Examples of finite-dimensional pointed Hopf algebras in positive characteristic

TL;DR

This paper introduces new finite-dimensional Nichols algebras over fields of positive characteristic, expanding the known examples of pointed Hopf algebras with non-diagonal braidings and finite Gelfand-Kirillov dimension.

Contribution

It provides novel examples of Nichols algebras in positive characteristic that are not of diagonal type and constructs associated pointed Hopf algebras via bosonization.

Findings

New finite-dimensional Nichols algebras over positive characteristic fields.

Existence of non-diagonal braided vector spaces realized as Yetter-Drinfeld modules.

Construction of finite-dimensional pointed Hopf algebras from these Nichols algebras.

Abstract

We present new examples of finite-dimensional Nichols algebras over fields of positive characteristic. The corresponding braided vector spaces are not of diagonal type, admit a realization as Yetter-Drinfeld modules over finite abelian groups and are analogous to braidings over fields of characteristic zero whose Nichols algebras have finite Gelfand-Kirillov dimension described in arXiv:1606.02521. We obtain nex examples of finite-dimensional pointed Hopf algebras by bosonization with group algebras of suitable finite abelian groups.

Tables

Table 1. Table 1. Finite-dimensional Nichols algebras of a block and a point

$V$	diagram	$q_{22}$	$𝒢$	$ℬ\left(V\right)$	$dimK$	$dimℬ\left(V\right)$
$𝔏(1,𝒢)$		$1$	discrete	§4.3.1	$p^{𝚛+1}$	$p^{𝚛+3}$
$𝔏(-1,𝒢)$		$-1$	discrete	§4.3.2	$2^{𝚛+1}$	$2^{𝚛+1}{p}^2$
$𝔏(\omega ,1)$		$\in {𝔾}_3^{\prime }$	1	§4.3.5	$3^3$	$3^3{p}^2$
${𝔏}_{-}(1,𝒢)$		$1$	discrete	§4.3.3	$2^{\frac{𝚛}{2}}{p}^{\frac{𝚛}{2}+1}$	$2^{\frac{𝚛}{2}+2}{p}^{\frac{𝚛}{2}+3}$
${𝔏}_{-}(-1,𝒢)$		$-1$	discrete	§4.3.4	$2^{\frac{𝚛}{2}+1}{p}^{\frac{𝚛}{2}}$	$2^{\frac{𝚛}{2}+3}{p}^{\frac{𝚛}{2}+2}$
${ℭ}_1$		$-1$	1	§4.4	$16$	$64p^2$

Table 2. Table 2. Finite-dimensional Nichols algebras of a block and several points, ω ∈ 𝔾 3 ′ 𝜔 subscript superscript 𝔾 ′ 3 \omega\in\mathbb{G}^{\prime}_{3} .

$V$	diagram	$ℬ\left(V\right)$	$dimℬ\left(V\right)$
$𝔏\left(A_{\theta -1}\right)$,		§5.2.7	$p^2{2}^6$
$\theta >2$	$\theta -1$ vertices		$p^2{2}^{\left(\theta -1\right)\left(\theta -2\right)}$
$𝔏(A_2,2)$		§5.2.6	$p^2{2}^{12}$
$𝔏(A{\left(1|0\right)}_2;\omega )$		§5.2.2	$p^2{2}^7{3}^4$
$𝔏(A{\left(1|0\right)}_1;\omega )$		§5.2.1	$p^2{2}^4{3}^2$
$𝔏(A{\left(1|0\right)}_3;\omega )$		§5.2.3	$p^2{2}^7{3}^4$
$𝔏(A{\left(1|0\right)}_1;r)$	, $r\in {𝔾}_N^{\prime },N>3$	§5.2.1	$p^2{2}^4{N}^2$
$𝔏(A{\left(2|0\right)}_1;\omega )$		§5.2.4	$p^2{2}^8{3}^9$
$𝔏(D\left(2|1\right);\omega )$		§5.2.5	$p^2{2}^8{3}^9$

Table 3. Table 3. Finite-dimensional Nichols algebras of a pale block and a point.

$V$	$ϵ$	${\tilde{q}}_{12}$	$q_{22}$	$ℬ\left(V\right)$	$dimK$	$dimℬ\left(V\right)$
${𝔈}_p\left(q\right)$	$1$	$1$	$-1$	§7.1	$2^p$	$2^p{p}^2$
${𝔈}_{+}\left(q\right)$	$-1$	$1$	$1$	§7.2	$2p$	$2^{3}p$
${𝔈}_{-}\left(q\right)$	$-1$	$1$	$-1$	§7.2	$2p$	$2^{3}p$
${𝔈}_{\star }\left(q\right)$	$-1$	$-1$	$-1$	§7.2	$2^4{p}^2$	$2^6{p}^2$

Table 4. Table 4. Pointed Hopf algebras K 𝐾 K from a block and a point

$V$	diagram	$\Gamma$	$q_{12}$	$dimA$
$𝔏(1,𝒢)$		$\mathbb{Z}/p\times \mathbb{Z}/p$	$1$	$p^{𝚛+5}$
$𝔏(-1,𝒢)$		$\mathbb{Z}/p\times \mathbb{Z}/2p$	$1$	$2^{𝚛+2}{p}^4$
$𝔏(\omega ,1)$		$\mathbb{Z}/p\times \mathbb{Z}/3p$	$1$	$3^4{p}^4$
${𝔏}_{-}(1,𝒢)$		$\mathbb{Z}/2p\times \mathbb{Z}/p$	$1$	$2^{\frac{𝚛}{2}+3}{p}^{\frac{𝚛}{2}+5}$
${𝔏}_{-}(-1,𝒢)$		$\mathbb{Z}/2p\times \mathbb{Z}/2p$	$\pm 1$	$2^{\frac{𝚛}{2}+5}{p}^{\frac{𝚛}{2}+4}$
${ℭ}_1$		$\mathbb{Z}/2p\times \mathbb{Z}/2p$	$\pm 1$	$256p^4$

Table 5. Table 5 . Pointed Hopf algebras from a block and several points

$V_J$	${𝒢}_J$	$\Gamma$	$dimA$
	$(1,0,\dots ,0)$	$\mathbb{Z}/p\times \mathbb{Z}/2p\times \mathbb{Z}/2$	$2^8{p}^4$
		$\mathbb{Z}/p\times \mathbb{Z}/2p\times {\left(\mathbb{Z}/2\right)}^{\theta -2}$	$2^{{\left(\theta -1\right)}^2}{p}^4$
	$(2,0)$	$\mathbb{Z}/p\times \mathbb{Z}/2p\times \mathbb{Z}/2$	$2^{14}{p}^4$
	$(1,0)$	$\mathbb{Z}/p\times \mathbb{Z}/2p\times \mathbb{Z}/6$	$2^9{3}^5{p}^4$
	$(1,0)$	$\mathbb{Z}/p\times \mathbb{Z}/2p\times \mathbb{Z}/3$	$2^5{3}^3{p}^4$
	$(0,1)$	$\mathbb{Z}/p\times \mathbb{Z}/6\times \mathbb{Z}/3p$	$2^8{3}^6{p}^4$
, $r\in {𝔾}_N^{\prime },N>3$	$(1,0)$	$\mathbb{Z}/p\times \mathbb{Z}/2p\times \mathbb{Z}/N$	$2^5{N}^3{p}^4$
	$(1,0,0)$	$\mathbb{Z}/p\times \mathbb{Z}/2p\times \mathbb{Z}/3\times \mathbb{Z}/3$	$2^9{3}^{11}{p}^4$
	$(1,0,0)$	$\mathbb{Z}/p\times \mathbb{Z}/2p\times \mathbb{Z}/3\times \mathbb{Z}/3$	$2^9{3}^{11}{p}^4$

Table 6. Table 6 . Pointed Hopf algebras K 𝐾 K from a pale block and a point

$V$	$(ϵ,{\tilde{q}}_{12},q_{22})$	$\Gamma$	$q_{12}$	$dimA$
${𝔈}_p\left(q\right)$	$(1,1,-1)$	$\mathbb{Z}/p\times \mathbb{Z}/2p$	$1$	$2^{p+1}{p}^4$
${𝔈}_{+}\left(q\right)$	$(-1,1,1)$	$\mathbb{Z}/2p\times \mathbb{Z}/p$	$1$	$2^4{p}^3$
${𝔈}_{-}\left(q\right)$	$(-1,1,-1)$	$\mathbb{Z}/2p\times \mathbb{Z}/2p$	$1$	$2^5{p}^3$
${𝔈}_{\star }\left(q\right)$	$(-1,-1,-1)$	$\mathbb{Z}/2p\times \mathbb{Z}/2p$	$\pm 1$	$2^8{p}^4$

Equations

$$\mathcal{B}(V)$$

$$c(x_i\otimes x_1)$$

$$V$$

$$c(V_i\otimes V_j)$$

$$(n{)}_{\mathtt{q}}$$

$${\partial }_i(w_j)$$

$$x_{ij}$$

$$x_2{x}_1-x_1{x}_2+\frac{1}{2}{x}_1^2,$$

$$x_1^p,$$

$$x_2^p.$$

$$c(x_i\otimes x_1)$$

$$x_{21}$$

$$x_1^2,$$

$$x_2{x}_{21}-x_{21}{x}_2-x_1{x}_{21},$$

$$x_{21}^p,$$

$$x_2^{2p}.$$

$$x_2^2{x}_1$$

$$x_{21}{x}_1$$

$${\partial }_1(x_{21}^n)$$

$$g\cdot x_2^2$$

$${\partial }_2(x_2^{2n})$$

$$a{x}_1$$

$${\partial }_2(x_2^{2n})$$

$${\left(b-a\right)}^n$$

$${\left(b-a\right)}^{n+1}$$

$$=b^{n+1}-nba{b}^{n-1}-a{b}^n+n{a}^2{b}^{n-1}=b^{n+1}-(n+1)a{b}^n$$

$$A_{n+1}$$

$${\partial }_2(x_2^{2n})$$

$${\partial }_1(x_2^c{x}_{21}^b)$$

$${\partial }_2(x_2^{2n+1})$$

$$\chi (h)g$$

$${\mathrm{Der}}_{\chi ,\chi }(H,\mathbb{k})$$

$$\eta (h)g_1$$

$$h\cdot x_1$$

Full text

We obtain new examples of finite-dimensional pointed Hopf algebras by bosonization with group algebras of suitable finite abelian groups.

2010 Mathematics Subject Classification. 16T20, 17B37.

The work of N. A. and I. A. was partially supported by CONICET, Secyt (UNC). The work of N. A. and I. A., respectively I. H., was partially done during visits to the University of Marburg, respectively Córdoba, supported by the Alexander von Humboldt Foundation through the Research Group Linkage Programme

1. Introduction

1.1. Overview

This is a contribution to the classification of finite-dimensional pointed Hopf algebras in positive characteristic. Beyond the classical theme of cocommutative Hopf algebras–see for instance [CF] and references therein–the problem was considered in several recent works [CLW, HW, NW, NWW1, NWW2, W]. As in various of these papers, the focus of our work is on finite-dimensional Nichols algebras over finite abelian groups. Let $\Bbbk$ be an algebraically closed field of characteristic $p\geq 0$. When $p=0$, such Nichols algebras are necessarily of diagonal type and their classification was achieved in [H2]. When $p>0$, finite-dimensional Nichols algebras of diagonal type of rank 2 and 3 were classified in [HW, W]. Notice that there are more examples than in characteristic 0: indeed, 1 in the diagonal is no longer excluded.

Example 1.1**.**

Assume that $p>0$. Given $\theta\in\mathbb{N}$, we set $\mathbb{I}_{\theta}=\{1,2,\dots,\theta\}$. Let $\mathbf{q}=(q_{ij})_{i,j\in\mathbb{I}_{\theta}}\in\Bbbk^{\theta\times\theta}$, be a matrix with $q_{ii}=1=q_{ij}q_{ji}$ for all $i\neq j\in\mathbb{I}_{\theta}$. Let$(V,c)$ be a braided vector space of dimension $\theta$, of diagonal type with matrix $\mathbf{q}$ with respect to a basis $(x_{i})_{i\in\mathbb{I}_{\theta}}$, that is $c:V\otimes V\to V\otimes V$ is given by $c(x_{i}\otimes x_{j})=q_{ij}x_{j}\otimes x_{i}$. Then the corresponding Nichols algebra is

[TABLE]

Clearly, $\dim\mathfrak{s}_{\mathbf{q}}(V)=p^{\theta}$.

Furthermore, if $p>0$, then there are finite-dimensional Nichols algebras over abelian groups that are not of diagonal type, a remarkable example being the Jordan plane that has dimension $p^{2}$ [CLW] (it gives rise to pointed Hopf algebras of order $p^{3}$, see [NW]), in contrast with characteristic 0, where it has Gelfand-Kirillov dimension $2$. In fact, Nichols algebras over abelian groups with finite Gelfand-Kirillov dimension and assuming $p=0$ were the subject of the recent papers [AAH1, AAH2]. Succinctly, the main relevant results in loc. cit. are:

• $\circ$

It was conjectured in [AAH1] that finite GK-dimensional Nichols algebras of diagonal type have arithmetic root system; the conjecture is true in rank 2 and also in affine Cartan type [AAH2].

• $\circ$

A class of braided vector spaces arising from abelian groups was introduced in [AAH1]; they are decomposable with components being points and blocks. Assuming the validity of the above Conjecture, the finite GK-dimensional Nichols algebras from this class were classified in [AAH1].

Beware that there are finite GK-dimensional Nichols algebras over abelian groups that do not belong to the referred class, see [AAH1, Appendix].

The braided vector spaces in the class alluded to above can be labelled with flourished Dynkin diagrams. The main result of [AAH1] says that the Nichols algebra of a braided vector space in the class has finite Gelfand-Kirillov dimension if and only if its flourished Dynkin diagram is admissible.

From now on we assume that $p>2$. (The case $p=2$ has to be treated separately). In the present paper, we show, adapting arguments from [AAH1], that the Nichols algebras of many braided vector spaces of admissible flourished Dynkin diagrams are finite-dimensional. This result extends Example 1.1 and the Jordan plane [CLW] and is reminiscent of a familiar phenomenon in Lie algebras in positive characteristic. By bosonization we obtain many new examples of finite-dimensional pointed Hopf algebras.

1.2. The main result

To describe more precisely our main Theorem we need first to discuss blocks.

For $k<\ell\in\mathbb{N}_{0}$, we set $\mathbb{I}_{k,\ell}=\{k,k+1,\dots,\ell\}$, $\mathbb{I}_{\ell}=\mathbb{I}_{1,\ell}$.

A block $\mathcal{V}(\epsilon,\ell)$, where $\epsilon\in\Bbbk^{\times}$ and $\ell\in\mathbb{N}_{\geq 2}$, is a braided vector space with a basis $(x_{i})_{i\in\mathbb{I}_{\ell}}$ such that for $i,j\in\mathbb{I}_{\ell}$, $1<j$:

[TABLE]

In characteristic 0, the only Nichols algebras of blocks with finite $\operatorname{GKdim}$ are the Jordan plane $\mathscr{B}(\mathcal{V}(1,2))$ and the super Jordan plane $\mathscr{B}(\mathcal{V}(-1,2))$; both have $\operatorname{GKdim}=2$. In our context with $p>2$, the Jordan plane $\mathscr{B}(\mathcal{V}(1,2))$ has dimension $p^{2}$ [CLW]; see Lemma 3.1. Our starting result is that the super Jordan plane $\mathscr{B}(\mathcal{V}(-1,2))$ has dimension $4p^{2}$, see Proposition 3.2. For simplicity a block $\mathcal{V}(\epsilon,2)$ of dimension 2 is called an $\epsilon$-block. We also prove that a block $\mathcal{V}(\epsilon,2)$ has finite-dimensional Nichols algebra only when $\epsilon=\pm 1$, see Proposition 3.3.

The braided vector spaces in this paper belong to the class analogous to the one considered in [AAH1]. Briefly, $(V,c)$ belongs to this class if

[TABLE]

where $V_{h}$ is a $\epsilon_{h}$-block, with $\epsilon_{h}^{2}=1$, for $h\in\mathbb{I}_{t}$; and $\dim V_{i}=1$ with braiding determined by $q_{ii}\in\Bbbk^{\times}$ (we say that $i$ is a point), $i\in\mathbb{I}_{t+1,\theta}$; the braiding between points $i$ and $j$ is given by $q_{ij}\in\Bbbk^{\times}$ while the braiding between a point and block, respectively two blocks, should have the form as in (4.1), respectively (6.6). For convenience, we attach to $(V,c)$ a flourished graph $\mathcal{D}$ with $\theta$ vertices, those corresponding to a $1$-block decorated with $\boxplus$, those to $-1$-block decorated with $\boxminus$ and the point $i$ with $\overset{q_{ii}}{\circ}$. If $i\neq j$ are points, and there is an edge between them decorated by $\widetilde{q}_{ij}:=q_{ij}q_{ji}$ when this is $\neq-1$, or no edge if $\widetilde{q}_{ij}=1$. If $h$ is a block and $j$ is a point, then there is an edge between $h$ and $j$ decorated either by $\mathscr{G}_{hj}$ if the interaction is weak and $\mathscr{G}_{hj}\neq 0$ is the ghost, cf. (4.2), or by $(-,\mathscr{G}_{hj})$ if the interaction is mild and $\mathscr{G}_{hj}$ is the ghost; but no edge if the interaction is weak and $\mathscr{G}_{hj}=0$. There are no edges between blocks and we assume that the diagram is connected by a well-known reduction argument.

This class of braided vector spaces together with those of diagonal type does not exhaust that of Yetter-Drinfeld modules arising from abelian groups; there are still those containing a pale block as in [AAH1, Chapter 8]. Synthetically our main result is the following.

Theorem 1.2**.**

Let $V$ be a braided vector space as in the following list, then $\dim\mathscr{B}(V)<\infty$.

1. (a)

$V$* has braiding (4.1) and is listed in Table 1, or* 2. (b)

$V$* has braiding (5.1) and is listed in Table 2, or* 3. (c)

$V$* has braiding (6.6), or* 4. (d)

$V$* has braiding (7.1) and is listed in Table 3.*

By bosonization with suitable abelian groups, we get examples of finite-dimensional pointed Hopf algebras in positive characteristic.

Concrete examples of such Hopf algebras are described in §3.3, §4.5, §5.3, §6.1 and §7.3. We also give a presentation by generators and relations of the Nichols algebras; references to this information and the dimensions are also given in the Tables.

All braided vector spaces in this Theorem belong to the class described above except those in (d) that contain a pale block.

1.3. Contents of the paper

Section 2 is devoted to preliminaries. The next Sections contain the examples of finite-dimensional Nichols algebras and some realizations over abelian groups; each Section describes a family of braided vector spaces with a certain decomposition as we describe now. In Section 3 we compute Nichols algebras of a block. In Section 4 we present examples of Nichols algebras corresponding to one block and one point, while in Section 5 we consider the case one block and several points. Section 6 is devoted to examples of several blocks and one point. Finally in Section 7 we give examples of finite-dimensional Nichols algebras whose braided vector spaces decompose as one pale block and one point.

2. Preliminaries

2.1. Conventions

The $\mathtt{q}$-numbers are the polynomials

[TABLE]

$n\in\mathbb{N}$, $0\leq i\leq n$. If $q\in\Bbbk$, then $(n)_{q}$, $(n)_{q}^{!}$, $\binom{n}{i}_{q}$ denote the evaluations of $(n)_{\mathtt{q}}$, $(n)_{\mathtt{q}}^{!}$, $\binom{n}{i}_{\mathtt{q}}$ at $\mathtt{q}=q$.

Let $\mathbb{G}_{N}$ be the group of $N$-th roots of unity, and $\mathbb{G}_{N}^{\prime}$ the subset of primitive roots of order $N$; $\mathbb{G}_{\infty}=\bigcup_{N\in\mathbb{N}}\mathbb{G}_{N}$. All the vector spaces, algebras and tensor products are over $\Bbbk$.

All Hopf algebras have bijective antipode.

2.2. Yetter-Drinfeld modules

Let $\Gamma$ be an abelian group. We denote by $\widehat{\Gamma}$ the group of characters of $\Gamma$. The category ${}^{\Bbbk\Gamma}_{\Bbbk\Gamma}\mathcal{YD}$ of Yetter-Drinfeld modules over the group algebra $\Bbbk\Gamma$ consists of $\Gamma$-graded $\Gamma$-modules, the $\Gamma$-grading being denoted by $V=\oplus_{g\in\Gamma}V_{g}$; that is, $hV_{g}=V_{g}$ for all $g,h\in\Gamma$. If $g\in\Gamma$ and $\chi\in\widehat{\Gamma}$, then the one-dimensional vector space $\Bbbk_{g}^{\chi}$, with action and coaction given by $g$ and $\chi$, is in ${}^{H}_{H}\mathcal{YD}$. Let $W\in{}^{\Bbbk\Gamma}_{\Bbbk\Gamma}\mathcal{YD}$ and $(w_{i})_{i\in I}$ a basis of $W$ consisting of homogeneous elements of degree $g_{i}$, $i\in I$, respectively. Then there are skew derivations $\partial_{i}$, $i\in I$, of $T(W)$ such that for all $x,y\in T(W)$, $i,j\in I$

[TABLE]

For a definition of Yetter-Drinfeld modules over arbitrary Hopf algebras we refer e.g. to [R, 11.6].

2.3. Nichols algebras

Nichols algebras are graded Hopf algebras $\mathscr{B}=\oplus_{n\geq 0}\mathscr{B}^{n}$ in ${}^{H}_{H}\mathcal{YD}$ coradically graded and generated in degree one. They are completely determined by $V:=\mathscr{B}^{1}\in{}^{H}_{H}\mathcal{YD}$ and it is customary to denote $\mathscr{B}=\mathscr{B}(V)$. If $W\in{}^{\Bbbk\Gamma}_{\Bbbk\Gamma}\mathcal{YD}$ as in Subsection 2.2, then the skew-derivations $\partial_{i}$ induce skew-derivations on $\mathscr{B}(W)$. Moreover, an element $w\in\mathscr{B}^{k}(W)$, $k\geq 1$, is zero if and only if $\partial_{i}(w)=0$ in $\mathscr{B}(W)$ for all $i\in I$. A pre-Nichols algebra of $V$ is a graded Hopf algebra in ${}^{H}_{H}\mathcal{YD}$ generated in degree one, with the one-component isomorphic to $V$.

Example 2.1**.**

Let $V$ be of dimension 1 with braiding $c=\epsilon\operatorname{id}$. Let $N$ be the smallest natural number such that $(N)_{\epsilon}=0$. Then $\mathscr{B}(V)=\Bbbk[T]/\langle T^{N}\rangle$, or $\mathscr{B}(V)=\Bbbk[T]$ if such $N$ does not exist.

A braided vector space $V$ is of diagonal type if there exist a basis $(x_{i})_{i\in\mathbb{I}_{\theta}}$ of $V$ and $\mathbf{q}=(q_{ij})_{i,j\in\mathbb{I}_{\theta}}\in\Bbbk^{\theta\times\theta}$ such that $q_{ij}\neq 0$ and $c(x_{i}\otimes x_{j})=q_{ij}x_{j}\otimes x_{i}$ for all $i,j\in\mathbb{I}=\mathbb{I}_{\theta}$. Given a braided vector space $V$ of diagonal type with a basis $(x_{i})$, we denote in $T(V)$, or $\mathscr{B}(V)$, or any intermediate Hopf algebra,

[TABLE]

for $i,j,i_{1},\dots,i_{M}\in\mathbb{I}$, $M\geq 2$. A braided vector space $V$ of diagonal type is of Cartan type if there exists a generalized Cartan matrix $\mathbf{a}=(a_{ij})$ such that $q_{ij}q_{ji}=q_{ii}^{a_{ij}}$ for all $i\neq j$.

Theorem 2.2**.**

If $V$ is of Cartan type with matrix $\mathbf{a}$ that is not finite, then $\dim\mathscr{B}(V)=\infty$.

Proof.

The argument in [AAH2, Proposition 3.1] is characteristic-free and applies here because there are infinite real roots in the root system of $\mathbf{a}$. ∎

3. Blocks

We consider braided vector spaces $\mathcal{V}(\epsilon,2)$ with braiding (1.1), $\epsilon^{2}=1$.

3.1. The Jordan plane

Here we deal with $\mathcal{V}=\mathcal{V}(1,2)$. In characteristic 0, $\mathscr{B}(\mathcal{V})$ is the well-known algebra presented by $x_{1}$ and $x_{2}$ with the relation (3.1). In positive characteristic, $\mathscr{B}(\mathcal{V})$ is a truncated version of that algebra.

Lemma 3.1**.**

[CLW]** $\mathscr{B}(\mathcal{V})$ is presented by generators $x_{1},x_{2}$ and relations

[TABLE]

Also $\dim\mathscr{B}(\mathcal{V})=p^{2}$ and $\{x_{1}^{a}x_{2}^{b}:0\leq a,b<p\}$ is a basis of $\mathscr{B}(\mathcal{V})$. ∎

In characteristic 2, the relations of $\mathscr{B}(\mathcal{V})$ are different.

Let $\Gamma=\mathbb{Z}/p=\langle g\rangle$. We realize $\mathcal{V}$ in ${}^{\Bbbk\Gamma}_{\Bbbk\Gamma}\mathcal{YD}$ by $g\cdot x_{1}=x_{1}$, $g\cdot x_{2}=x_{2}+x_{1}$, $\deg x_{i}=g$, $i\in\mathbb{I}_{2}$. Thus the Hopf algebra $\mathscr{B}(\mathcal{V})\#\Bbbk\Gamma$ has dimension $p^{3}$.

3.2. The super Jordan plane

Let $\mathcal{V}=\mathcal{V}(-1,2)$ be the braided vector space with basis $x_{1}$, $x_{2}$ and braiding

[TABLE]

Let $g$ be a generator of the cyclic group $\mathbb{Z}$. We realize $\mathcal{V}(-1,2)$ in ${}^{\Bbbk\mathbb{Z}}_{\Bbbk\mathbb{Z}}\mathcal{YD}$ by $g\cdot x_{1}=-x_{1}$, $g\cdot x_{2}=-x_{2}+x_{1}$, $\deg x_{i}=g$, $i\in\mathbb{I}_{2}$. As in (2.2),

[TABLE]

The Nichols algebra $\mathscr{B}(\mathcal{V}(-1,2))=T(\mathcal{V}(-1,2))/\mathcal{J}(\mathcal{V}(-1,2))$ (called the super Jordan plane) was studied in [AAH1, 3.3] over fields of characteristic 0. Assuming $p>2$, the basic features of $\mathscr{B}(\mathcal{V}(-1,2))$ are summarized here:

Proposition 3.2**.**

The defining ideal $\mathcal{J}(\mathcal{V}(-1,2))$ is generated by

[TABLE]

The set $B=\{x_{1}^{a}x_{21}^{b}x_{2}^{c}:a\in\mathbb{I}_{0,1},b\in\mathbb{I}_{0,p-1},c\in\mathbb{I}_{0,2p-1}\}$ is a basis of $\mathscr{B}(\mathcal{V})$ and $\dim\mathscr{B}(\mathcal{V})=4p^{2}$.

Proof.

Since $\partial_{2}(x_{21})=0$, we have that $\partial_{2}(x_{21}^{n})=0$ for every $n\in\mathbb{N}$. Also $\partial_{1}(x_{21})=x_{1}$ and $g\cdot x_{21}=x_{21}$. Both (3.6) and (3.7) are [math] in $\mathscr{B}(\mathcal{V})$ being annihilated by $\partial_{1}$ and $\partial_{2}$, cf. (2.1). From (3.6) and (3.7) we see that in $\mathscr{B}(\mathcal{V})$

[TABLE]

By the preceding, we have

[TABLE]

Hence $x_{21}^{p}=0$, i.e. (3.8) holds. Next we prove (3.9). Clearly $\partial_{1}(x_{2}^{n})=0$ for every $n\in\mathbb{N}$. We observe that

[TABLE]

Setting for simplicity $a:=x_{21}$ and $b:=x_{2}^{2}$, we have for any $n\in\mathbb{N}$:

[TABLE]

By (3.7), (3.10) and (3.11) we have

[TABLE]

hence

[TABLE]

We prove recursively that for all $n\in\mathbb{N}$

[TABLE]

The case $n=1$ is evident. We start with the first identity:

[TABLE]

as desired. For the second identity we use the first:

[TABLE]

The claim is proved; summarizing we have

[TABLE]

In particular, this implies (3.9).

We now argue as in [AAH1, 3.3.1]. The quotient $\widetilde{\mathcal{B}}$ of $T(\mathcal{V})$ by (3.6), (3.7), (3.8) and (3.9) projects onto $\mathscr{B}(\mathcal{V})$ and the subspace $I$ spanned by $B$ is a left ideal of $\widetilde{\mathcal{B}}$, by (3.7), (3.11). Since $1\in I$, $\widetilde{\mathcal{B}}$ is spanned by $B$. To prove that $\widetilde{\mathcal{B}}\simeq\mathscr{B}(\mathcal{V})$, we just need to show that $B$ is linearly independent in $\mathscr{B}(\mathcal{V})$. We claim that this is equivalent to prove that $B^{\prime}=\{x_{2}^{c}x_{21}^{b}x_{1}^{a}:a\in\{0,1\},b\in\mathbb{I}_{0,p-1},c\in\mathbb{I}_{0,2p-1}\}$ is linearly independent. Indeed, $\widetilde{\mathcal{B}}$ is spanned by $B^{\prime}$ since the subspace spanned $B^{\prime}$ is also a left ideal; if $B^{\prime}$ is linearly independent, then the dimension of $\widetilde{\mathcal{B}}$ is $4p^{2}$, so $B$ should be linearly independent and vice versa. Suppose that there is a non-trivial linear combination of elements of $B^{\prime}$ in $\mathscr{B}(\mathcal{V})$ of minimal degree. As

[TABLE]

such linear combination does not have terms with $a$ or $b$ greater than 0. We claim that the elements $x_{2}^{c}$, $c\in\mathbb{I}_{0,2p-1}$, are linearly independent, yielding a contradiction. By homogeneity it is enough to prove that they are $\neq 0$. If $c$ is even this follows from (3.14). If $c=2n+1$ with $n<p$, then

[TABLE]

Again a degree argument gives the desired claim. ∎

Let $\Gamma=\mathbb{Z}/2p$. We may realize $\mathcal{V}(-1,2)$ in ${}^{\Bbbk\Gamma}_{\Bbbk\Gamma}\mathcal{YD}$ by the same formulas as above; thus $\mathscr{B}(\mathcal{V})\#\Bbbk\Gamma$ is a pointed Hopf algebra of dimension $8p^{3}$.

3.3. Realizations

Let $H$ be a Hopf algebra. A YD-pair for $H$ is a pair $(g,\chi)\in G(H)\times\operatorname{Hom}_{\text{alg}}(H,\Bbbk)$ such that

[TABLE]

Let $\Bbbk_{g}^{\chi}$ be a one-dimensional vector space with $H$-action and $H$-coaction given by $\chi$ and $g$ respectively; then (3.16) says that $\Bbbk_{g}^{\chi}\in{}^{H}_{H}\mathcal{YD}$.

If $\chi\in\operatorname{Hom}_{\text{alg}}(H,\Bbbk)$, then the space of $(\chi,\chi)$-derivations is

[TABLE]

A YD-triple for $H$ is a collection $(g,\chi,\eta)$ where $(g,\chi)$, is a YD-pair for $H$, $\eta\in\operatorname{Der}_{\chi,\chi}(H,\Bbbk)$, $\eta(g)=1$ and

[TABLE]

Given a YD-triple $(g,\chi,\eta)$ we define $\mathcal{V}_{g}(\chi,\eta)\in{}^{H}_{H}\mathcal{YD}$ as the vector space with a basis $(x_{i})_{i\in\mathbb{I}_{2}}$, whose $H$-action and $H$-coaction are given by

[TABLE]

the compatibility is granted by (3.16), (3.17). As a braided vector space, $\mathcal{V}_{g}(\chi,\eta)\simeq\mathcal{V}(\epsilon,2)$, $\epsilon=\chi(g)$.

Consequently, if $H$ is finite-dimensional and $\epsilon^{2}=1$, then $\mathscr{B}(\mathcal{V}_{g}(\chi,\eta))\#H$ is a Hopf algebra satisfying

[TABLE]

3.4. Exhaustion in rank $2$

We recall some facts from [AAH1, §3.4].

Let $H$ be a Hopf algebra with bijective antipode and $V\in{}^{H}_{H}\mathcal{YD}$. Let $0=V_{0}\subsetneq V_{1}\dots\subsetneq V_{d}=V$ be a flag of Yetter-Drinfeld submodules with $\dim V_{i}=\dim V_{i-1}+1$ for all $i$. Then $V^{\textrm{diag}}:=\operatorname{gr}V$ is of diagonal type. If $\mathscr{B}$ is a pre-Nichols algebra of $V$, then it is a graded filtered Hopf in ${}^{H}_{H}\mathcal{YD}$ and $\mathcal{B}^{\mathrm{diag}}:=\operatorname{gr}\mathscr{B}$ is a pre-Nichols algebra of $V^{\textrm{diag}}$.

Proposition 3.3**.**

Let $\epsilon\in\Bbbk^{\times}$. If $\dim\mathscr{B}(\mathcal{V}(\epsilon,2))<\infty$, then $\epsilon^{2}=1$.

Proof.

Let $\mathcal{V}=\mathcal{V}(\epsilon,2)$; it has a flag as above and $\mathcal{V}^{\mathrm{diag}}$ is the braided vector space of diagonal type with matrix $(q_{ij})_{i,j\in\mathbb{I}_{2}}$, $q_{ij}=\epsilon$ for all $i,j\in\mathbb{I}_{2}$. Hence

[TABLE]

Step 1*.*

If $\epsilon\notin\mathbb{G}_{\infty}$, then $\dim\mathscr{B}(\mathcal{V}(\epsilon,2))=\infty$.

Proof.

Here $\dim\mathscr{B}(\mathcal{V}^{\mathrm{diag}})=\infty$ by Example 2.1 and (3.19) applies. ∎

Step 2*.*

If $\epsilon\in\mathbb{G}^{\prime}_{N}$, $N\geq 4$, then $\dim\mathscr{B}(\mathcal{V}(\epsilon,\ell))=\infty$ for all $\ell\geq 2$.

Proof.

Here $\mathcal{V}^{\mathrm{diag}}$ is of Cartan type with Cartan matrix $\begin{pmatrix}2&2-N\\ 2-N&2\end{pmatrix}$. Thus Theorem 2.2 and (3.19) apply. ∎

Step 3*.*

Let $\epsilon\in\mathbb{G}^{\prime}_{3}$. Then $\dim\mathscr{B}(\mathcal{V}(\epsilon,2))=\infty$.

Proof.

The proof of [AAH1, §3.5 – Step 3] holds verbatim. ∎

The Proposition is proved. ∎

4. One block and one point

4.1. The setting and the statement

Let $(q_{ij})_{1\leq i,j\leq 2}$ be a matrix of invertible scalars and $a\in\Bbbk$. We assume that $\epsilon:=q_{11}$ satisfies $\epsilon^{2}=1$. Let $V$ be a braided vector space with a basis $(x_{i})_{i\in\mathbb{I}_{3}}$ and a braiding given by

[TABLE]

Let $V_{1}=\langle x_{1},x_{2}\rangle$ (the block) and $V_{2}=\langle x_{3}\rangle$ (the point). Let $\Gamma=\mathbb{Z}^{2}$ with canonical basis $g_{1},g_{2}$. We realize $(V,c)$ in ${}^{\Bbbk\Gamma}_{\Bbbk\Gamma}\mathcal{YD}$ as $\mathcal{V}_{g_{1}}(\chi_{1},\eta)\oplus\Bbbk_{g_{2}}^{\chi_{2}}$ with suitable $\chi_{1},\chi_{2}$ and $\eta$, where $V_{1}=\mathcal{V}_{g_{1}}(\chi_{1},\eta)$, while $V_{2}=\Bbbk_{g_{2}}^{\chi_{2}}$. Thus $V_{1}\simeq\mathcal{V}(\epsilon,2)$; thus we use the notations and results from §3.2.

The interaction between the block and the point is $q_{12}q_{21}$; it is

[TABLE]

In characteristic 0, we introduced a normalized version of $a$ called the ghost, which is discrete when it belongs to $\mathbb{N}$. In our context, $p>2$, we need a variant of this notion. First we say that $V$ has discrete ghost if $a\in\mathbb{F}_{p}^{\times}$. When this is the case, we pick a representative $\mathtt{r}\in\mathbb{Z}$ of $2a$ by imposing

[TABLE]

Then $\mathscr{G}$ is called the ghost. In this Section we consider the following braided vectors spaces with braiding (4.1), where the ghost is discrete and $q_{22}\in\mathbb{G}_{\infty}$:

[TABLE]

In this Section, we shall prove part (a) of Theorem 1.2.

Theorem 4.1**.**

Let $V$ be a braided vector space with braiding (4.1). If $V$ is as in Table 1, then $\dim\mathscr{B}(V)<\infty$.

To prove the Theorem, we consider $K=\mathscr{B}(V)^{\mathrm{co}\,\mathscr{B}(V_{1})}$. By [HS, Proposition 8.6], $\mathscr{B}(V)\simeq K\#\mathscr{B}(V_{1})$ and $K$ is the Nichols algebra of

[TABLE]

Now $K^{1}\in{}^{\mathscr{B}(V_{1})\#\Bbbk\Gamma}_{\mathscr{B}(V_{1})\#\Bbbk\Gamma}\mathcal{YD}$ with the adjoint action and the coaction given by

[TABLE]

In order to describe $K^{1}$, we set

[TABLE]

4.2. Weak interaction

Here $q_{12}q_{21}=1$. In general,

[TABLE]

If $a=0$, then

[TABLE]

Here $\underline{\otimes}$ denotes the braided tensor product of Hopf algebras (the structure of Hopf algebra in ${}^{\Bbbk G}_{\Bbbk G}\mathcal{YD}$)

From now on we assume that the ghost is discrete, in particular $\neq 0$. We follow the exposition in [AAH1, §4.2].

Lemma 4.2**.**

The following formulae hold in $\mathscr{B}(V)$ for all $n\in\mathbb{N}_{0}$:

[TABLE]

Proof.

The proof of [AAH1, Lemma 4.2.1] is valid in any characteristic. ∎

Let $(\mu_{n})_{n\in\mathbb{N}_{0}}$ be the family of elements of $\Bbbk$ defined recursively by

[TABLE]

This can be reformulated as

[TABLE]

Thus $\mu_{n}=0\iff n>|\mathtt{r}|$.

Lemma 4.3**.**

For all $k\in\{1,\dots,p-1\}$, $\partial_{1}(z_{k})=\partial_{2}(z_{k})=0$,

[TABLE]

Therefore, $z_{n}=0\iff n>|\mathtt{r}|$.

Proof.

The proof of [AAH1, Lemma 4.2.2] is valid in any characteristic, taking into the account the conventions on $\mathtt{r}$. ∎

If $\epsilon=1$, then (4.11) says that

[TABLE]

Recall that $K=\mathscr{B}(V)^{\mathrm{co}\,\mathscr{B}(V_{1})}\simeq\mathscr{B}(K^{1})$, $K^{1}=\operatorname{ad}\mathscr{B}(V_{1})(V_{2})$.

Lemma 4.4**.**

The family $(z_{n})_{0\leq n\leq|\mathtt{r}|}$ is a basis of $K^{1}$.

Proof.

The family is linearly independent, because the $z_{n}$’s are homogeneous of distinct degrees, and are $\neq 0$ by (4.11). We have for all $n\in\mathbb{N}_{0}$

[TABLE]

The Lemma follows. ∎

If $\epsilon=1$, then we define recursively $\nu_{k,n}$ as follows: $\nu_{n,n}=1$,

[TABLE]

Lemma 4.5**.**

The coaction (4.4) on $z_{n}$, $0\leq n\leq|\mathtt{r}|$, is given by (4.15), when $\epsilon=1$, and by (4.16), (4.17), when $\epsilon=-1$:

[TABLE]

[TABLE]

Proof.

The proof of [AAH1, Lemma 4.2.1] can be adapted since $\binom{n}{k}\neq 0$ by assumption. ∎

We are ready to prove the finite-dimensionality in the case of weak interaction. We claim that the braided vector space $K^{1}$ is of diagonal type with braiding matrix

[TABLE]

Hence, the corresponding generalized Dynkin diagram has labels

[TABLE]

Indeed, by Lemma 4.4 it is enough to compute

[TABLE]

by Lemmas 4.5 and 4.2, (4.13) and (4.14). We proceed then case by case.

Case 1*.*

$q_{22}^{2}=1$.

Here the Dynkin diagram of $K^{1}$ is totally disconnected with vertices $i\in\mathbb{I}_{0,|\mathtt{r}|}$ labelled with $\epsilon^{i}q_{22}$. The vertices with label $1$, respectively $-1$, contribute with $p$, respectively $2$, to $\dim\mathscr{B}(K^{1})$.

Case 2*.*

$\epsilon=1$, $q_{22}\in\mathbb{G}_{3}^{\prime}$, $|\mathtt{r}|=1$ (provided that $p>3$).

The Dynkin diagram is of Cartan type $A_{2}$, so $\dim\mathscr{B}(K^{1})<\infty$.

4.3. The presentation by generators and relations

We still assume that the interaction is weak. We start by some general Remarks that are proved exactly as in [AAH1, §4.3].

Remark 4.6*.*

Let

[TABLE]

By Lemma 4.2

[TABLE]

Lemma 4.7**.**

Assume that $\epsilon^{2}=q_{22}^{2}=1$. In $\mathscr{B}(\mathfrak{L}(q_{22},\mathscr{G}))$, or correspondingly $\mathscr{B}_{-}(\mathfrak{L}(q_{22},\mathscr{G}))$,

[TABLE]

Lemma 4.8**.**

Let $\mathscr{B}$ be a quotient algebra of $T(V)$. Assume that $x_{1}x_{3}=q_{12}x_{3}x_{1}$, and either

1. (a)

(3.1), or else 2. (b)

(3.7), $x_{21}x_{3}=q_{12}^{2}x_{3}x_{21}$

hold in $\mathscr{B}$. Then for all $n\in\mathbb{N}_{0}$,

[TABLE]

Lemma 4.9**.**

Let $\mathscr{B}$ be a quotient algebra of $T(V)$, $\epsilon^{2}=q_{22}^{2}=1$.

(i) ** Assume that (4.20) and (4.21) hold in $\mathscr{B}$. Then for $0\leq t<k\leq|\mathtt{r}|$,

[TABLE]

(ii) ** Assume that $z_{t}^{2}=0$ in $\mathscr{B}$ for $t\in\mathbb{N}_{0}$ such that $\epsilon^{t}q_{22}=-1$. Then $z_{t}z_{t+1}=q_{21}q_{22}z_{t+1}z_{t}$ in $\mathscr{B}$.

In other words, (ii) says that (4.21) for a specific $t$ implies (4.20) for $t$.

4.3.1. The Nichols algebra $\mathscr{B}(\mathfrak{L}(1,\mathscr{G}))$

Proposition 4.10**.**

Let $\mathscr{G}\in\mathbb{I}_{p-1}$. The algebra $\mathscr{B}(\mathfrak{L}(1,\mathscr{G}))$ is presented by generators $x_{1},x_{2},x_{3}$ and relations (3.1), (3.2), (3.3), together with

[TABLE]

The dimension of $\mathscr{B}(\mathfrak{L}(1,\mathscr{G}))$ is $p^{\mathscr{G}+3}$, since it has a PBW-basis

[TABLE]

. ∎

4.3.2. The Nichols algebra $\mathscr{B}(\mathfrak{L}(-1,\mathscr{G}))$

Proposition 4.11**.**

Let $\mathscr{G}\in\mathbb{I}_{p-1}$. The algebra $\mathscr{B}(\mathfrak{L}(-1,\mathscr{G}))$ is presented by generators $x_{1},x_{2},x_{3}$ and relations (3.1), (3.2), (3.3), (4.26), (4.27) and

[TABLE]

The dimension of $\mathscr{B}(\mathfrak{L}(1,\mathscr{G}))$ is $2^{\mathscr{G}+1}p^{2}$, since it has a PBW-basis

[TABLE]

∎

4.3.3. The Nichols algebra $\mathscr{B}(\mathfrak{L}_{-}(1,\mathscr{G}))$

Proposition 4.12**.**

Let $\mathscr{G}\in\mathbb{I}_{2p-1}\cap 2\mathbb{Z}$. The algebra $\mathscr{B}(\mathfrak{L}_{-}(1,\mathscr{G}))$ is presented by generators $x_{1},x_{2},x_{3}$ and relations (3.6), (3.7), (3.8), (3.9), (4.26) and

[TABLE]

The dimension of $\mathscr{B}(\mathfrak{L}(1,\mathscr{G}))$ is $2^{\frac{\mathscr{G}}{2}+2}p^{\frac{\mathscr{G}}{2}+3}$, since it has a PBW-basis

[TABLE]

∎

4.3.4. The Nichols algebra $\mathscr{B}(\mathfrak{L}_{-}(-1,\mathscr{G}))$

Proposition 4.13**.**

Let $\mathscr{G}\in\mathbb{I}_{2p-1}\cap 2\mathbb{Z}$. The algebra $\mathscr{B}(\mathfrak{L}_{-}(-1,\mathscr{G}))$ is presented by generators $x_{1},x_{2},x_{3}$ and relations (3.6), (3.7), (3.8), (3.9), (4.26), (4.31), (4.32) and

[TABLE]

The dimension of $\mathscr{B}(\mathfrak{L}(1,\mathscr{G}))$ is $2^{\frac{\mathscr{G}}{2}+3}p^{\frac{\mathscr{G}}{2}+2}$, since it has a PBW-basis

[TABLE]

∎

4.3.5. The Nichols algebra $\mathscr{B}(\mathfrak{L}(\omega,1))$

Proposition 4.14**.**

Let $\omega\in\mathbb{G}^{\prime}_{3}$. The algebra $\mathscr{B}(\mathfrak{L}(\omega,1))$ is presented by generators $x_{1},x_{2},x_{3}$ and relations (3.1), (3.2), (3.3),(4.26),

[TABLE]

The dimension of $\mathscr{B}(\mathfrak{L}(\omega,1))$ is $3^{3}p^{2}$, since it has a PBW-basis

[TABLE]

∎

4.4. Mild interaction

We assume in this Subsection that $q_{12}q_{21}=-1=\epsilon$, $a=1$, $q_{22}=-1$. The corresponding braided vector space is denoted $\mathfrak{C}_{1}$, as above. We proceed as above but now the elements $z_{n}=(\operatorname{ad}_{c}x_{2})^{n}x_{3}$ are not enough to describe $K^{1}$ and we need $f_{n}=\operatorname{ad}_{c}x_{1}(z_{n})$, $n=0,1$. Then

[TABLE]

Proposition 4.15**.**

The Nichols algebra $\mathscr{B}(\mathfrak{C}_{1})$ is presented by generators $x_{1},x_{2},x_{3}$ and relations (3.6), (3.7), (3.8), (3.9),

[TABLE]

The dimension of $\mathscr{B}(\mathfrak{C}_{1})$ is $64p^{2}$, since it has a PBW-basis

[TABLE]

∎

4.5. Realizations

Let $H$ be a Hopf algebra, $(g_{1},\chi_{1},\eta)$ a YD-triple and $(g_{2},\chi_{2})$ a YD-pair for $H$, see §3.3. Let $(V,c)$ be a braided vector space with braiding (4.1). Then $\mathcal{V}_{g_{1}}(\chi_{1},\eta)\oplus\Bbbk_{g_{2}}^{\chi_{2}}\in{}^{H}_{H}\mathcal{YD}$ is a principal realization of $(V,c)$ over $H$ if

[TABLE]

Thus $(V,c)\simeq\mathcal{V}_{g_{1}}(\chi_{1},\eta)\oplus\Bbbk_{g_{2}}^{\chi_{2}}$ as braided vector space. Hence, if $H$ is finite-dimensional and $(V,c)$ is as in Table 1, then \mathscr{B}\big{(}\mathcal{V}_{g_{1}}(\chi_{1},\eta)\oplus\Bbbk_{g_{2}}^{\chi_{2}}\big{)}\#H is a finite-dimensional Hopf algebra. Examples of finite-dimensional pointed Hopf algebras A=\mathscr{B}\big{(}\mathcal{V}_{g_{1}}(\chi_{1},\eta)\oplus\Bbbk_{g_{2}}^{\chi_{2}}\big{)}\#\Bbbk\Gamma like this are listed in Table 4. In all cases $\Gamma$ is a product of two cyclic groups, $g_{1}=(1,0)$, $g_{2}=(0,1)$ and $\chi_{j}(g_{i})=1$ if $i\neq j$; hence it remains to fix the value of $q_{12}$.

5. One block and several points

5.1. The setting and the main result

Let $\theta\in\mathbb{N}_{\geq 3}$, $\mathbb{I}_{2,\theta}=\mathbb{I}_{\theta}-\{1\}$, $\mathbb{I}^{\dagger}_{\theta}=\mathbb{I}_{\theta}\cup\{\frac{3}{2}\}$. Let $\lfloor i\rfloor$ be the largest integer $\leq i$. We start from the data

[TABLE]

We assume that $q_{11}=1=:a_{1}$. Let $(V,c)$ be the braided vector space of dimension $\theta+1$, with a basis $(x_{i})_{i\in\mathbb{I}^{\dagger}_{\theta}}$ and braiding given by

[TABLE]

We say that the block and the points have discrete ghost if $a_{j}\in\mathbb{F}_{p}^{\mathbb{I}_{2,\theta}}$, $(a_{j})\neq 0$. When this is the case, we pick the representative $\mathtt{r}_{j}\in\mathbb{Z}$ of $2a_{j}$ by imposing $\mathtt{r}_{j}\in\{1-p,\dots,-1,0\}$, and set $\mathscr{G}_{j}=-\mathtt{r}_{j}$. The ghost between the block and the points is the vector $\mathscr{G}=(\mathscr{G}_{j})_{j\in\mathbb{I}_{2,\theta}}$ given by

[TABLE]

The braided subspace $V_{1}$ spanned by $x_{1},x_{\frac{3}{2}}$ is $\simeq\mathcal{V}(1,2)$, while $V_{\operatorname{diag}}$ spanned by $(x_{i})_{i\in\mathbb{I}_{2,\theta}}$ is of diagonal type. Obviously,

[TABLE]

Let $\mathcal{X}$ be the set of connected components of the Dynkin diagram of the matrix $\mathbf{q}=(q_{ij})_{i,j\in\mathbb{I}_{2,\theta}}$. If $J\in\mathcal{X}$, then we set $J^{\prime}=\mathbb{I}_{2,\theta}-J$,

[TABLE]

We shall use the results and notations from the preceding Sections, but with $\frac{3}{2}$ replacing 2 when appropriate, e. g. $x_{\frac{3}{2}1}=x_{\frac{3}{2}}x_{1}-x_{1}x_{\frac{3}{2}}$. Let

[TABLE]

Let

[TABLE]

For all $i,j\in\mathbb{I}_{2,\theta}$, $n\in\mathbb{N}_{0}$, we have as in [AAH1, §5.2.1] that

[TABLE]

Lemma 5.1**.**

The braided vector space $K^{1}$ is of diagonal type in the basis

[TABLE]

with braiding matrix

[TABLE]

Hence, the corresponding generalized Dynkin diagram has labels

[TABLE]

Proof.

The proof in [AAH1, Lemma 7.2.5] applies as the combinatorial numbers appearing there are not zero. ∎

Let $K_{J}$ be the braided vector subspace of $K^{1}$ spanned by $(z_{j,n})_{\begin{subarray}{c}j\in J,\\ 0\leq n\leq\mathscr{G}_{j}\end{subarray}}$.

Corollary 5.2**.**

The braided subspaces corresponding to the connected components of the Dynkin diagram of $K^{1}$ are $K_{J}$, $J\in\mathcal{X}$. Hence

[TABLE]

Observe that if $\mathscr{G}_{J}=0$, then $K_{J}=V_{J}$.

In this Section, we shall prove part (b) of Theorem 1.2.

Theorem 5.3**.**

Let $V$ be a braided vector space with braiding (5.1). Assume that for every $J\in\mathcal{X}$, either $\mathscr{G}_{J}=0$, or else $\dim V_{J}=1$ and $V_{1}\oplus V_{j}$ is as in Table 1, or else $V_{J}$ is as in Table 2. Then

[TABLE]

Proof.

By Corollary 5.2 we reduce to connected components in $\mathcal{X}$. If $J\in\mathcal{X}$ has weak interaction and $\mathscr{G}_{J}=0$, then $\mathscr{B}(V)\simeq\mathscr{B}(V_{1}\oplus V_{J^{\prime}})\,\underline{\otimes}\,\mathscr{B}(V_{J})$, hence $\dim\mathscr{B}(V)=\dim\mathscr{B}(V_{1}\oplus V_{J^{\prime}})\dim\mathscr{B}(V_{J})$.

If $J\in\mathcal{X}$ is a point, then Theorem 4.1 applies. We need to analyze those $J$ with $|J|\geq 2$ and $\mathscr{G}_{J}\neq 0$.

Below we denote $\imath\,=\sqrt{-1}$.

$\mathfrak{L}(A_{2})$, $\mathscr{G}_{J}=(1,0)$: Here $K_{J}$ is of Cartan type $A_{3}$ and $\dim\mathscr{B}(K_{J})=2^{6}$.

$\mathfrak{L}(A_{j})$, $j>2$, $\mathscr{G}_{J}=(1,0)$: Here $K_{J}$ is of Cartan type $D_{j-1}$ and $\dim\mathscr{B}(K_{J})=2^{j(j+1)}$.

$\mathfrak{L}(A_{2},2)$: Here $K_{J}$ is of Cartan type $D_{4}$ and $\dim\mathscr{B}(K_{J})=2^{12}$.

$\mathfrak{L}(A(1|0)_{2};\omega)$, provided that $p>3$: Here $K_{J}$ is of diagonal type in the basis $1=z_{i,0}$, $2=z_{i,1}$, $3=z_{j,0}$ with diagram

[TABLE]

Then $\dim\mathscr{B}(K_{J})<\infty$ by [H2, Table 2, row 15]; it has the same root system as $\mathfrak{g}(2,3)$ and dimension $2^{7}3^{4}$, see [AA, 8.3.4].

$\mathfrak{L}(A(1|0)_{1};\omega)$ and $\mathfrak{L}(A(1|0)_{3};\omega)$, provided that $p>3$: Here $K_{J}$ is of diagonal type in the basis $1=z_{i,0}$, $2=z_{j,1}$, $3=z_{j,0}$, respectively $1=z_{i,0}$, $2=z_{i,1}$, $3=z_{j,0}$, and its diagram is

[TABLE]

Then $\dim\mathscr{B}(K_{J})<\infty$ by [H2, Table 2, row 8], respectively [H2, Table 2, row 15]. In the first case it has the same root system as $\mathfrak{sl}(2|2)$ and dimension $2^{4}3^{2}$, see [AA, 5.1.8]. In the second case it has the same root system as $\mathfrak{g}(2,3)$ and dimension $2^{7}3^{4}$, see [AA, 8.3.4].

R$\mathfrak{L}(A(1|0)_{1};r)$: analogous to $\mathfrak{L}(A(1|0)_{1};\omega)$; same root system as $\mathfrak{sl}(2|2)$ and dimension $2^{4}N^{2}$, see [AA, 5.1.8].

$\mathfrak{L}(A(2|0)_{1};\omega)$ and $\mathfrak{L}(D(2|1);\omega)$, $p>3$: In both cases, $\dim\mathscr{B}(K_{J})=2^{8}3^{9}$; it has the same root system as $\mathfrak{g}(3,3)$, see [H2, Table 3, row 18], [AA, 8.4.5]. ∎

5.2. The presentation of the Nichols algebras

We give defining relations and an explicit PBW basis of $\mathscr{B}(V)$, for all $V$ as in Theorem 5.3, assuming that the Dynkin diagram of $V_{\operatorname{diag}}$ is connected, i.e. $V_{\operatorname{diag}}=V_{J}$, where $J=\mathbb{I}_{2,\theta}$. Essentially the relations are the same as in [AAH1] up to adding the suitable $p$-powers; we omit the proofs as they are minor variations of those in loc. cit. The passage from connected $V_{\operatorname{diag}}$ to the general case is standard, just add the quantum commutators between points in different components. Since the case $|J|=1$ was treated in §4, we also suppose that $|J|>1$. These braided vector spaces have names given in [AAH1], see Table 2. The braided vector subspace $V_{1}\oplus\Bbbk x_{2}$ of such $V$ is of type

• •

$\mathfrak{L}(-1,2)$ when $V$ is of type $\mathfrak{L}(A_{2},2)$,

• •

$\mathfrak{L}(\omega,1)$ when $V$ is of type $\mathfrak{L}(A(1|0)_{3};\omega)$, or

• •

$\mathfrak{L}(-1,1)$ for all the other cases.

Thus the subalgebra generated by $V_{1}\oplus\Bbbk x_{2}$ is a Nichols algebra. We recall its relations up to the change of index with respect to §4; the $2$ and $3$ there are now $\frac{3}{2}$ and $2$. As in (2.2), we set $x_{i_{1}i_{2}\dots i_{M}}=\operatorname{ad}_{c}x_{i_{1}}\,x_{i_{2}\dots i_{M}}$. Also, we have now $z_{n}=(\operatorname{ad}_{c}x_{\frac{3}{2}})^{n}x_{2}$, $n\in\mathbb{N}_{0}$.

First, the defining relations of $\mathscr{B}(\mathfrak{L}(-1,1))$ are

[TABLE]

Second, the defining relations of $\mathscr{B}(\mathfrak{L}(-1,2))$ are (5.10), (3.2), (5.11), (5.12), (5.14) and

[TABLE]

Third, the defining relations of $\mathscr{B}(\mathfrak{L}(\omega,1))$ are (5.10), (3.2), (5.11), (5.12), (5.13) and

[TABLE]

We also observe that, since $q_{1j}q_{j1}=1$ and $\mathscr{G}_{j}=0$, we have

[TABLE]

5.2.1. The Nichols algebra $\mathscr{B}(\mathfrak{L}(A(1|0)_{1};r))$, $r\in\mathbb{G}_{N}^{\prime}$, $N\geq 3$

Let

[TABLE]

Proposition 5.4**.**

The algebra $\mathscr{B}(\mathfrak{L}(A(1|0)_{1};r))$ is presented by generators $x_{i}$, $i\in\mathbb{I}^{\dagger}_{3}$, and relations (5.10), (3.2), (5.11), (5.12), (5.13), (5.14) (5.20), and

[TABLE]

The set

[TABLE]

is a basis of $\mathscr{B}(\mathfrak{L}(A(1|0)_{1};r))$ and $\dim\mathscr{B}(\mathfrak{L}(A(1|0)_{1};r))=p^{2}2^{4}N^{2}$. ∎

5.2.2. The Nichols algebra $\mathscr{B}(\mathfrak{L}(A(1|0)_{2};\omega))$

Proposition 5.5**.**

The algebra $\mathscr{B}(\mathfrak{L}(A(1|0)_{2};\omega))$ is presented by generators $x_{i}$, $i\in\mathbb{I}^{\dagger}_{3}$, and relations (5.10), (3.2), (5.11), (5.12), (5.13), (5.14), (5.20), and

[TABLE]

The set

[TABLE]

is a basis of $\mathscr{B}(\mathfrak{L}(A(1|0)_{2};\omega))$ and $\dim\mathscr{B}(\mathfrak{L}(A(1|0)_{2};\omega))=p^{2}2^{7}3^{4}$. ∎

5.2.3. The Nichols algebra $\mathscr{B}(\mathfrak{L}(A(1|0)_{3};\omega))$

Proposition 5.6**.**

The algebra $\mathscr{B}(\mathfrak{L}(A(1|0)_{3};\omega))$ is presented by generators $x_{i}$, $i\in\mathbb{I}^{\dagger}_{3}$, and relations (5.10), (3.2), (5.11), (5.12), (5.13), (5.17), (5.18), (5.19) (5.20),

[TABLE]

The set

[TABLE]

is a basis of $\mathscr{B}(\mathfrak{L}(A(1|0)_{3};\omega))$ and $\dim\mathscr{B}(\mathfrak{L}(A(1|0)_{3};\omega))=p^{2}2^{7}3^{4}$. ∎

5.2.4. The Nichols algebra $\mathscr{B}(\mathfrak{L}(A(2|0)_{1};\omega))$

Proposition 5.7**.**

The algebra $\mathscr{B}(\mathfrak{L}(A(2|0)_{1};\omega))$ is presented by generators $x_{i}$, $i\in\mathbb{I}^{\dagger}_{4}$, and relations (5.10), (3.2), (5.11), (5.12), (5.13), (5.14), (5.20), and

[TABLE]

The set

[TABLE]

is a basis of $\mathscr{B}(\mathfrak{L}(A(2|0)_{1};\omega))$ and $\dim\mathscr{B}(\mathfrak{L}(A(2|0)_{1};\omega))=p^{2}2^{8}3^{9}$. ∎

5.2.5. The Nichols algebra $\mathscr{B}(\mathfrak{L}(D(2|1);\omega))$

Proposition 5.8**.**

The algebra $\mathscr{B}(\mathfrak{L}(D(2|1);\omega))$ is presented by generators $x_{i}$, $i\in\mathbb{I}^{\dagger}_{4}$, and relations (5.10), (3.2), (5.11), (5.12), (5.13), (5.14), (5.20), and

[TABLE]

The set

[TABLE]

is a basis of $\mathscr{B}(\mathfrak{L}(D(2|1);\omega))$ and $\dim\mathscr{B}(\mathfrak{L}(D(2|1);\omega))=p^{2}2^{8}3^{9}$. ∎

5.2.6. The Nichols algebra $\mathscr{B}(\mathfrak{L}(A_{2},2))$

Proposition 5.9**.**

The algebra $\mathscr{B}(\mathfrak{L}(A_{2},2))$ is presented by generators $x_{i}$, $i\in\mathbb{I}^{\dagger}_{3}$, and relations (5.10), (3.2), (5.11), (5.12), (5.14), (5.15), (5.16), (5.20), and

[TABLE]

The set $B=$

[TABLE]

is a basis of $\mathscr{B}(\mathfrak{L}(A_{2},2))$ and $\dim\mathscr{B}(\mathfrak{L}(A_{2},2))=p^{2}2^{12}$. ∎

5.2.7. The Nichols algebra $\mathscr{B}(\mathfrak{L}(A_{\theta-1}))$

For details of the following result we refer to **[AAH1, §5.3.8]**. In particular, one defines $x_{ij}$, $2\leq i\leq j$, as usual, $\mbox{я}_{1\ell}=[x_{\frac{3}{2}2},x_{3\ell}]_{c}$ and $\mbox{я}_{k\ell}=[\mbox{я}_{1\ell},x_{2k}]$, $k>1$.

Proposition 5.10**.**

The algebra $\mathscr{B}(\mathfrak{L}(A_{\theta-1}))$ is presented by generators $x_{i}$, $i\in\mathbb{I}^{\dagger}_{\theta}$, and relations (5.10), (3.2), (5.11), (5.12), (5.13), (5.14), (5.20), and

[TABLE]

Furthermore there is a PBW-basis in terms of the positive roots of the root system of type $D_{\theta}$ and $\dim\mathscr{B}(\mathfrak{L}(A_{\theta-1}))=p^{2}2^{\theta(\theta-1)}$. ∎

5.3. Realizations

Let $H$ be a Hopf algebra, $(g_{1},\chi_{1},\eta)$ a YD-triple and $(g_{j},\chi_{j})$, $j\in\mathbb{I}_{2,\theta}$, a family of YD-pairs for $H$, see §3.3. Let $(V,c)$ be a braided vector space with braiding (5.1). Then

[TABLE]

is a principal realization of $(V,c)$ over $H$ if

[TABLE]

Thus $(V,c)\simeq\mathcal{V}$ as braided vector space. Consequently, if $H$ is finite-dimensional and $(V,c)$ is as in Table 2, then $\mathscr{B}(\mathcal{V})\#H$ is a finite-dimensional Hopf algebra. Examples of finite-dimensional pointed Hopf algebras $A=\mathscr{B}(V)\#\Bbbk\Gamma$ with $\Gamma$ abelian like this are listed in Table 5 where the interaction is weak, $\epsilon=1$ and $\omega\in\mathbb{G}^{\prime}_{3}$. As in §4.5, $\Gamma$ is a product of $\theta$ cyclic groups, $g_{i}$ is the $i$-th canonical generator, $\chi_{j}(g_{i})=1$ if $i\neq j$; we set $q_{ij}=1$, $i<j$.

6. Several blocks, one point

Let $t\geq 2$ and $\theta=t+1$. We shall use the following notation:

[TABLE]

The Poseidon braided vector space $\mathfrak{P}(\mathbf{q},\mathscr{G})$ depends on a datum

• •

$\mathbf{q}\in\Bbbk^{\theta\times\theta}$* such that $\epsilon_{i}:=q_{ii}=\pm 1$, $q_{ij}q_{ji}=1$ for all $i\neq j\in\mathbb{I}_{\theta}$.*

• •

$\mathscr{G}=(\mathscr{G}_{j})\in\mathbb{N}^{t}$, $(a_{j})$ such that $\mathscr{G}_{j}=\begin{cases}-\mathtt{r}_{j},&\epsilon_{j}=1,\\ \mathtt{r}_{j},&\epsilon_{j}=-1;\end{cases}$ cf. (4.2);

it has a basis $(x_{i})_{i\in\mathbb{I}^{\ddagger}}$ and braiding

[TABLE]

We shall use the elements

[TABLE]

Let $\mathcal{A}:=\{\mathbf{n}\in\mathbb{N}^{t}_{0}:0\leq\mathbf{n}\leq\mathbf{a}=(|\mathtt{r}_{1}|,\dots,|\mathtt{r}_{t}|)\}$, ordered lexicographically. For $\mathbf{m},\mathbf{n}\in\mathcal{A}$ we set

[TABLE]

We also need the following notation:

[TABLE]

The main result of this Section is the following.

Proposition 6.1**.**

The algebra $\mathscr{B}(\mathfrak{P}(\mathbf{q},\mathscr{G}))$ is presented by generators $x_{i}$, $i\in\mathbb{I}^{\ddagger}$, and relations

[TABLE]

A basis of $\mathscr{B}(\mathfrak{P}(\mathbf{q},\mathscr{G}))$ is given by

[TABLE]

Hence $\dim\mathscr{B}(\mathfrak{P}(\mathbf{q},\mathscr{G}))=2^{2t_{-}+M_{-}}p^{2t+M_{+}}$. ∎

6.1. Realizations

Let $H$ be a Hopf algebra, $(g_{i},\chi_{i},\eta_{i})$, $i\in\mathbb{I}_{t}$, a family of YD-triples and $(g_{\theta},\chi_{\theta})$ a YD-pair for $H$, see §3.3. Let $(V,c)$ be a braided vector space with braiding (6.6). Then

[TABLE]

is a principal realization of $(V,c)$ over $H$ if

[TABLE]

Thus $(V,c)\simeq\mathcal{V}$ as braided vector space. Consequently, if $H$ is finite-dimensional, then $\mathscr{B}(\mathcal{V})\#H$ is a finite-dimensional Hopf algebra.

If $q_{ij}=1$ for $i\neq j$, then we may choose $H=\Bbbk\Gamma$, where

[TABLE]

Thus $\mathscr{B}(\mathfrak{P}(\mathbf{q},\mathscr{G}))\#\Bbbk\Gamma$ is a Hopf algebra of dimension $2^{3t_{-}+M_{-}+\delta_{\epsilon_{\theta},-1}}p^{3t+M_{+}+1}$.

7. A pale block and a point

Let $V$ be a braided vector space of dimension 3 with braiding given in the basis $(x_{i})_{i\in\mathbb{I}_{3}}$ by

[TABLE]

Let $V_{1}=\langle x_{1},x_{2}\rangle$, $V_{2}=\langle x_{3}\rangle$. Let $\Gamma=\mathbb{Z}^{2}$ with a basis $g_{1},g_{2}$. We realize $V$ in ${}^{\Bbbk\Gamma}_{\Bbbk\Gamma}\mathcal{YD}$ by $V_{1}=V_{g_{1}}$, $V_{2}=V_{g_{2}}$, $g_{1}\cdot x_{1}=\epsilon x_{1}$, $g_{2}\cdot x_{1}=q_{21}x_{1}$, $g_{1}\cdot x_{2}=\epsilon x_{2}$, $g_{2}\cdot x_{2}=q_{21}(x_{2}+x_{1})$, $g_{i}\cdot x_{3}=q_{i2}x_{3}$.

*As usual, let $\widetilde{q}_{12}=q_{12}q_{21}$; in particular the Dynkin diagram of the braided subspace $\langle x_{1},x_{3}\rangle$ is * \textstyle{\overset{\epsilon}{\circ}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\widetilde{q}_{12}}$$\textstyle{\overset{q_{22}}{\circ}} .

As for other cases, we consider $K=\mathscr{B}(V)^{\mathrm{co}\,\mathscr{B}(V_{1})}$; then $K=\oplus_{n\geq 0}K^{n}$ inherits the grading of $\mathscr{B}(V)$; $\mathscr{B}(V)\simeq K\#\mathscr{B}(V_{1})$ and $K$ is the Nichols algebra of $K^{1}=\operatorname{ad}_{c}\mathscr{B}(V_{1})(V_{2})$. Now $K^{1}\in{}^{\mathscr{B}(V_{1})\#\Bbbk\Gamma}_{\mathscr{B}(V_{1})\#\Bbbk\Gamma}\mathcal{YD}$ with the adjoint action and the coaction given by (4.4), i.e. $\delta=(\pi_{\mathscr{B}(V_{1})\#\Bbbk\Gamma}\otimes\operatorname{id})\Delta_{\mathscr{B}(V)\#\Bbbk\Gamma}$. Next we introduce $\mbox{ш}_{m,n}=(\operatorname{ad}_{c}x_{1})^{m}(\operatorname{ad}_{c}x_{2})^{n}x_{3}$; we distinguish two cases:

[TABLE]

By direct computation,

[TABLE]

7.1. The block has $\epsilon=1$

Here $\mathscr{B}(V_{1})\simeq S(V_{1})$ is a polynomial algebra, so that $x_{1}$ and $x_{2}$ commute, and

[TABLE]

Thus $\mbox{ш}_{m,n}$, $m,n\in\mathbb{N}_{0}$ generate $K^{1}$. As in **[AAH1, §8.1]**, we have that

[TABLE]

For $q\in\Bbbk^{\times}$, let $\mathfrak{E}_{p}(q)=V$ be the braided vector space as in (7.1) under the assumptions that $\epsilon=1$, $q_{12}=q=q_{21}^{-1}$, $q_{22}=-1$. We call $\mathscr{B}(\mathfrak{E}_{p}(q))$ and the Nichols algebras $\mathscr{B}(\mathfrak{E}_{\pm}(q))$, $\mathscr{B}(\mathfrak{E}_{\star}(q))$ studied in Propositions 7.2, 7.3 and 7.4 the Endymion algebras.

Proposition 7.1**.**

The algebra $\mathcal{B}(\mathfrak{E}_{p}(q))$ is presented by generators $x_{1},x_{2},x_{3}$ and relations

[TABLE]

The dimension of $\mathscr{B}(\mathfrak{E}_{p}(q))$ is $2^{p}p^{2}$, since it has a PBW-basis

[TABLE]

Proof.

We claim that $\dim K^{1}=p$ and $K\simeq\Lambda(K^{1})$. In fact, by (7.4), (7.5) and the hypothesis $\widetilde{q}_{12}=1$, $w_{m}=0$ for all $m>0$; thus $\mbox{ш}_{m,n}=0$ for all $m>0$ by (7.6). Using this fact and (7.7),

[TABLE]

We have that for all $n\in\mathbb{N}_{0}$:

[TABLE]

Indeed the proof follows as in [AAH1, Lemma 8.1.4]. As $x_{1}^{p}=0$, we have $z_{p}=0$. Thus the set $(z_{n})_{n\in\mathbb{I}_{0,p-1}}$ is a basis of $K^{1}$. The braiding is

[TABLE]

That is, $K^{1}$ is of diagonal type and the Dynkin diagram consists of disconnected $p$ points labeled with $-1$. Hence $\mathscr{B}(K^{1})=\Lambda(K^{1})$. Moreover, $B$ is a basis of $\mathscr{B}(V)$ since $\mathscr{B}(V)\simeq K\#\mathscr{B}(V_{1})$.

The presentation follows as in [AAH1, Proposition 4.3.7]. ∎

7.2. The block has $\epsilon=-1$

Here $\mathscr{B}(V_{1})\simeq\Lambda(V_{1})$ is an exterior algebra and consequently $\mbox{ш}_{m,n}$, $m,n\in\{0,1\}$ generates $K^{1}$. By direct computation,

[TABLE]

7.2.1. Case 1: $\widetilde{q}_{12}=1$

Here $w_{1}=0$ by (7.4) and (7.5), so

[TABLE]

Thus $z_{0}=x_{3}$ and $z_{1}$ form a basis of $K^{1}$ and the braiding of $K^{1}$ is given by

[TABLE]

*That is, $K^{1}$ is of diagonal type with Dynkin diagram * \textstyle{\overset{q_{22}}{\circ}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{q_{22}^{2}}$$\textstyle{\overset{-q_{22}}{\circ}} .

For $q\in\Bbbk^{\times}$, let $\mathfrak{E}_{\pm}(q)=V$ be the braided vector space as in (7.1) under the assumptions that $\epsilon=-1$, $q_{12}=q=q_{21}^{-1}$, $q_{22}=\pm 1$.

Proposition 7.2**.**

The algebra $\mathcal{B}(\mathfrak{E}_{+}(q))$ is presented by generators $x_{1},x_{2},x_{3}$ and relations

[TABLE]

Let $z_{1}=x_{2}x_{3}-qx_{3}x_{2}$. Then $\mathcal{B}(\mathfrak{E}_{+}(q))$ has a PBW-basis

[TABLE]

hence $\dim\mathcal{B}(\mathfrak{E}_{+}(q))=2^{3}p$.

Proof.

Notice that $x_{3}^{p}=0$ since $x_{3}$ is a point labeled with $q_{22}=1$ in $K^{1}$. Also, $B$ is a basis thanks to the isomorphism $\mathscr{B}(\mathfrak{E}_{+}(q))\simeq\mathscr{B}(K^{1})\#\mathscr{B}(V_{1})$. The rest of the proof follows as in [AAH1, Proposition 8.1.6]. ∎

Proposition 7.3**.**

The algebra $\mathcal{B}(\mathfrak{E}_{-}(q))$ is presented by generators $x_{1},x_{2},x_{3}$ and relations (7.17), (7.20),

[TABLE]

Let $z_{1}=x_{2}x_{3}-qx_{3}x_{2}$. Then $\mathcal{B}(\mathfrak{E}_{-}(q))$ has a PBW-basis

[TABLE]

hence $\dim\mathcal{B}(\mathfrak{E}_{-}(q))=2^{3}p$.

Proof.

Notice that $z_{1}^{p}=0$ since $z_{1}$ is a point labeled with $1$ in $K^{1}$. Also, $B$ is a basis thanks to the isomorphism $\mathscr{B}(\mathfrak{E}_{-}(q))\simeq\mathscr{B}(K^{1})\#\mathscr{B}(V_{1})$. The rest of the proof follows as in [AAH1, Proposition 8.1.6]. ∎

7.2.2. Case 2: $\widetilde{q}_{12}=-1$

We consider now a fixed choice of $q_{22}$ and $\widetilde{q}_{12}$, which is the corresponding one to the example of finite $\operatorname{GKdim}$ over a field of characteristic 0.

For $q\in\Bbbk^{\times}$, let $\mathfrak{E}_{\star}(q)=V$ be the braided vector space as in (7.1) under the assumptions that $\epsilon=-1$, $q_{22}=-1$, $q_{12}=q$, $q_{21}=-q^{-1}$. Recall (2.2).

Proposition 7.4**.**

The algebra $\mathcal{B}(\mathfrak{E}_{\star}(q))$ is presented by generators $x_{1},x_{2},x_{3}$ and relations (7.17),

[TABLE]

Moreover $\mathcal{B}(\mathfrak{E}_{\star}(q))$ has a PBW-basis

[TABLE]

hence $\dim\mathcal{B}(\mathfrak{E}_{\star}(q))=2^{6}p^{2}$.

Proof.

Relations (7.17) are 0 in $\mathcal{B}(\mathfrak{E}_{\star}(q))$ because $\mathscr{B}(V_{1})\simeq\Lambda(V_{1})$; (7.23) are 0 since $x_{1}$, $x_{3}$ generate a Nichols algebra of Cartan type $A_{2}$ at $-1$.

Notice that $[x_{23},x_{13}]_{c}=x_{23}x_{13}+x_{13}x_{23}$. By (7.17) and (7.23),

[TABLE]

As in [AAH1, Proposition 8.1.8] we check that

[TABLE]

Now we prove that (7.25) holds in $\mathscr{B}(V)$. We check that $\partial_{i}$ annihilates these terms for $i=1,2,3$. To simplify the notation, let $u=[x_{23},x_{13}]_{c}$. As $\partial_{1}$, $\partial_{2}$ annihilate $x_{23}$, $x_{213}$ and $u$, it remains the case $i=3$. Using (7.26)-(7.31),

[TABLE]

For the remaining relation, we check that $\partial_{3}(x_{23}^{2})=q^{-1}x_{213}-x_{13}(2x_{2}+x_{1})$ and the following equalities hold (using (7.26)-(7.31)):

[TABLE]

Using the previous computations and $x_{13}^{2}=0$,

[TABLE]

We apply this equality to compute:

[TABLE]

Using (7.24), (7.29) and (7.30) we get $ux_{23}^{2}=(u+x_{23}^{2})u$. Hence $a=x_{23}^{2}+u$ and $b=u$ satisfy the last equation of (3.12), so (3.13) applies and we have

[TABLE]

The rest of the proof follows as in [AAH1, Proposition 8.1.8]. ∎

7.3. Realizations

Here we present a realization of a braided vector space as in (7.1) over a group algebra $H=\Bbbk\Gamma$, with $\Gamma$ a finite abelian group. We consider $V_{1}=\langle x_{1},x_{2}\rangle$, $V_{2}=\langle x_{3}\rangle$. We realize $V$ in ${}^{\Bbbk\Gamma}_{\Bbbk\Gamma}\mathcal{YD}$ by $V_{1}=V_{g_{1}}$, $V_{2}=V_{g_{2}}$, $g_{1}\cdot x_{1}=\epsilon x_{1}$, $g_{2}\cdot x_{1}=q_{21}x_{1}$, $g_{1}\cdot x_{2}=\epsilon x_{2}$, $g_{2}\cdot x_{2}=q_{21}(x_{2}+x_{1})$, $g_{i}\cdot x_{3}=q_{i2}x_{3}$. In all the cases $\Gamma$ will be a product of two cyclic groups, $g_{1}=(1,0)$, $g_{2}=(0,1)$. Examples of finite-dimensional pointed Hopf algebras A=\mathscr{B}\big{(}V_{g_{1}}\oplus V_{g_{2}}\big{)}\#H are listed in Table 6.