_{332}, x_{32}]_c \\ &- (1-q_{33})^2q_{23} \, x_3^2\otimes [x_{32}, x_{2}]_c - (1-q_{33})q_{33}q_{23} \, x_{3332} \otimes x_2. \end{align*}$$

The element $[x_{32}, x_2]_c$ is primitive in $\mathscr{B}$, so it vanishes by the same reason that $x_{223}$ does (cf. proof of Lemma 4.12). So $[x_{332}, x_{32}]_c \in \mathcal{P}(\mathscr{B})$ by Step 1. If it is non-zero, consider $\mathbb{k}x_1 \oplus \mathbb{k}[x_{332},x_{32}]_c \subset \mathcal{P}(\mathscr{B})$ where the braiding is

$$\begin{align*} \vcenter{\img[][100pt][23pt][{$\xymatrix{ \underset{x_1}{\overset{q^2}{\circ}} \ar@{-}[rr]^{q^{-1}} & & \underset{[x_{332}, x_{32}]}{\overset{q^2}{\circ} }}$}]{Images/img980f09159cde237196009b8d319a3441.svg}}, && \text{ Cartan type } \begin{pmatrix} 2 & -2 \\ -2 & 2 \end{pmatrix}, && \text{ affine type } A_1^{(1)}, \end{align*}$$

thus we get the same contradiction as with $x_{33321}$.

$\heartsuit$ $q_{13}q_{23}q_{33} [x_{3321},x_2]_c +[x_{321},x_{32}]_c=0$. Denote this element by $r$. Then

$$\begin{align*} \Delta (r) =& r \otimes 1 + 1\otimes r + (1-q_{33})q_{22}q_{12}q_{13} \, x_{32}\otimes (x_{321}-[x_{32}, x_{1}])\\ & + (1-q_{33})q_{33}q_{12}q_{13}q_{23} \, \big [x_3, [x_{32},x_2]\big ] \otimes x_1\\ &- (1-q_{33})q_{22}q_{12}q_{13}q_{23} \, x_3 \otimes \big [ [x_{32},x_2], x_1\big ]. \end{align*}$$

Since $[x_{32},x_2]=0$ and $x_{321}-[x_{32}, x_{1}]=0$ in $\mathscr{B}$, it follows that $r$ is primitive. If $r\neq 0$ we consider $\mathbb{k}x_2 \oplus \mathbb{k}r \subset \mathcal{P}(\mathscr{B})$. The Dynkin diagram is computed:

$$\begin{align*} \vcenter{\img[][75pt][21pt][{$\xymatrix{ \underset{2}{\overset{q^2}{\circ}} \ar@{-}[rr]^{q^{-1}} & & \underset{r}{\overset{q^2}{\circ} }}$}]{Images/imgf3a89db896083235a11a4c6eac452da4.svg}}, && \text{ Cartan type } \begin{pmatrix} 2 & -2 \\ -2 & 2 \end{pmatrix}, && \text{ affine type } A_1^{(1)}, \end{align*}$$

thus we get the same contradiction as before.

Using these three $\heartsuit$ we get $[x_{3321}, x_{32}]_c \in \mathcal{P}(\mathscr{B})$. If this element is non-zero, consider $U=\mathbb{k}x_3 \oplus \mathbb{k}[x_{3321}, x_{32}]_c\subset \mathcal{P}(\mathscr{B})$. We compute the braiding:

$$\begin{align*} \mathfrak{q}(\alpha _1+2\alpha _2+3\alpha _3,\alpha _1+2\alpha _2+3\alpha _3) &=1, && \widetilde{\mathfrak{q}}(\alpha _1+2\alpha _2+3\alpha _3,\alpha _3) = q^{-1}\neq 1. \end{align*}$$

From AAH1, Lemma 2.8 it follows that $\operatorname {GK-dim}\mathscr{B}(U)=\infty$, but this contradicts $\operatorname {GK-dim}\mathscr{B} < \infty$. Then $[x_{3321}, x_{32}]_c=0$ in $\mathscr{B}$ and Step 2 holds.

■

Step 3.

If $\mathbf{a}$ is of Cartan type $B_3$ with $N=4$, then $\widetilde{\mathscr{B}}_{\mathfrak{q}}$ is eminent.

Proof.

By AA2, pp. 399, 400, $\widetilde{\mathscr{B}}_{\mathfrak{q}}$ is presented by the quantum Serre relations and $[x_{123}, x_2]_c=0$. We claim that this element is primitive in any $\mathscr{B} \in \mathfrak{Pre}_{\mathrm{fGK}}$. Indeed, using that $x_{13}=0$ in $\mathscr{B}$, we get

$$\begin{align*} \Delta ([x_{123}, x_2]_c) =& [x_{123}, x_2]_c\otimes 1 + 1\otimes [x_{123}, x_2]_c \\ &-(1-\widetilde{q_{12}})q_{32} x_1 \otimes x_{223} + (1-\widetilde{q_{23}})q_{32} [x_{12}, x_2]_c \otimes x_3. \end{align*}$$

By straightforward computations, $[x_{12}, x_2]_c \in \mathcal{P} (\mathscr{B})$ and it vanishes by the same reason that $x_{221}$ does (cf. proof of Lemma 4.7). Since $x_{223}=0$, the claim follows.

Assume $[x_{123}, x_2]_c\neq 0$. Inside $\mathcal{P} (\mathscr{B})$ we have the 2-dimensional subspace $U=\mathbb{k}x_3 \oplus \mathbb{k}[x_{123}, x_2]_c$ where the braiding is given by

$$\begin{align*} \vcenter{\img[][97pt][20pt][{$\xymatrix{ \underset{x_3}{\overset{q}{\circ}} \ar@{-}[rr]^{-1} & & \underset{[x_{123}, x_2]}{\overset{-q}{\circ} }}$}]{Images/img2eec6ff9148210fc01c6612d0c80f47c.svg}}, && \text{ Cartan type } \begin{pmatrix} 2 & -2 \\ -2 & 2 \end{pmatrix}. \end{align*}$$

Since this matrix is of affine type $A_1^{(1)}$, from AAH2, Theorem 1.2(b) it follows $\operatorname {GK-dim}\mathscr{B}(U)=\infty$, contradicting $\mathscr{B} \in \mathfrak{Pre}_{\mathrm{fGK}}$.

■

Step 4.

If $\mathbf{a}$ is of Cartan type $C_3$, then the quantum Serre relations hold in any finite $\operatorname {GK-dim}$ pre-Nichols algebra.

Proof.

Now $\mathbb{k}x_1 \oplus \mathbb{k}x_2$ has Dynkin diagram $\vcenter{\img[][39pt][14pt][{$\xymatrix{\overset{q}{\circ}\ar@{-}[r]^{q^{-1}} &\overset{q}{\circ}}$}]{Images/img9fc9853690a84238ed6767f3a844afcc.svg}}$, type $A_2$. If $N>3$, then the quantum Serre relations in $x_1$ and $x_2$ hold by Lemma 4.7. For the case $N=3$, let $i, j$ such that $\{i, j\}=\{1,2\}$ and suppose $x_{iij}\neq 0$ in $\mathscr{B}$. Since $\mathfrak{q}(2\alpha _i + \alpha _j, 2\alpha _i + \alpha _j) = q^5 q^{-2}=1$ and $\widetilde{\mathfrak{q}}(2\alpha _i + \alpha _j, \alpha _3)=\widetilde{q_{i3}}^2\widetilde{q_{j3}}\neq 1$, we get $\operatorname {GK-dim}\mathscr{B} = \infty$ by AAH1, Lemma 2.8.

■

The assertion (ii) for $N>4$ follows since, in that case, $\widetilde{\mathscr{B}}_{\mathfrak{q}}$ is presented by the quantum Serre relations, see Remark 4.1.

Step 5.

If $\mathbf{a}$ is of Cartan type $C_3$ with $N=3$, then $\widetilde{\mathscr{B}}_{\mathfrak{q}}$ is eminent.

Proof.

Following AA2, pp. 401, 402) we see that $\widetilde{\mathscr{B}}_{\mathfrak{q}}$ is presented by the quantum Serre relations and $\big [ [x_{123}, x_2]_c, x_2\big ]_c=0$. Given $\mathscr{B} \in \mathfrak{Pre}_{\mathrm{fGK}}$, let us show that this element is primitive in $\mathscr{B}$. Using $x_{13}=0$ it follows that

$$\begin{align*} \Delta ([x_{123}, x_2]_c) =& [x_{123}, x_2]_c\otimes 1 + 1\otimes [x_{123}, x_2]_c + (1-\widetilde{q_{12}}) x_{123} \otimes x_2 \\ &+ (1 - q^2) x_{12}\otimes x_{32} + (1-\widetilde{q_{12}})x_1 \otimes (x_{23}x_2-q_{32}x_2x_{23}) \\ &+ (1-\widetilde{q_{23}})q_{32} [x_{12}, x_2]_c \otimes x_3. \end{align*}$$

By straightforward computations, $[x_{12}, x_2]_c = q_{12}^2qx_{221}$ in $T(V)$, and so $[x_{12}, x_2]_c$ vanishes in $\mathscr{B}$ by Step 4. Then we obtain

$$\begin{align*} \Delta &\big ( \big [ [x_{123}, x_2]_c, x_2\big ]_c\big ) = \big [ [x_{123}, x_2]_c, x_2\big ]_c \otimes 1 + 1\otimes \big [ [x_{123}, x_2]_c, x_2\big ]_c \\ &+ (1-q^2)q_{32}q_{22}[x_{12}, x_2]_c \otimes x_{32} + (1-\widetilde{q_{12}})q_{22}q_{32}^2 x_1 \otimes x_{2223}, \end{align*}$$

and now the claim follows from Step 4.

If $\big [ [x_{123}, x_2]_c, x_2\big ]_c\neq 0$, consider $U=\mathbb{k}x_1\oplus \mathbb{k}\big [ [x_{123}, x_2]_c, x_2\big ]_c\subset \mathcal{P}(\mathscr{B})$. By AAH1, Lemma 2.8, since $\mathfrak{q}(\alpha _1+3\alpha _2+\alpha _3,\alpha _1+3\alpha _2+\alpha _3)= q^{12}q^{-9} = 1$ and $\widetilde{\mathfrak{q}}(\alpha _1+3\alpha _2+\alpha _3,\alpha _1)=q^2q^{-3}\neq 1$, we have $\operatorname {GK-dim}\mathscr{B}(U)=\infty$. This contradicts $\mathscr{B} \in \mathfrak{Pre}_{\mathrm{fGK}}$.

■

Step 6.

If $\mathbf{a}$ is of Cartan type $C_3$ with $N=4$, then $\widetilde{\mathscr{B}}_{\mathfrak{q}}$ is eminent.

Proof.

By An1, Theorem 3.1, $\widetilde{\mathscr{B}}_{\mathfrak{q}}$ is presented by the quantum Serre relations and $[x_{123}, x_{23}]_c=0$. Let us show that this element is primitive in any pre-Nichols algebra $\mathscr{B}$ of finite $\operatorname {GK-dim}$.

First we claim that $[x_{123}, x_3]_c=0$ in $\mathscr{B}$: using that $x_{13}=0$ we compute

$$\begin{align*} \Delta \big ( [x_{123}, x_3]_c\big ) =& [x_{123}, x_3]_c \otimes 1 + 1\otimes [x_{123}, x_3]_c \\ &+ (1-\widetilde{q_{12}}q_{23}q_{33})x_{13}\otimes x_{23} + (1-\widetilde{q_{12}}) x_1 \otimes [x_{23},x_3]_c. \end{align*}$$

Since $[x_{23}, x_3]_c \in \mathcal{P} (\mathscr{B})$, it vanishes in $\mathscr{B}$ by the same reason that $x_{332}$ does (cf. proof of Lemma 4.12). So $[x_{123}, x_3]_c \in \mathcal{P}(\mathscr{B})$. Hence, if it is non-zero we get a subspace $U=\mathbb{k}x_1 \oplus [x_{123}, x_3]_c \subset \mathcal{P}(\mathscr{B})$ where the braiding is given by

$$\begin{align*} \vcenter{\img[][97pt][23pt][{$\xymatrix{ \underset{x_3}{\overset{q}{\circ}} \ar@{-}[rr]^{q^{-3}} & & \underset{[x_{123}, x_3]}{\overset{q}{\circ} }}$}]{Images/img0a926c478c6662af72d99feef0bab253.svg}}, &&\text{ indefinite Cartan type } \begin{pmatrix} 2 & -3 \\ -3 & 2 \end{pmatrix}. \end{align*}$$

But then $\operatorname {GK-dim}\mathscr{B}(U)=\infty$ by AAH2, Theorem 1.2(b), a contradiction.

Next we compute

$$\begin{align*} \Delta &\big ( [x_{123}, x_{23}]_c\big ) =[x_{123}, x_{23}]_c \otimes 1 + 1\otimes [x_{123}, x_{23}]_c \\ &+ (1-\widetilde{q_{23}})q_{12}q_{22}q_{32}x_{2}\otimes [x_{123}, x_3] + (1-\widetilde{q_{23}})^2q_{32} x_1 \otimes [x_{12},x_2]_c. \end{align*}$$

Using the previous claim and the fact $[x_{12},x_2]_c=q_{12}^2qx_{221}=0$ (by Step 4), we get $[x_{123}, x_{23}]_c \in \mathcal{P}(\mathscr{B})$. If $[x_{123}, x_{23}]_c \neq 0$, consider the subspace $W = \mathbb{k}x_1 \oplus \mathbb{k}x_2 \oplus \mathbb{k}[x_{123}, x_{23}]_c \subset \mathcal{P}(\mathscr{B})$, where the braiding is

$$\begin{align*} \vcenter{\img[][160pt][23pt][{$\xymatrix{ \underset{1}{\overset{q}{\circ}} \ar@{-}[rr]^{q^{-1}} & & \underset{2}{\overset{q}{\circ} } \ar@{-}[rr]^{q^{-1}} & & \underset{[x_{123}, x_{23}]}{\overset{q^{3}}{\circ} }}$}]{Images/img1196c50b79a6630582153337fda233f4.svg}}, && \text{ Cartan type }\begin{pmatrix} 2 & -1 & 0\\ -1&2&-1\\ 0&-3 & 2 \end{pmatrix}. \end{align*}$$

Since the Cartan matrix is of affine type $G^{(1)}_2$, it follows $\operatorname {GK-dim}\mathscr{B}(W) =\infty$ by AAH2, Theorem 1.2(a). This contradicts $\mathscr{B} \in \mathfrak{Pre}_{\mathrm{fGK}}$.

■

The result follows.

■

4.7. Some cases in rank $> 3$

Here we assume that $\theta \ge 4$.

Lemma 4.16.

In any of the following cases, $\widetilde{\mathscr{B}}_{\mathfrak{q}}$ is eminent.

(a)

$\mathbf{a}$ is of Cartan type with simply laced Dynkin diagram and $N>2$.

(b)

$\mathbf{a}$ is of type $B_\theta$, $C_\theta$ ($\theta \ge 4$) or $F_4$, and $N>4$.

Proof.

By Remark 4.1 and the restrictions on $N$, $\widetilde{\mathscr{B}}_{\mathfrak{q}}$ is presented by the quantum Serre relations. Let $\mathscr{B} \in \mathfrak{Pre}_{\mathrm{fGK}}(\mathfrak{q})$. If $a_{ij} = 0$, then $x_{ij}=0$ holds in $\mathscr{B}$ by Proposition 3.2. If $a_{ij} \neq 0$, then there is $k \in \mathbb{I}$ such that $\{i,j,k\}$ span a subdiagram of type $A_3$, $B_3$ or $C_3$. Then $(\operatorname {ad}x_i)^{1-a_{ij}} (x_j)=0$ by Lemmas 4.14 or 4.15. Thus $\widetilde{\mathscr{B}}_{\mathfrak{q}} \twoheadrightarrow \mathscr{B}$.

■

In the next subsections we treat some remaining cases with small $N$.

4.8. Types $B_{\theta }$, $C_{\theta }$, $F_4$, $\theta > 3$, $N = 3, 4$

Lemma 4.17.

If $\mathbf{a}$ is of types $B_{\theta }$, $C_{\theta }$, with $\theta > 3$, or $F_4$, and $N = 3$ or $4$, then $\widetilde{\mathscr{B}}_{\mathfrak{q}}$ is eminent.

Proof.

We split the proof according to the type. Let $\mathscr{B} \in \mathfrak{Pre}_{\mathrm{fGK}}$.

$\heartsuit$ Type $F_4$. Here $\widetilde{\mathscr{B}}_{\mathfrak{q}}$ is presented by the quantum Serre relations and

$$\begin{align} [x_{123}, x_{23}]_c=[x_{432} , x_3]_c= 0 \text{ if } N=4; && [x_{2234} , x_{23}]_c = 0 \text{ if } N=3. {\tag{4.12}} \end{align}$$

Since $N>2$ we get $x_{14}=0$ in $\mathscr{B}$ from Proposition 3.2. The subdiagram spanned by $\{1,2,3\}$ is of type $C_3$ thus the quantum Serre relations involving these indices hold in $\mathscr{B}$ by Lemma 4.15 (ii). Finally, $\{4,3,2\}$ span a diagram of type $B_3$ so the quantum Serre relations involving these indices hold in $\mathscr{B}$ by Lemma 4.15 (i). Moreover 4.12 are defining relations of the distinguished pre-Nichols algebra of type $B_3$ or $C_3$ for the corresponding $N$, hence Lemma 4.15 implies that these also vanish in $\mathscr{B}$.

$\heartsuit$ Type $B_{\theta }$. Here $\widetilde{\mathscr{B}}_{\mathfrak{q}}$ is presented by the quantum Serre relations and

$$\begin{align} [x_{(i \, i+2)}, x_{i+1}]_c, \, i<\theta -1, \text{ if } N=4; && [x_{\theta \theta (\theta -1) (\theta -2)}, x_{\theta \theta -1}]_c, \, \text{ if } N=3. {\tag{4.13}} \end{align}$$

The relations involving the indices $\{\theta -2, \theta -1, \theta \}$ hold in $\mathscr{B}$ by Lemma 4.15 (i); also $x_{i \theta }=0$ for any $i<\theta -1$ by Proposition 3.2. We are left to treat the relations involving $\{1, \dots , \theta -1\}$. If $N=3$ we only have the quantum Serre relations, which hold by Lemma 4.14. Turn to $N=4$. Now $\{1, \dots , \theta -1\}$ form a subdiagram of type $A_{\theta -1}$ at a root of order $2$. If $\theta -1 \geq 4$ we apply Lemma 5.6 to get all the Serre relations except for $x_{221}$ and $x_{\theta -2 \,\theta -2\, \theta -1}$. The last one holds by Lemma 4.15 (i). For the first one, we apply AAH2, Theorem 1.2 since the diagram

$$\begin{align*} \vcenter{\img[][231pt][56pt][{$\xymatrix@C40pt @R-15pt { \overset{-1}{\underset{221 }{\circ}}&&&&\\ \overset{-1}{\underset{1 }{\circ}}\ar@{-}[r]^{-1} & \overset{-1}{\underset{2 }{\circ}}\ar@{-}[r]^{-1} \ar@{-}[ul]_{-1}& \overset{-1}{\underset{3 }{\circ}}\ar@{.}[r] & \overset{-1}{\underset{\theta-1}{\circ}} \ar@{-}[r]^{-1} & \overset{q}{\underset{\theta}{\circ}} }$}]{Images/img3bbbe0f42c8086204a4fa8d852bc6c68.svg}} \end{align*}$$

is of affine Cartan type. Now $[x_{(i \, i+2)}, x_{i+1}]_c=0$ for $i<\theta -2$ hold by Lemma 5.6 (e). We treat separately the last case standing.

$\heartsuit \heartsuit$ Type $B_4$ with $N=4$. The relations $x_{221}$ and $x_{\theta -2 \,\theta -2\, \theta -1}$ hold by the same reason as above. Moreover, we also have $x_{13}=0$. This follows from AAH1, Lemma 2.8 since $\mathfrak{q}(\alpha _1+\alpha _3,\alpha _1+\alpha _3)=1$ and $\widetilde{\mathfrak{q}}(\alpha _1+\alpha _3,\alpha _4)\neq 1$. Finally, using 5.1 and the relations deduced so far, we get that $[x_{(13)}, x_{2}]_c$ is primitive in $\mathscr{B}$. Notice that $\mathfrak{q}(\alpha _1+2\alpha _2+\alpha _3,\alpha _1+2\alpha _2+\alpha _3)=1$ and $\widetilde{\mathfrak{q}}(\alpha _1+2\alpha _2+\alpha _3,\alpha _4)\neq 1$, so AAH1, Lemma 2.8 applies again.

$\heartsuit$ Type $C_{\theta }$. Here $\widetilde{\mathscr{B}}_{\mathfrak{q}}$ is presented by the quantum Serre relations and

$$\begin{align} [x_{(\theta -2 \, \theta )}, x_{\theta -1 \, \theta }]_c, \text{ if } N=4; && \big [ [x_{(\theta -2 \, \theta )}, x_{\theta -1}]_c, x_{\theta -1} \big ], \text{ if } N=3. {\tag{4.14}} \end{align}$$

As before, Proposition 3.2 gives $x_{i \theta }=0$ for any $i<\theta -1$; all the relations involving the indices $\{\theta -2, \theta -1, \theta \}$ hold in $\mathscr{B}$ by Lemma 4.15 (ii). It remains to verify the relations involving $\{1, \dots , \theta -1\}$. Here we only have the Serre relations. But these indices span a subdiagram of type $A_{\theta -1}$, $\theta -1 \geq 3$, at a root of unity of order $3$ or $4$, so they hold by Lemma 4.14.

■

4.9. Types $E_{6}, E_{7}$ and $E_{8}$ with $N=2$

By AA2, p. 407 the distinguished pre-Nichols algebra is presented by the quantum Serre relations and

$$\begin{align*} [x_{ijk}, x_j]_c=0 \, \text{ if } i, j, k \text{ are all different and }\widetilde{q_{ij}}, \widetilde{q_{jk}} \neq 1. \end{align*}$$

Lemma 4.18.

Assume that Conjecture 2.6 is true. If $\mathbf{a}$ is of type $E_{6}, E_{7}$ or $E_{8}$ with $N=2$, then $\widetilde{\mathscr{B}}_{\mathfrak{q}}$ is eminent.

We point out that Conjecture 2.6 is needed only for a $5$-dimensional braided vector space of indefinite Cartan type.

Proof.

Let $\mathscr{B} \in \mathfrak{Pre}_{\mathrm{fGK}} (\mathfrak{q})$. First we deal with the quantum Serre relations, which are always primitive. Fix $i\neq j \in \mathbb{I}_{\theta }$. Consider two possibilities.

$\heartsuit$ $\widetilde{q_{ij}} = 1$. In this case choose $k \in \mathbb{I}_{\theta }$ different from $i$ and $j$ such that $\widetilde{q_{i}} = 1$ but $\widetilde{q_{ik}} \neq 1$. We get $\mathfrak{q}(\alpha _{i}+\alpha _{j}, \alpha _{i}+\alpha _{j})= 1$ and $\widetilde{\mathfrak{q}}(\alpha _{i}+\alpha _{j}, \alpha _{k})\neq 1$. By AAH1, Lemma 2.8, this warranties $x_{ij}=0$ in $\mathscr{B}$.

$\heartsuit$ $\widetilde{q_{ij}} \neq 1$. In this case $i$ and $j$ are consecutive vertices in a subdiagram of type $A_4$ with $N=2$. By Lemma 5.6 (b) below, it follows that $x_{iij}=0$ except in the following cases: $(i,j) \in \{ (2,1), (\theta -3, \theta ), (\theta -2, \theta -1)\}$. Fix such $(i,j)$, assume $x_{iij}\neq 0$ and consider $\mathbb{k}x_1 \oplus \dots \oplus \mathbb{k}x_{\theta } \oplus \mathbb{k}x_{iij} \subset \mathcal{P}(\mathscr{B})$. Then the Dynkin diagram of this braided vector space is of indefinite Cartan type. We illustrate the case $(i,j)=(\theta -3, \theta )$, the other cases being similar.

$$\begin{align*} \vcenter{\img[][305pt][71pt][{$\xymatrix@C50pt { & & \overset{-1}{\underset{\theta}{\circ}} & \overset{-1}{\underset{\theta-3 \, \theta-3 \, \theta}{\circ}} & \\ \overset{-1}{\underset{1 }{\circ}}\ar@{-}[r]^{-1} & \overset{-1}{\underset{2 }{\circ}} \ar@{.}[r] & \overset{-1}{\underset{\theta-3 }{\circ}} \ar@<0.3ex> @{-}[u]_{-1}^{} \ar@{-}[r]^{-1} \ar@{-}[ur]_{-1}& \overset{-1}{\underset{\theta-2 }{\circ}} \ar@{-}[r]^{-1}& \overset{-1}{\underset{\theta-1 }{\circ}}.}$}]{Images/img3d25487a9ba868a8fcf9423d344fc3b0.svg}} \end{align*}$$

Thus Conjecture 2.6 and Lemma 2.8 imply $\operatorname {GK-dim}\mathscr{B} = \infty$.

Finally, fix $i,j,k$ different such that $\widetilde{q_{ij}}, \widetilde{q_{jk}} \neq 1$. These are consecutive vertices in a suitable chosen subdiagram of type $A_4$. The Serre relations hold in $\mathscr{B}$, so by Lemma 5.6 (c) below we get that also $[x_{ijk}, x_j]_c=0$ in $\mathscr{B}$.

■

5. On the open cases

This section contain partial results towards those braidings of finite Cartan type which are still open. The detailed proofs can be found in S.

5.1. Type $A_2$ with $N=2$

Lemma 5.1.

Assume $\mathbf{a}$ is of Cartan type $A_2$ with $N=2$. Let $\mathscr{B}$ be a finite GK-dimensional pre-Nichols algebra of $\mathfrak{q}$. The following hold:

(a)

if $\mathscr{B} \in \mathfrak{Pre}_{\mathrm{fGK}}^{\mathbb{Z}^2}$, then either $x_{112}=0$ or $x_{221}=0$ in $\mathscr{B}$;

(b)

for different $i, j \in \mathbb{I}_2$, $(\operatorname {ad}_c x_i)^4x_j = 0$ in $\mathscr{B}$.■

Question 5.2.

Let $\widehat{\mathscr{B}}_1 = \mathbb{k}\langle x_1, x_2 | x_{221}, x_{11112} \rangle$. By Lemma 5.1 any $\mathscr{B} \in \mathfrak{Pre}_{\mathrm{fGK}}^{\mathbb{Z}^2}(V)$ is a quotient of either $\widehat{\mathscr{B}}_1$ or $\widehat{\mathscr{B}}_2 : = \mathbb{k}\langle x_1, x_2 | x_{112}, x_{22221} \rangle$. Clearly $\widehat{\mathscr{B}}_1\simeq \widehat{\mathscr{B}}_2$ as algebras. Is $\operatorname {GK-dim}\widehat{\mathscr{B}}_1 < \infty$?

5.2. Type $A_3$ with $N=2$

Lemma 5.3.

Assume $\mathbf{a}$ is of Cartan type $A_3$ with $N=2$. Let $\mathscr{B}$ be a finite GK-dimensional pre-Nichols algebra of $\mathfrak{q}$. Then the following hold in $\mathscr{B}$:

(a)

$x_{112}=0=x_{332}$, $x_{213}\overset{\star }{=}0$,

(b)

$x_{22221}=0= x_{22223}$, $x_{11113}=0=x_{33331}$,

(c)

if $\mathscr{B} \in \mathfrak{Pre}_{\mathrm{fGK}}^{\mathbb{Z}^3}$, then at most one of $x_{113}, x_{331}, x_{221}, x_{223}$ is non-zero.■

Remark 5.4.

The relation $\star$ is relevant because in the tensor algebra

$$\begin{align} \begin{aligned} \Delta ([x_{(13)}, x_2]_c)=[x_{(13)}, x_2]\otimes 1 + 1 \otimes [x_{(13)}, x_2] -2 q_{32} x_1 \otimes x_{223} \\ - 2 q_{12}^2q_{32} x_{221} \otimes x_3 -2q_{12}^2q_{32} x_2 \otimes x_{213} + 4 q_{12}^2q_{32} x_2^2 \otimes x_{13}. \end{aligned} {\tag{5.1}} \end{align}$$

Question 5.5.

By Lemma 5.3 every $\mathscr{B} \in \mathfrak{Pre}_{\mathrm{fGK}}^{\mathbb{Z}^3}$ is covered by one of

$$\begin{align*} \widehat{\mathscr{B}}&= \mathbb{k}\langle x_1, x_2, x_3 | x_{112},x_{332},x_{22221}, x_{22223},x_{11113},x_{33331},x_{213} \rangle , \\ &\begin{aligned} \widehat{\mathscr{B}}_1 &= \widehat{\mathscr{B}}/\langle x_{113}\rangle , & \widehat{\mathscr{B}}_2 &= \widehat{\mathscr{B}}/\langle x_{331}\rangle , & \widehat{\mathscr{B}}_3 &= \widehat{\mathscr{B}}/\langle x_{221}\rangle , & \widehat{\mathscr{B}}_4 &= \widehat{\mathscr{B}}/\langle x_{223}\rangle . \end{aligned} \end{align*}$$

Are $\operatorname {GK-dim}\widehat{\mathscr{B}}_1$ or $\operatorname {GK-dim}\widehat{\mathscr{B}}_3 < \infty$? ($\widehat{\mathscr{B}}_1 \simeq \widehat{\mathscr{B}}_2$ and $\widehat{\mathscr{B}}_3 \simeq \widehat{\mathscr{B}}_4$ as algebras).

5.3. Type $A_{\theta }$, $\theta \geq 4$ with $N=2$

In this setting $\widetilde{\mathscr{B}}_{\mathfrak{q}}$ is presented by

$$\begin{align*} x_{ij}&=0, & |i-j|&>1; & x_{iij} &= 0, & |i-j| &= 1; & [x_{(i i+2)}, x_{i+1}]_c &= 0, & i \in \mathbb{I}_{\theta -2}. \end{align*}$$

Lemma 5.6.

Assume $\mathbf{a}$ is of Cartan type $A_{\theta }$, $\theta \geq 4$, with $N=2$. The following hold in any finite GK-dimensional pre-Nichols algebra $\mathscr{B}$ of $\mathfrak{q}$:

(a)

$x_{ij}=0$ for any $|i-j|> 1$;

(b)

$x_{iij}=0$ for $|i-j|=1$ and $(i,j) \neq (2,1), \, (\theta -1, \theta )$;

(c)

$x_{iiiij}=0$ for $(i,j) = (2,1), \, (\theta -1, \theta )$;

(d)

if $\mathscr{B} \in \mathfrak{Pre}_{\mathrm{fGK}}^{\mathbb{Z}^{\theta }}$, then either $x_{221}=0$ or $x_{\theta -1 \theta -1 \theta }=0$;

(e)

if $(i,j) \in \{(2,1), \, (\theta -1, \theta )\}$ and $x_{iij}=0$, then $[x_{(i-1 \ i+1)}, x_{i}]_c=0$.■

Question 5.7.

Let $\widehat{\mathscr{B}}_1$ denote the quotient of $T(V)$ by the relations

$$\begin{align*} x_{ij}&=0, \quad |i-j|>1; && (\operatorname {ad}_cx_{\theta -1})^4x_{\theta } = 0; \\ x_{iij} &= 0, \quad |i-j|=1, (i,j) \neq (\theta -1,\theta ); && [x_{(13)}, x_{2}]_c=0. \end{align*}$$

Similarly, define $\widehat{\mathscr{B}}_2$ by the relations

$$\begin{align*} x_{ij}&=0, \quad |i-j|>1; && (\operatorname {ad}_cx_{2})^4x_1= 0; \\ x_{iij} &= 0, \quad |i-j|=1, (i,j) \neq (2,1); && [x_{(\theta -2 \ \theta )}, x_{\theta -1}]_c=0. \end{align*}$$

(Clearly $\widehat{\mathscr{B}}_2 \simeq \widehat{\mathscr{B}}_1$ as algebras). Is $\operatorname {GK-dim}\widehat{\mathscr{B}}_1 < \infty$?

5.4. Type $D_{\theta }$ with $N=2$

Here (cf. AA2, p. 404) the distinguished pre-Nichols algebra $\widetilde{\mathscr{B}}_{\mathfrak{q}}$ is presented by the quantum Serre relations and a bunch of $q$-brackets coming from the several subdiagrams of type $A_3$, namely:

$$\begin{align} [x_{(i\, i+2)}, x_{i+1}]_c, \, i\leq \theta -3; && [x_{\theta -3 \, \theta -2 \, \theta }, x_{\theta -2}]_c; && [x_{\theta \, \theta -2 \, \theta -1}, x_{\theta -2 }]_c. {\tag{5.2}} \end{align}$$

Lemma 5.8.

Assume $\mathbf{a}$ is of Cartan type $D_4$ with $N=2$. The following relations hold in any $\mathscr{B}\in \mathfrak{Pre}_{\mathrm{fGK}}$:

(a)

if $i\neq j$ and $\widetilde{\mathfrak{q}}_{ij}=-1$, then $x_{iij}=0$;

(b)

if $i\neq j$ and $\widetilde{\mathfrak{q}}_{ij}=1$, then $x_{kij}=0$ for all $k \in \mathbb{I}_4$;

(c)

if $r$ is one of the elements in 5.2, then $(\operatorname {ad}_c x_k) r=0$ for all $k \in \mathbb{I}_4$.■

Question 5.9.

Let $\widehat{\mathscr{B}}$ denote the quotient of $T(V)$ by the relations (a), (b) and (c) . Is $\operatorname {GK-dim}\widehat{\mathscr{B}} < \infty$?

Lemma 5.10.

Assume $\mathbf{a}$ is of Cartan type $D_{\theta }$ with $\theta >4$ and $N=2$. The following relations hold in any $\mathscr{B}\in \mathfrak{Pre}_{\mathrm{fGK}}(V)$:

(a)

all the defining relations of $\widetilde{\mathscr{B}}_{\mathfrak{q}}$ except $x_{\theta \, \theta -1}$ and $[x_{\theta \, \theta -2 \, \theta -1}, x_{\theta -2 }]_c$;

(b)

the relations $x_{k \, \theta \, \theta -1}$ and $(\operatorname {ad}_c x_k)[x_{\theta \, \theta -2 \, \theta -1}, x_{\theta -2 }]_c$ for all $k \in \mathbb{I}_{\theta }$.■

Question 5.11.

Let $\widehat{\mathscr{B}}$ denote the quotient of $T(V)$ by the relations

$$\begin{align*} x_{ij}&=0, \ \widetilde{\mathfrak{q}}_{ij}=1, \ (i,j)\neq (\theta , \theta -1); && [x_{\theta -3 \, \theta -2 \, \theta }, x_{\theta -2}]_c=0; \\ x_{iij}& = 0, \ \widetilde{\mathfrak{q}}_{ij}=-1;&& [x_{(i\, i+2)}, x_{i+1}]_c=0, \ i\leq \theta -3; \\ x_{k \, \theta \, \theta -1} &= 0, \ k \in \mathbb{I}_{\theta }; && [x_k, [x_{\theta \, \theta -2 \, \theta -1}, x_{\theta -2 }]]=0, \ k \in \mathbb{I}_{\theta }. \end{align*}$$

Is $\operatorname {GK-dim}\widehat{\mathscr{B}} <\infty$? We conjecture that $\operatorname {GK-dim}\widehat{\mathscr{B}}= \operatorname {GK-dim}\widetilde{\mathscr{B}}_{\mathfrak{q}} +2$. This will be treated in a subsequent paper.