# On the finite generation of the cohomology of abelian extensions of Hopf
algebras

Nicolás Andruskiewitsch and Sonia Natale  FaMAF-CIEM (CONICET), Universidad
Nacional de Córdoba, Medina Allende s/n, Ciudad Universitaria, 5000 Córdoba,
Argentina<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

A finite-dimensional Hopf algebra is called quasi-split if it is Morita
equivalent to a split abelian extension of Hopf algebras. Combining results of
Schauenburg and Negron, it is shown that every quasi-split finite-dimensional
Hopf algebra satisfies the finite generation cohomology conjecture of Etingof
and Ostrik. This is applied to a family of pointed Hopf algebras in odd
characteristic introduced by Angiono, Heckenberger and the first author,
proving that they satisfy the aforementioned conjecture.

2020 _Mathematics Subject Classification._ 16T05; 17B37; 18M05. The work of N.
A. was partially supported by Secyt (UNC), CONICET (PIP 11220200102916CO),
FONCyT-ANPCyT (PICT-2019-03660) and by the Shenzhen International Center of
Mathematics, SUSTech. The research of S. N. was partially supported by Secyt
(UNC)

##  1\. Introduction

###  1.1. The problem

Let $\Bbbk$ be an algebraically closed field. We say that an augmented
$\Bbbk$-algebra $A$ has finitely generated cohomology (fgc for short) when

  1. (a)

the cohomology ring
$\operatorname{H}(A,\Bbbk)=\oplus_{n\in\mathbb{N}_{0}}\operatorname{Ext}^{n}_{A}(\Bbbk,\Bbbk)$
is finitely generated, and

  2. (b)

for any finitely generated $A$-module $M$,
$\operatorname{H}(A,M)=\oplus_{n\in\mathbb{N}_{0}}\operatorname{Ext}^{n}_{A}(\Bbbk,M)$
is a finitely generated $\operatorname{H}(A,\Bbbk)$-module.

This definition was extended in [16] as follows. A finite tensor category
${\mathcal{C}}$ (with unit object $\mathbb{1}$) has finite generation of
cohomology (or fgc) when

  1. (a)

the $\Bbbk$-algebra
$\operatorname{H}({\mathcal{C}})=\oplus_{n\in\mathbb{N}_{0}}\operatorname{Ext}^{n}(\mathbb{1},\mathbb{1})$
is finitely generated, and

  2. (b)

for any object $V$ in ${\mathcal{C}}$ the
$\operatorname{H}({\mathcal{C}})$-module
$\operatorname{H}(V)=\oplus_{n\in\mathbb{N}_{0}}\operatorname{Ext}^{n}(\mathbb{1},V)$
is finitely generated.

In [16], Etingof and Ostrik, drawing on fundamental results of [44, 23, 18, 1,
20, 21, 22], conjectured that finite tensor categories have fgc, in particular
that finite-dimensional Hopf algebras have fgc; this was verified in many
cases [10, 24, 13, 14, 31, 36, 17, 41, 19, 34, 35, 5]; see [5, § 1.1] for
background.

###  1.2. Morita equivalence

Let $H$ and $U$ be finite-dimensional Hopf algebras. By [35, Theorem 3.4], see
also [5, Theorem 3.2.1], if the Drinfeld double $D(H)$ has fgc, then $H$ has
fgc. The following argument was intensively used in [5].

We say that $H$ and $U$ are _Morita equivalent_ , denoted
$H\sim_{\text{Mor}}U$, if the finite tensor categories $\operatorname{rep}H$
and $\operatorname{rep}U$ are Morita equivalent as in [33, 15]. Equivalently,
$H\sim_{\text{Mor}}U$ if there exists an equivalence of braided tensor
categories between the Drinfeld centers $\mathcal{Z}(\operatorname{rep}H)$ and
$\mathcal{Z}(\operatorname{rep}U)$ [15]. In other words, $H\sim_{\text{Mor}}U$
iff $D(H)$ and $D(U)$ are twist equivalent quasitriangular Hopf algebras (this
does not imply that $H$ and $U$ are Morita equivalent as algebras). Thus to
prove that $H$ has fgc it suffices to find $U$ such that

$\displaystyle\textrm{(i)}\ D(U)$ has fgc and (ii) $\displaystyle
H\sim_{\text{Mor}}U.$

###  1.3. Extensions

In this paper we consider the question of fgc for a Hopf algebra $H$ fitting
into an exact sequence of finite-dimensional Hopf algebras

(1) $\displaystyle\Bbbk\to K\rightarrow H\rightarrow L\to\Bbbk.$

If $H$ has fgc, then $K$ has fgc by [5, Theorem 3.2.1] while it is unclear
whether we could infer that $L$ has fgc. Thus, it is natural to ask:

######  Question 1.1.

Given an extension (1) such that $K$ and $L$ have fgc, does $H$ also have fgc?

For instance, if $K$ is semisimple and $L$ has fgc, then $H$ has fgc, see [5,
Lemma 3.2.5]. The proof uses a variation of the classical Hochschild-Serre
spectral sequence but it is not clear (to us) how to proceed when the
semisimplicity assumption on $K$ is dropped.

The extension (1) is _abelian_ when $K$ is commutative and $L$ is
cocommutative. In this case there are suitable actions of $L$ on $K$ and of
$K^{*}$ on $L$, and a pair $(\sigma,\tau)$ of compatible cocycles that
determine the possible extensions $H$. The suitable actions give rise to a
double complex $\mathscr{E}$. The abelian extension (1) is _split_ if the pair
$(\sigma,\tau)$ is trivial in the total complex associated to $\mathscr{E}$.
Furthermore, by an argument due to G. I. Kac there is a long exact sequence
(the Kac exact sequence) in which the pair $(\sigma,\tau)$ is sent to a
3-cocycle. See [32] for details.

We shall say that an abelian extension is _quasi-split_ if it is Morita
equivalent to the split extension (with respect to the same suitable actions);
see Definition 3.1.

Tautologically, split abelian extensions are quasi-split; also, abelian
extensions for which the Kac 3-cocycle is trivial are quasi-split (see
Corollary 3.5). The starting point of the paper is the following result.

######  Theorem 1.2.

If (1) is a quasi-split abelian extension, then $H$ and $D(H)$ have fgc.

See Theorem 3.6 for a precise formulation. As a consequence, the dual, any
twist and any cocycle deformation of $H$ have fgc. In characteristic 0,
Theorem 1.2 is trivial, as a finite-dimensional abelian extension is
semisimple.

Theorem 1.2 follows combining the next two results due to Negron and
Schauenburg respectively.

######  Theorem 1.3.

[34] If $U$ is a cocommutative Hopf algebra, $\dim U<\infty$, then $D(U)$ has
fgc.

Notice that Theorem 1.3 is stated in [34] in the language of finite group
schemes.

######  Theorem 1.4.

[39] Let (1) be a split abelian extension of finite-dimensional Hopf algebras.
Then there exists a cocommutative Hopf algebra $U$ such that
$H\sim_{\text{Mor}}U$.

Observe that the Drinfeld double of a finite-dimensional cocommutative Hopf
algebra is a split abelian extension.

###  1.4. Pointed Hopf algebras

In the paper [3] (assuming $\operatorname{char}\Bbbk=0$) Nichols algebras over
abelian groups with finite Gelfand-Kirillov dimension of a certain kind were
classified. It was observed later in [4] that many of the Nichols algebras
over the analogous braided vector spaces in odd characteristic have finite
dimension, hence give rise to finite dimensional pointed Hopf algebras by
bosonization with group algebras of finite abelian groups. Theorem 6.7, the
main result of this paper, shows that these pointed Hopf algebras have fgc
when the groups are well chosen. Concretely, in the notation of Section 6, we
consider a family of braided vector spaces
$\mathscr{V}(\mathfrak{q},\mathbf{a})$, we fix a suitable finite group
$\Gamma$ such that the Nichols algebra
$\mathscr{B}(\mathscr{V}(\mathbf{1},\mathbf{a}))$ can be realized as a Hopf
algebra in the category ${}^{\Bbbk\Gamma}_{\Bbbk\Gamma}\mathcal{YD}$ of
Yetter-Drinfeld modules over $\Gamma$ and proceed in two stages:

  1. (i)

we show that the bosonization
$H=\mathscr{B}(\mathscr{V}(\mathbf{1},\mathbf{a}))\\#\Bbbk\Gamma$ fits into a
_split_ abelian exact sequence, hence $D(H)$ and $H$ have fgc by Theorem 1.2;

  2. (ii)

for a general $\mathfrak{q}$, we present
$\mathscr{B}(\mathscr{V}(\mathfrak{q},\mathbf{a}))\\#\Bbbk\Gamma$ as a cocycle
deformation of $H$, hence
$\mathscr{B}(\mathscr{V}(\mathfrak{q},\mathbf{a}))\\#\Bbbk\Gamma$ has fgc too.

We observe:

  * $\circ$

Not all Nichols algebras in [4] belong to the family treated here; most of the
remaining ones arise from abelian extensions of Hopf superalgebras and will be
dealt with elsewhere.

  * $\circ$

Many realizations of $\mathscr{B}(\mathscr{V}(\mathbf{1},\mathbf{a}))$ fit
into abelian exact sequences which are not split.

  * $\circ$

Many realizations of $\mathscr{B}(\mathscr{V}(\mathfrak{q},\mathbf{a}))$ fit
into exact sequences which are not abelian.

###  1.5. Organization

Abelian extensions are discussed in Section 2. Section 3 is devoted to Theorem
1.4. The analysis of the Nichols algebras
$\mathscr{B}(\mathscr{V}(\mathfrak{q},\mathbf{a}))$ is delicate as it involves
a number of distinct combinatorial features developped in [3]. For clarity, we
first deal with the two simplest examples, namely the restricted Jordan plane
and the first Laestrygonian algebra, in Sections 4 and 5 respectively. In
Section 6, we work out the strategy outlined above for the general Nichols
algebras $\mathscr{B}(\mathscr{V}(\mathfrak{q},\mathbf{a}))$.

###  1.6. Conventions

For $\ell<\theta\in\mathbb{N}_{0}$, we set
$\mathbb{I}_{\ell,\theta}=\\{\ell,\ell+1,\dots,\theta\\}$,
$\mathbb{I}_{\theta}=\mathbb{I}_{1,\theta}$. Let $\mathbb{G}_{N}$ be the group
of roots of unity of order $N$ in $\Bbbk$ and $\mathbb{G}_{N}^{\prime}$ the
subset of primitive roots of order $N$;
$\mathbb{G}_{\infty}=\bigcup_{N\in\mathbb{N}}\mathbb{G}_{N}$ and
$\mathbb{G}^{\prime}_{\infty}=\mathbb{G}_{\infty}-\\{1\\}$. If
$L\in\mathbb{N}$ and $q\in\Bbbk^{\times}$, then
$(L)_{q}\coloneqq\sum_{j=0}^{L-1}q^{j}$.

All vector spaces, algebras and tensor products are over $\Bbbk$. We use
$V^{*}$ to denote the linear dual of a vector space $V$,
$V^{*}=\operatorname{Hom}_{k}(V,k)$.

By abuse of notation, $\langle a_{i}:i\in I\rangle$ denotes either the group,
the subgroup or the vector subspace generated by all $a_{i}$ for $i$ in an
indexing set $I$, the meaning being clear from the context. Instead, the
subalgebra generated by all $a_{i}$ for $i\in I$ is denoted by $\Bbbk\langle
a_{i}:i\in I\rangle$.

The notation for Hopf algebras is standard: $\Delta$, $\varepsilon$,
$\mathcal{S}$ denote the comultiplication, the counit, the antipode (always
assumed bijective), respectively. We use the Sweedler notation for the
comultiplication and the coactions. Our reference for the theory of Hopf
algebras is [37]. Generalities on Nichols algebras can be found in [2].

##  2\. Extensions of Hopf algebras

This Section contains a crash exposition of extensions of Hopf algebras.

###  2.1. Exact sequences

Recall that a sequence of morphisms of Hopf algebras

(2) $\displaystyle\Bbbk\to A\overset{\iota}{\to}C\overset{\pi}{\to}B\to\Bbbk$

is exact [6, 40, 25] if the following conditions holds:

  1. (i)

$\iota$ is injective.

  2. (ii)

$\pi$ is surjective.

  3. (iii)

$\ker\pi=C\iota(A)^{+}$.

  4. (iv)

$\iota(A)=C^{\operatorname{co}\pi}$.

The exact sequence (2) is _abelian_ if $A$ is commutative and $B$ is
cocommutative. We shall also refer to $C$ in (2) as an (abelian when it
corresponds) extension of $B$ by $A$.

The exact sequence (2) is _cleft_ (see [6, Definition 3.2.13]) if there exist

  1. (i)

a unit preserving right $B$-colinear section $\mathpzc{s}:B\to C$ of the
projection $\pi$ and

  2. (ii)

a counit preserving left $A$-linear retraction $\mathpzc{r}:C\to A$ of the
inclusion $\iota$,

both $\mathpzc{s}$ and $\mathpzc{r}$ being invertible with respect to the
convolution product, such that the following equivalent conditions hold, for
all $c\in C$:

  1. (a)

$\mathpzc{s}^{-1}(\pi(c))=\mathcal{S}(c_{(1)})\mathpzc{r}(c_{(2)})$.

  2. (b)

$\mathpzc{s}(\pi(c))=\mathpzc{r}^{-1}(c_{(1)})c_{(2)}$.

  3. (c)

$\mathpzc{r}^{-1}(c)=\mathpzc{s}(\pi(c_{(1)}))\mathcal{S}(c_{(2)})$.

  4. (d)

$\mathpzc{r}(c)=c_{(1)}\mathpzc{s}^{-1}(\pi(c_{(2)}))$.

  5. (e)

$\mathpzc{r}\mathpzc{s}=\epsilon_{B}1_{A}$.

The maps $\mathpzc{s}$ and $\mathpzc{r}$ are called _cleaving maps_ of (2).

######  Remark 2.1.

[40] If $C$ is finite dimensional, then the extension (2) is cleft.

######  Definition 2.2.

[39, Definition 6.5.2] The exact sequence (2) is called _split_ if there exist
cleaving maps $\mathpzc{s}$ and $\mathpzc{r}$ as above, called _splittings_ of
(2), such that $\mathpzc{s}$ is an algebra map and $\mathpzc{r}$ is a
coalgebra map.

Cleft extensions can be described using suitable linear maps. A _bicrossed
product datum_ of Hopf algebras is a collection
$(A,B,\rightharpoonup,\rho,\sigma,\tau)$ where $A$ and $B$ are Hopf algebras,

$\displaystyle\rightharpoonup:B\otimes A$ $\displaystyle\to A,$
$\displaystyle\rho:B$ $\displaystyle\to B\otimes A,$
$\displaystyle\sigma:B\otimes B$ $\displaystyle\to A$ and
$\displaystyle\tau:B$ $\displaystyle\to A\otimes A$

are maps, called, respectively, a weak action, a weak coaction, a cocycle and
a dual cocycle, obeying the conditions in [6, Theorem 2.20].

A bicrossed product datum $(A,B,\rightharpoonup,\rho,\sigma,\tau)$ gives rise
to a Hopf algebra $A\\#{}_{\sigma}^{\tau}B$, called a _bicrossed product_ :
the underlying vector space is $A\otimes B$, while the multiplication,
comultiplication and antipode are determined by the formulas

$\displaystyle(k\\#h)(t\\#g)$ $\displaystyle=k\left(h_{(1)}\rightharpoonup
t\right)\sigma(h_{(2)},g_{(1)})\\#h_{(3)}g_{(2)},$
$\displaystyle\Delta(k\\#h)$
$\displaystyle=k_{(1)}\tau^{(1)}(h_{(1)})\\#\rho(h_{(2)})_{i}\otimes
k_{(2)}\tau^{(2)}(h_{(1)})\rho(h_{(2)})^{i}\\#h_{(3)},$
$\displaystyle\mathcal{S}(a\\#b)$
$\displaystyle=((\mathcal{S}((\rho(b)_{i})_{(2)})\rightharpoonup\mathcal{S}(\rho(b)^{i})\\#\mathcal{S}((\rho(b)_{i})_{(1)}))\mathcal{S}(a)\\#1.$

The natural maps $\iota:A\to A\\#{}_{\sigma}^{\tau}B$ and
$\pi:A\\#{}_{\sigma}^{\tau}B\to B$ fit into a cleft exact sequence

$\Bbbk\to
A\overset{\iota}{\to}A\\#{}_{\sigma}^{\tau}B\overset{\pi}{\to}B\to\Bbbk,$

of Hopf algebras, with cleaving maps

$\displaystyle\mathpzc{s}:B$ $\displaystyle\to A\\#{}_{\sigma}^{\tau}B,$
$\displaystyle\mathpzc{s}(b)$ $\displaystyle=1\\#b,$
$\displaystyle\mathpzc{r}:A\\#{}_{\sigma}^{\tau}B$ $\displaystyle\to A,$
$\displaystyle\mathpzc{r}(a\\#b)$ $\displaystyle=\epsilon(b)a,$ $\displaystyle
a$ $\displaystyle\in A,\,b\in B.$

Conversely, given a cleft exact sequence (2), there exists a bicrossed product
datum $(A,B,\rightharpoonup,\rho,\sigma,\tau)$ such that $C\cong
A\\#{}_{\sigma}^{\tau}B$; it arises from the cleaving maps $\mathpzc{s}$ and
$\mathpzc{r}$ as follows:

$\displaystyle b\rightharpoonup a$
$\displaystyle=\mathpzc{s}(b_{(1)})a\mathpzc{s}^{-1}(b_{(2)}),\quad\sigma(b\\#b^{\prime})=\mathpzc{s}(b_{(1)})\mathpzc{s}(b^{\prime}_{(1)})\mathpzc{s}^{-1}(b_{(2)}b^{\prime}_{(2)}),$
$\displaystyle\rho(\pi(c))$
$\displaystyle=\pi(c_{(2)})\otimes\mathpzc{r}^{-1}(c_{(1)})\mathpzc{r}(c_{(3)}),\quad\tau(\pi(c))=\Delta(\mathpzc{r}^{-1}(c_{(1)}))\;\mathpzc{r}(c_{(2)})\otimes\mathpzc{r}(c_{(3)}),$

for all $a\in A$, $b\in B$, $c\in C$. See [6, Theorem 3.2.14].

A special instance of a bicrossed product datum
$(A,B,\rightharpoonup,\rho,\sigma,\tau)$ occurs where the cocycle $\sigma$ and
the dual cocycle $\tau$ are trivial maps; in this case, we omit the mention of
$\sigma$ and $\tau$ and call the bicrossed product datum
$(A,B,\rightharpoonup,\rho)$ a _bismash datum_. Notice that in a bismash
datum, $A$ is a left $B$-module algebra and $B$ is a right $A$-comodule
algebra with action $\rightharpoonup$ and coaction $\rho$.

Given a bismash datum $(A,B,\rightharpoonup,\rho)$ , the associated bicrossed
product is denoted $A\\#B$. The canonical cleaving maps imply that the
associated exact sequence

(3) $\Bbbk\to A\overset{\iota}{\to}A\\#B\overset{\pi}{\to}B\to\Bbbk$

is split. Conversely, if (2) is a split exact sequence, then the corresponding
bicrossed product datum $(A,B,\rightharpoonup,\rho,\sigma,\tau)$ is in fact a
bismash datum $(A,B,\rightharpoonup,\rho)$ such that $C\cong A\\#B$.

A bismash datum $(A,B,\rightharpoonup,\rho)$ is called a _Singer pair_ if $A$
is commutative and $B$ is cocommutative. In this case (3) is a split abelian
extension of Hopf algebras. Observe that every abelian cleft exact sequence
(2) gives rise to a Singer pair through the actions $\rightharpoonup$, $\rho$.

###  2.2. Matched pairs

We now present a way to produce extensions due to G. I. Kac for abelian
extensions and to S. Majid in general. We start by the definition, see [29,
§7.2]. An _exact factorization_ of a Hopf algebra $S$ consists of a pair
$(G,L)$ of Hopf subalgebras of $S$ such that the restriction of the
multiplication map

$\displaystyle\operatorname{mult}:G\otimes L\to S$

is a linear isomorphism. Exact factorizations are classified through the
following notion.

######  Definition 2.3.

[29, 7.2.1] A _matched pair_ of Hopf algebras is a collection
$(G,L,\triangleright,\triangleleft)$ where $L$ and $G$ Hopf algebras, $G$ is a
left $L$-module coalgebra with action $\triangleright$, $L$ is a right
$G$-module coalgebra with action $\triangleleft$ such that for all $\ell,m\in
L$, $x,y\in G$:

(4) $\displaystyle(\ell m)\triangleleft x$
$\displaystyle=\left(\ell\triangleleft(m_{(1)}\triangleright
x_{(1)})\right)(m_{(2)}\triangleleft x_{(2)}),$ (5)
$\displaystyle\ell\triangleright(xy)$
$\displaystyle=\left(\ell_{(1)}\triangleright
x_{(1)}\right)\left((\ell_{(2)}\triangleleft x_{(2)})\triangleright y\right),$
(6) $\displaystyle\ell_{(1)}\triangleleft
x_{(1)}\otimes\ell_{(2)}\triangleright x_{(2)}$
$\displaystyle=\ell_{(2)}\triangleleft x_{(2)}\otimes\ell_{(1)}\triangleright
x_{(1)}.$

######  Proposition 2.4.

  1. (i)

[29, 7.2.2] Given a matched pair $(G,L,\triangleright,\triangleleft)$, the
coalgebra $G\otimes L$ with the multiplication

$\displaystyle(x\otimes\ell)(y\otimes m)$
$\displaystyle=x\left(\ell_{(1)}\triangleright
y_{(1)}\right)\otimes\left(\ell_{(2)}\triangleleft y_{(2)}\right)m,$
$\displaystyle\ell,m$ $\displaystyle\in L,\ x,y\in G,$

is a Hopf algebra denoted $G\bowtie L$. The natural inclusions
$G\hookrightarrow G\bowtie L\hookleftarrow L$ form an exact factorization of
$G\bowtie L$.

  1. (ii)

[29, 7.2.3] If $(G,L)$ is an exact factorization of a Hopf algebra $S$, then
there are actions $\triangleright$ and $\triangleleft$ such that
$(G,L,\triangleright,\triangleleft)$ is a matched pair and

$\displaystyle S$ $\displaystyle\simeq G\bowtie L.$

The Hopf algebra $G\bowtie L$ is called the _double crossproduct_ associated
to the actions $\triangleleft,\triangleright$. Matched pairs of Hopf algebras
give rise to split exact sequences in the following way.

######  Proposition 2.5.

[29, 7.2.3] If $(G,L,\triangleright,\triangleleft)$ is a matched pair of a
Hopf algebras such that $\dim L<\infty$, then $(L^{*},G,\rightharpoonup,\rho)$
is a bismash datum, where $\rightharpoonup$ and $\rho$ are obtained by
dualization; and vice versa.

In conclusion, finite-dimensional split abelian extensions are determined by
exact factorizations of _cocommutative_ Hopf algebras (that can be thought of
as finite group schemes). We illustrate these notions with some examples, see
Examples 3.7 and 3.8 for a discussion of the fgc property.

#### Group algebras

The exact factorizations of a group algebra $\Bbbk\varSigma$ are in bijective
correspondence with the exact factorizations of the group $\varSigma$, [27,
28, 43]. Also, the matched pairs of Hopf algebras
$(\Bbbk\varGamma,\Bbbk\varLambda,\triangleright,\triangleleft)$ are the
linearizations of the matched pairs of groups
$(\varGamma,\varLambda,\triangleright,\triangleleft)$.

#### Lie algebras

The exact factorizations of an enveloping algebra $U(\mathfrak{s})$ are in
bijective correspondence with those of the Lie algebra $\mathfrak{s}$; matched
pairs of Hopf algebras
$(U(\mathfrak{g}),U(\mathfrak{l}),\triangleright,\triangleleft)$ are in
bijective correspondence with matched pairs of Lie algebras
$(\mathfrak{g},\mathfrak{l},\triangleright,\triangleleft)$. See [29, §8.3].
More precisely,

  * $\circ$

an exact factorization of a Lie algebra $\mathfrak{s}$ consists of a pair
$(\mathfrak{g},\mathfrak{l})$ of Lie subalgebras such that
$\mathfrak{s}=\mathfrak{g}\oplus\mathfrak{l}$ (as vector spaces);

  * $\circ$

a _matched pair_ of Lie algebras is a collection
$(\mathfrak{g},\mathfrak{l},\triangleright,\triangleleft)$ where
$\mathfrak{g}$ and $\mathfrak{l}$ are Lie algebras, $\triangleright$ and
$\triangleleft$ are left and right actions
$\textstyle{\mathfrak{l}}$$\textstyle{\mathfrak{l}\times\mathfrak{g}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\triangleright}$$\scriptstyle{\triangleleft}$$\textstyle{\mathfrak{g}}$
satisfying (12) and (13) below.

Given such a matched pair,
$\mathfrak{g}\bowtie\mathfrak{l}\coloneqq\mathfrak{g}\oplus\mathfrak{l}$ with
the multiplication given by (16) is a Lie algebra. Up to identifications,
$(\mathfrak{g},\mathfrak{l})$ is an exact factorization of
$\mathfrak{g}\bowtie\mathfrak{l}$. Also, (16) is equivalent to $\mathfrak{g}$
and $\mathfrak{l}$ being Lie subalgebras and

(7) $\displaystyle\left[\ell,y\right]$ $\displaystyle=\ell\triangleright
y+\ell\triangleleft y,$ $\displaystyle\ell$ $\displaystyle\in\mathfrak{l},\
y\in\mathfrak{g}.$

Conversely if $(\mathfrak{g},\mathfrak{l})$ is an exact factorization of a Lie
algebra $\mathfrak{s}$, then (7) defines the actions $\triangleright$ and
$\triangleleft$, $(\mathfrak{g},\mathfrak{l},\triangleright,\triangleleft)$ is
a matched pair and $\mathfrak{s}\simeq\mathfrak{g}\bowtie\mathfrak{l}$.

#### Restricted Lie algebras

In this example $\operatorname{char}\Bbbk=p>0$. Let $\mathfrak{s}$ be a
restricted Lie algebra with $p$-operation $s\mapsto s^{[p]}$, see [26, (65),
p. 187]. Concretely, given $i\in\mathbb{I}_{p-1}$, recall that
$\mathtt{s}_{i}:\mathfrak{s}\times\mathfrak{s}\to\mathfrak{s}$ is the
homogeneous polynomial of degree $p$ defined on a pair $(s,t)$ as the
coefficient of $X^{i-1}$ in $\operatorname{ad}(Xs+t)^{p-1}(s)$, where $X$ is a
formal variable. Then

(8) $\displaystyle(s+t)^{p}$
$\displaystyle=s^{p}+t^{p}+\sum_{i=1}^{p-1}\dfrac{\mathtt{s}_{i}(s,t)}{i},$
$\displaystyle s,t\in\mathfrak{s}.$

By definition the $p$-operation satisfies for all $k\in\Bbbk$,
$s,t\in\mathfrak{s}$ the identities

(9) $\displaystyle(ks)^{[p]}$ $\displaystyle=k^{p}s^{[p]},$ (10)
$\displaystyle\operatorname{ad}(s^{[p]})$
$\displaystyle=\operatorname{ad}(s)^{p},$ (11) $\displaystyle(s+t)^{[p]}$
$\displaystyle=s^{[p]}+t^{[p]}+\sum_{i=1}^{p-1}\dfrac{\mathtt{s}_{i}(s,t)}{i}.$

The following definitions are natural.

  * $\circ$

An _exact factorization_ of $\mathfrak{s}$ is a pair
$(\mathfrak{g},\mathfrak{l})$ of restricted Lie subalgebras such that
$\mathfrak{s}=\mathfrak{g}\oplus\mathfrak{l}$ (as vector spaces).

  * $\circ$

A _matched pair_ of restricted Lie algebras is a collection
$(\mathfrak{g},\mathfrak{l},\triangleright,\triangleleft)$ where
$\mathfrak{g}$ and $\mathfrak{l}$ are restricted Lie algebras,
$\triangleright$ and $\triangleleft$ are left and right $p$-actions
$\textstyle{\mathfrak{l}}$$\textstyle{\mathfrak{l}\times\mathfrak{g}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\triangleright}$$\scriptstyle{\triangleleft}$$\textstyle{\mathfrak{g}}$
satisfying for all $\ell,m\in\mathfrak{l}$, $x,y\in\mathfrak{g}$ the
identities

(12) $\displaystyle[\ell,m]\triangleleft x$ $\displaystyle=[\ell\triangleleft
x,m]+[\ell,m\triangleleft x]+\ell\triangleleft(m\triangleright
x)-m\triangleleft(\ell\triangleright x),$ (13)
$\displaystyle\ell\triangleright[x,y]$ $\displaystyle=\left[\ell\triangleright
x,y\right]+[x,\ell\triangleright y]+(\ell\triangleleft x)\triangleright
y-(\ell\triangleleft y)\triangleright x.$ (14)
$\displaystyle\ell^{[p]}\triangleleft y$ $\displaystyle=\sum_{1\leq i\leq
p-1}(\operatorname{ad}\ell)^{i}\left(\ell\triangleleft(\ell^{p-i}\triangleright
y)\right),$ (15) $\displaystyle\ell\triangleright y^{[p]}$
$\displaystyle=\sum_{1\leq i\leq
p-1}(-1)^{p-i}(\operatorname{ad}y)^{i}\left((\ell\triangleleft
y^{p-1-i})\triangleright y\right).$

######  Lemma 2.6.

Let $\mathfrak{g}$ and $\mathfrak{l}$ be restricted Lie algebras.

  1. (i)

If $(\mathfrak{g},\mathfrak{l},\triangleright,\triangleleft)$ is a matched
pair of restricted Lie algebras, then
$\mathfrak{g}\bowtie\mathfrak{l}\coloneqq\mathfrak{g}\oplus\mathfrak{l}$ is a
restricted Lie algebra with the multiplication

(16) $\displaystyle\left[(x,\ell),(y,m)\right]$
$\displaystyle=\left([x,y]+\ell\triangleright y-m\triangleright
x,[\ell,m]+\ell\triangleleft y-m\triangleleft x\right)$

$\ell,m\in\mathfrak{l}$, $x,y\in\mathfrak{g}$; and with $p$-operation
extending those of $\mathfrak{g}$ and $\mathfrak{l}$ and

(17) $\displaystyle(y+\ell)^{[p]}$
$\displaystyle=y^{[p]}+\ell^{[p]}+\sum_{i=1}^{p-1}\dfrac{\mathtt{s}_{i}(y,\ell)}{i},$
$\displaystyle y\in\mathfrak{g},\ $ $\displaystyle\ell\in\mathfrak{l},$

where
$\mathtt{s}_{i}:\mathfrak{g}\bowtie\mathfrak{l}\times\mathfrak{g}\bowtie\mathfrak{l}\to\mathfrak{g}\bowtie\mathfrak{l}$
is defined from the Lie bracket (16).

  2. (ii)

Let $(\mathfrak{g},\mathfrak{l})$ be an exact factorization of a restricted
Lie algebra $\mathfrak{s}$. Then
$(\mathfrak{g},\mathfrak{l},\triangleright,\triangleleft)$ with the actions
given by (7) is a matched pair of restricted Lie algebras and
$\mathfrak{s}\simeq\mathfrak{g}\bowtie\mathfrak{l}$ as restricted Lie
algebras.

###### Proof.

(i): Since $\triangleright,\triangleleft$ are actions that satisfy (12) and
(13), $\mathfrak{g}\bowtie\mathfrak{l}$ with the bracket (16) is a Lie algebra
[29, §8.3]. Thus the maps $\mathtt{s}_{i}$ are defined and we just have to
check that (17) gives a $p$-operation. (9) holds because $\mathtt{s}_{i}$ is
homogeneous of degree $p$. We first verify (10) for $s=y\in\mathfrak{g}$.
Since both sides are linear operators and the restriction to $\mathfrak{g}$ is
a $p$-operation, it is enough to see that
$\operatorname{ad}(y^{[p]})(\ell)\overset{\star}{=}\operatorname{ad}(y)^{p}(\ell)$
for $\ell\in\mathfrak{l}$. Arguing by induction from (7) we prove that for
every $N\in\mathbb{N}$ and $\ell\in\mathfrak{l}$

$\displaystyle(\operatorname{ad}\ell)^{N}(y)$
$\displaystyle=\ell^{N}\triangleright y+\sum_{0\leq i\leq
N-1}(\operatorname{ad}\ell)^{i}\left(y\triangleleft(\ell^{N-1-i}\triangleright
y)\right).$

Taking $N=p$ and again by (7), we conclude that $\star$ holds iff (14) is
true. Similarly we prove that for every $N\in\mathbb{N}$, $y\in\mathfrak{g}$
and $\ell\in\mathfrak{l}$

$\displaystyle(\operatorname{ad}y)^{N}(\ell)$ $\displaystyle=\sum_{0\leq i\leq
N-1}(-1)^{N+i}(\operatorname{ad}y)^{i}\left((\ell\triangleleft
y^{N-1-i})\triangleright y)\right)+(-1)^{N}\ell\triangleleft y^{N}.$

(Here (7) says that $(\operatorname{ad}y)(\ell)=-\ell\triangleright
y-\ell\triangleleft y$). Taking $N=p$, we conclude that
$\operatorname{ad}(\ell^{[p]})(y)=\operatorname{ad}(\ell)^{p}(y)$ holds iff
(15) is true. Finally

$\displaystyle\operatorname{ad}(y+\ell)^{p}$
$\displaystyle=\operatorname{ad}(y)^{p}+\operatorname{ad}(\ell)^{p}+\sum_{i=1}^{p-1}\dfrac{\operatorname{ad}\mathtt{s}_{i}(y,\ell)}{i}$
$\displaystyle=\operatorname{ad}(y^{[p]})+\operatorname{ad}(\ell^{[p]})+\sum_{i=1}^{p-1}\dfrac{\operatorname{ad}\mathtt{s}_{i}(y,\ell)}{i}=\operatorname{ad}((y+\ell)^{[p]}),$

where the first equality is by (8), the second because these are
$p$-operations on the subalgebras and the third is by definition. Thus (10)
holds.

We proceed with (11). Let $s=x+\ell$ and $t=y+m$, where $x,y\in\mathfrak{g}$,
$\ell,m\in\mathfrak{l}$. Then

$\displaystyle(s+t)^{[p]}$ $\displaystyle=\left((x+y)+(\ell+m)\right)^{[p]}$
$\displaystyle=(x+y)^{[p]}+(\ell+m)^{[p]}+\sum_{i=1}^{p-1}\dfrac{\mathtt{s}_{i}((x+y),(\ell+m))}{i},$
$\displaystyle=x^{[p]}+y^{[p]}+\sum_{i=1}^{p-1}\dfrac{\mathtt{s}_{i}(x,y)}{i}+\ell^{[p]}+m^{[p]}+\sum_{i=1}^{p-1}\dfrac{\mathtt{s}_{i}(\ell,m)}{i}$
$\displaystyle\qquad+\sum_{i=1}^{p-1}\dfrac{\mathtt{s}_{i}((x+y),(\ell+m))}{i}.$

On the other hand,

$\displaystyle s^{[p]}+t^{[p]}+\sum_{i=1}^{p-1}\dfrac{\mathtt{s}_{i}(s,t)}{i}$
$\displaystyle=x^{[p]}+\ell^{[p]}+\sum_{i=1}^{p-1}\dfrac{\mathtt{s}_{i}(x,\ell)}{i}$
$\displaystyle+y^{[p]}+m^{[p]}+\sum_{i=1}^{p-1}\dfrac{\mathtt{s}_{i}(y,m)}{i}+\sum_{i=1}^{p-1}\dfrac{\mathtt{s}_{i}(s,t)}{i}$

To show that these two expressions are equal just apply (8) to both sides of
the equality
$\left((x+y)+(\ell+m)\right)^{p}=\left((x+\ell)+(y+m)\right)^{p}$. ∎

Let $\mathfrak{u}(\mathfrak{s})$ be the restricted enveloping algebra of the
restricted Lie algebra $\mathfrak{s}$; it has a PBW-basis with the powers of
the generators truncated at $p$.

######  Lemma 2.7.

Let $\mathfrak{s}$ be a restricted Lie algebra. The following are equivalent:

  1. (i)

Exact factorizations of the Hopf algebra $\mathfrak{u}(\mathfrak{s})$.

  2. (ii)

Exact factorizations of the restricted Lie algebra $\mathfrak{s}$.

###### Proof.

(i) $\Rightarrow$ (ii). If $(\mathfrak{G},\mathfrak{L})$ is an exact
factorization of $\mathfrak{u}(\mathfrak{s})$, then take
$\mathfrak{g}={\mathcal{P}}(\mathfrak{G})$,
$\mathfrak{l}={\mathcal{P}}(\mathfrak{L})$. As a consequence of [42,
Proposition 13.2.3], $\mathfrak{G}\simeq\mathfrak{u}(\mathfrak{g})$,
$\mathfrak{L}\simeq\mathfrak{u}(\mathfrak{l})$, and by the PBW theorem,
$(\mathfrak{g},\mathfrak{l})$ is an exact factorization of $\mathfrak{s}$.

(i) $\Leftarrow$ (ii): If $(\mathfrak{g},\mathfrak{l})$ is an exact
factorization of $\mathfrak{s}$, then the multiplication
$\mathfrak{u}(\mathfrak{g})\otimes\mathfrak{u}(\mathfrak{l})\to\mathfrak{u}(\mathfrak{s})$
is a linear isomorphism–apply the PBW-theorem to the union of bases of
$\mathfrak{g}$ and $\mathfrak{l}$. ∎

Here are some concrete examples of factorizations of restricted Lie algebras.

#### Restricted Lie bialgebras

Let us say that a finite-dimensional Lie bialgebra $\mathfrak{b}$ is
restricted if and only if its Manin triple
$(\mathfrak{p},\mathfrak{b},\mathfrak{b}^{*})$ is restricted, meaning that
$\mathfrak{p}$ is restricted and $\mathfrak{b}$, $\mathfrak{b}^{*}$ are
restricted subalgebras. In particular $(\mathfrak{b},\mathfrak{b}^{*})$ is an
exact factorization of $\mathfrak{p}$.

#### Restricted Lie algebras with triangular decompositions

A triangular decomposition of a Lie algebra $\mathfrak{a}$ is a collection
$\left(\mathfrak{a}_{0},\mathfrak{a}_{+},\mathfrak{a}_{-},(\,|\,)\right)$
where $\mathfrak{a}_{0},\mathfrak{a}_{-},\mathfrak{a}_{+}$ are subalgebras of
$\mathfrak{a}$ and $(\,|\,):\mathfrak{a}\times\mathfrak{a}\rightarrow\Bbbk$ is
a non-degenerate symmetric $\mathfrak{a}$-invariant bilinear form such that
$\mathfrak{a}_{0}$ is abelian,

$\displaystyle\mathfrak{a}$
$\displaystyle=\mathfrak{a}_{-}\oplus\mathfrak{a}_{0}\oplus\mathfrak{a}_{+},$
$\displaystyle\left[\mathfrak{a}_{\pm},\mathfrak{a}_{0}\right]$
$\displaystyle\subset\mathfrak{a}_{\pm},$ and
$\displaystyle\left(\mathfrak{a}_{+}|\mathfrak{a}_{+}\right)=\left(\mathfrak{a}_{-}|\mathfrak{a}_{-}\right)$
$\displaystyle=\left(\mathfrak{a}_{+}|\mathfrak{a}_{0}\right)=\left(\mathfrak{a}_{0}|\mathfrak{a}_{-}\right)=0.$

A triangular decomposition gives rise to a Manin triple
$(\mathfrak{p},\mathfrak{p}_{1},\mathfrak{p}_{2})$ defined by

$\displaystyle\mathfrak{p}$ $\displaystyle=\mathfrak{a}\times\mathfrak{a},$
$\displaystyle\mathfrak{p}_{1}$
$\displaystyle=\operatorname{diag}\mathfrak{a},$ and
$\displaystyle\mathfrak{p}_{2}$
$\displaystyle=\left\\{\left(a_{-}+a_{0},a_{+}-a_{0}\right):a_{\star}\in\mathfrak{a}_{\star},\star\in\\{+,0,-\\}\right\\}.$

If $\mathfrak{a}$ is restricted and
$\mathfrak{a}_{0},\mathfrak{a}_{-},\mathfrak{a}_{+}$ are restricted
subalgebras, then $\mathfrak{p}$ is restricted and
$(\mathfrak{p}_{1},\mathfrak{p}_{2})$ is an exact factorization. There are
other factorizations:

  * •

$(\mathfrak{a}_{\pm},\mathfrak{a}_{0}\oplus\mathfrak{a}_{\mp})$ are exact
factorizations of $\mathfrak{a}$;

  * •

$(\mathfrak{a}_{-}\oplus\mathfrak{a}_{0},\mathfrak{a}_{0}\oplus\mathfrak{a}_{+})$
is an exact factorization of $\mathfrak{a}\times\mathfrak{a}_{0}$.

#### Restricted $\mathbb{Z}$-graded Lie algebras.

A finite-dimensional $\mathbb{Z}$-graded Lie algebra
$\mathfrak{s}=\oplus_{i=-r}^{t}\mathfrak{s}_{i}$ (where
$r,t\in\mathbb{N}_{0}$) is _restricted_ if the underlying Lie algebra
$\mathfrak{s}$ is restricted and
$\mathfrak{s}_{i}^{[p]}\subseteq\mathfrak{s}_{pi}$ for all $i$. Then
$\mathfrak{s}_{+}\coloneqq\oplus_{i=1}^{t}\mathfrak{s}_{i}$ and
$\mathfrak{s}_{\leq 0}\coloneqq\oplus_{i=-r}^{0}\mathfrak{s}_{i}$ form an
exact factorization of $\mathfrak{s}$.

##  3\. Quasi-split extensions

###  3.1. Morita equivalence

Two finite-dimensional Hopf algebras $H$ and $U$ are _Morita equivalent_
($H\sim_{\text{Mor}}U$) iff there exists an equivalence of braided tensor
categories between the Drinfeld centers $\mathcal{Z}(\operatorname{rep}H)$ and
$\mathcal{Z}(\operatorname{rep}U)$, iff $D(H)$ and $D(U)$ are twist equivalent
quasitriangular Hopf algebras.111Observe that this differs from [5], where it
was claimed that $H\sim_{\text{Mor}}U\Leftrightarrow D(H)\simeq D(U)$ as
quasitriangular Hopf algebras; we point out that this discrepancy does not
affect the results of _loc. cit._

In other words, $H\sim_{\text{Mor}}U$ if the tensor categories
$\operatorname{rep}H$ and $\operatorname{rep}U$ are Morita equivalent. Notice
that our defining condition is in fact a characterization of the original
definition of Morita equivalence of tensor categories in [33], [15].

For instance, if either $U\simeq H^{*}$, or $U\simeq H^{J}$ is obtained from
$H$ by twisting the comultiplication by $J\in H\otimes H$, or $U\simeq
H_{\sigma}$ is obtained from $H$ by twisting the multiplication by
$\sigma:H\otimes H\to\Bbbk$, then $H\sim_{\text{Mor}}U$.

From now on, we assume that all Hopf algebras in (2) are finite-dimensional.
In this Section we study the following notion.

######  Definition 3.1.

We shall say that a cleft abelian exact sequence (2) is _quasi-split_ if the
Hopf algebra $C$ is Morita equivalent to the bismash product $A\\#B$
associated to the Singer pair arising from any cleaving maps.

###  3.2. Coquasi-Hopf algebras

Recall that quasi-Hopf algebras were introduced by Drinfeld as generalizations
of Hopf algebras, where the main difference is that the coassociativity of the
comultiplication holds up to a 3-tensor called the associator. Dually, a
coquasi-bialgebra or coquasi-Hopf algebra $H$ is a generalization of a
bialgebra or a Hopf algebra where the main difference is that the
associativity of the multiplication holds up to a dual 3-tensor
$\varphi:H\otimes H\otimes H\to\Bbbk$, called the coassociator. See e.g. [39]
for details.

Here is the starting point of our analysis.

######  Definition 3.2.

[39] Let $K$ and $Q$ be Hopf algebras. A _generalized product coquasi-Hopf
algebra_ of $K$ and $Q$ is a co-quasi Hopf algebra $H$ together with coquasi-
Hopf algebra maps

$\displaystyle i:K$ $\displaystyle\to H$ and $\displaystyle j:Q$
$\displaystyle\to H$ such that $\displaystyle\operatorname{mult}(i\otimes
j):K$ $\displaystyle\otimes Q\to H$

is a linear isomorphism.

Let us say that a finite-dimensional Hopf algebra $L$ and a coquasi-Hopf
algebra $U$ are _Morita equivalent_ iff there exists an equivalence of braided
tensor categories between $\mathcal{Z}(\operatorname{corep}L)$ and
$\mathcal{Z}(\operatorname{corep}U)$ or equivalently that the quantum doubles
$D(L)$ and $D(U)$ are twist-equivalent.

By the results of [39, Section 6], a cleft exact sequence of Hopf algebras (2)
gives rise to a generalized product coquasi-Hopf algebra $H$ of $A^{*}$ and
$B$, where

  * $\circ$

$H=B\otimes A^{*}$ as a coalgebra;

  * $\circ$

the multiplication is given by

$\displaystyle(x\otimes\ell)(y\otimes m)$
$\displaystyle=x\left(\ell_{(1)}\triangleleft
y_{(1)}\right)\otimes\left(\ell_{(2)}\triangleright y_{(2)}\right)m,$
$\displaystyle\ell,m$ $\displaystyle\in A^{*},\ x,y\in B,$

where the maps $\triangleleft:A^{*}\otimes B\to A^{*}$ and
$\triangleright:A^{*}\otimes B\to B$ are determined by the associated weak
action $\rightharpoonup$ and the weak coaction $\rho$ by

$\displaystyle(\ell\triangleleft x)(a)$ $\displaystyle=\ell(x\rightharpoonup
a),$ $\displaystyle\ell\triangleright x$
$\displaystyle=\rho(x)_{i}\,\ell(\rho(x)^{i}),$ $\displaystyle\ell$
$\displaystyle\in A^{*},x\in B;$

  * $\circ$

the coassociator $\varphi:H\otimes H\otimes H\to\Bbbk$ is determined by the
cocycle $\sigma$ and the dual cocycle $\tau$ in the form

$\varphi(x\otimes\ell\otimes y\otimes m\otimes z\otimes
r)=\epsilon(x)\,\ell(y\rightharpoonup\tau^{(1)}(z_{(1)})\sigma(y_{(2)},\rho(z_{(2)})_{i})\;m(\tau^{(1)}(z_{(1)})\rho(z_{(2)})^{i}))\,\epsilon(r).$

######  Proposition 3.3.

Given a cleft exact sequence of Hopf algebras (2), the Hopf algebra $C$ is
Morita equivalent to the coquasi-Hopf algebra $H$.

###### Proof.

The main result of [39] implies the existence of an equivalence of monoidal
categories

${}_{A}(\operatorname{corep}C)_{A}\simeq\operatorname{corep}H,$

which amounts to the Morita equivalence of the categories
$\operatorname{corep}C$, $\operatorname{corep}H$, and _a fortiori_ of $C$ and
$H$. Indeed, an equivalence of braided tensor categories between the Drinfeld
centers of $\operatorname{corep}C$ and $\operatorname{corep}H$ was established
in [38]. ∎

Combining Propositions 2.5 and 3.3, we obtain:

######  Corollary 3.4.

Let $S$ be a finite-dimensional Hopf algebra and suppose that $S$ is a double
crossproduct of its Hopf subalgebras $G$ and $L$. Then $S$ is Morita
equivalent to a bismash product $L^{*}\\#G$. In particular, if $D(S)$ has fgc
so does $L^{*}\\#G$. ∎

###  3.3. Abelian extensions

Suppose now that the exact sequence (2) is abelian. Then the coquasi-Hopf
algebra $H$ described above turns out to be the double crossproduct associated
to the Singer pair $(A,B)$ with a possibly nontrivial coassociator determined
by the cocycles $\sigma$ and $\tau$. However, the coalgebra $H$ is
cocommutative in this case.

######  Corollary 3.5.

An abelian extension of $B$ by $A$ is quasi-split provided that the
coassociator $\varphi$ is trivial.

###### Proof.

It follows from the fact that the split extension $A\\#B$ is Morita equivalent
to $A^{*}\bowtie B$ (with trivial associator) by Proposition 3.3. ∎

Exactness of the Kac sequence of [39] implies that, since $A$ is finite-
dimensional, $H$ is isomorphic as a coquasi-Hopf algebra to the double
crossproduct $A^{*}\bowtie B$ (as generalized products of $A^{*}$ and $B$) if
and only if (2) is isomorphic (as a $B$-extension of $A$) to a twisting
$(A\\#B)_{\chi}^{J}$ of the bismash product, where

  * $\circ$

$J\in A\otimes A$ is a twist in $A$, and

  * $\circ$

$\chi:B\otimes B$ is a 2-cocycle on $B$,

regarded respectively as a twist in $C$ and a 2-cocycle on $C$, see [39,
Theorem 6.3.6].

The next theorem is the main result of this section.

######  Theorem 3.6.

Let $(A,B)$ be a Singer pair of finite-dimensional Hopf algebras. Given a
quasi-split abelian extension $C$ of $B$ by $A$, the double $D(C)$ and a
fortiori $C$ have fgc. In particular, $A\\#B$ has fgc.

###### Proof.

We have that $C$ is Morita equivalent to the cocommutative Hopf algebra
$A^{*}\bowtie B$. Whence $D(C)$ is twist equivalent to $D(A^{*}\bowtie B)$,
which has fgc by the main result of [34]. Hence $D(C)$ and therefore also $C$
have fgc. ∎

######  Example 3.7.

Let $\varLambda$ and $\varGamma$ be finite groups. Consider an exact sequence
of Hopf algebras $\Bbbk\to\Bbbk^{\varLambda}\to C\to\Bbbk\varGamma\to\Bbbk$
where $\Bbbk^{\varLambda}$ is the algebra of functions on $\varLambda$. Then
$C\simeq\Bbbk^{\varLambda}\\#_{\sigma}^{\tau}\Bbbk\varGamma$ is a bicrossed
product. The relevant (weak) actions in this case are determined by actions by
permutations $\triangleright:\varGamma\times\varLambda\to\varLambda$ and
$\triangleleft:\varGamma\times\varLambda\to\varGamma$ that make
$(\varGamma,\varLambda,\triangleright,\triangleleft)$ into a matched pair of
finite groups. Let $\varSigma=\varGamma\bowtie\varLambda$ be the associated
double crossproduct group.

The Hopf algebra $C$ is Morita equivalent to a quasi-Hopf algebra
$(\Bbbk{\varSigma},\omega)$, where $\omega\in H^{3}(\varSigma,\Bbbk^{\times})$
is the 3-cocycle attached to the class of $C$ under the Kac exact sequence
(hence in particular, the restriction of $\omega$ to $\varGamma$ and
$\varLambda$ is trivial). It follows from [34] that $C$ has fgc whenever
$\omega$ is trivial.

For instance, we have
$\Bbbk\varSigma\sim_{\text{Mor}}\Bbbk^{\varLambda}\\#\Bbbk\varGamma$, hence
$\Bbbk^{\varLambda}\\#\Bbbk\varGamma$ has fgc. This is evident if
$\operatorname{char}\Bbbk$ is $0$ o coprime to $|\varSigma|$; otherwise it
follows alternatively from [5, Lemma 3.2.5] since $\Bbbk^{\varLambda}$ is
semisimple.

######  Example 3.8.

Let $(\mathfrak{g},\mathfrak{l},\triangleright,\triangleleft)$ be a matched
pair of restricted Lie algebras and
$\mathfrak{s}=\mathfrak{g}\bowtie\mathfrak{l}$. The corresponding matched pair
of Hopf algebras
$(\mathfrak{u}(\mathfrak{g}),\mathfrak{u}(\mathfrak{l}),\triangleright,\triangleleft)$
gives rise to an exact sequence
$\Bbbk\to\mathfrak{u}(\mathfrak{l})^{*}\to\mathfrak{u}(\mathfrak{l})^{*}\\#\mathfrak{u}(\mathfrak{g})\to\mathfrak{u}(\mathfrak{g})\to\Bbbk$.
We have
$\mathfrak{u}(\mathfrak{s})\sim_{\text{Mor}}\mathfrak{u}(\mathfrak{l})^{*}\\#\mathfrak{u}(\mathfrak{g})$,
hence $\mathfrak{u}(\mathfrak{l})^{*}\\#\mathfrak{u}(\mathfrak{g})$ has fgc by
[34].

##  4\. The restricted Jordan plane

_In this Section and the next two, $\operatorname{char}\Bbbk=p$ is an odd
prime_ (except when explicitly stated otherwise). In Section 6 we consider a
subclass of the finite-dimensional Nichols algebras introduced in [4] and show
that their bosonizations with suitable group algebras are split abelian
extensions, therefore they have fgc. In this Section and in the next, we work
out the two simplest examples for illustration.

###  4.1. The property fgc for bosonizations

In this Subsection, $\operatorname{char}\Bbbk$ is arbitrary. We record a
useful result, a variation of [5, Theorem 3.1.6].

######  Theorem 4.1.

Let $F$ be a finite group.

  1. (i)

Assume that $\Bbbk F$ is semisimple. If $R$ is a finite-dimensional Hopf
algebra in ${}^{\Bbbk F}_{\Bbbk F}\mathcal{YD}$ that has fgc, then $R\\#\Bbbk
F$ has fgc.

  2. (ii)

If $R$ is a finite-dimensional Hopf algebra in
${}^{\Bbbk^{F}}_{\Bbbk^{F}}\mathcal{YD}$ that has fgc, then $R\\#\Bbbk^{F}$
has fgc.

###### Proof.

We sketch the proof for the reader’s convenience. Let $K$ be either $\Bbbk F$
as in (i) or $\Bbbk^{F}$ as in (ii), so clearly $K$ is semisimple. Let $R$ be
as in (i) or (ii) accordingly.

Since the proofs of [31, Corollary 3.13] and [5, Lemma 3.1.4] just require
that $\Bbbk$ is a field, we conclude that the algebra
$\operatorname{H}(R,\Bbbk)$ is Noetherian. Now [5, Lemma 3.1.1] also holds for
any field, hence $\operatorname{H}(R,\Bbbk)^{K}$ is finitely generated.

On the other hand, there is an isomorphism
$\operatorname{H}(R\\#K,\Bbbk)\simeq\operatorname{H}(R,\Bbbk)^{K}$, see [41,
Theorem 2.17]. Hence $\operatorname{H}(R\\#K,\Bbbk)$ is finitely generated.

Next, given a finitely generated $R\\#K$-module $M$, one can prove that
$\operatorname{H}(R\\#K,M)$ is finitely generated as an
$\operatorname{H}(R\\#K,\Bbbk)$-module repeating word-by-word the proof of the
analogous fact in [5, Theorem 3.1.6]. ∎

For further developments, it would be useful to extend Theorem 4.1 to an
arbitrary semisimple Hopf algebra $K$.

######  Lemma 4.2.

Let $A$ be a finite-dimensional Hopf algebra and $U\in{}^{A}_{A}\mathcal{YD}$
such that $\mathscr{B}(U)$ is finite-dimensional. If $\mathscr{B}(U)\\#A$ has
fgc, then so does the Nichols algebra $\mathscr{B}(U)$.

###### Proof.

Since $\mathscr{B}(U)\\#A$ is free over $\mathscr{B}(U)$, [7, Theorem 3.2.1]
implies the claim. ∎

###  4.2. The minimal bosonization

We begin with the following basic example. The _block_ $\mathcal{V}(1,2)$ is
the braided vector space with a basis $\\{x,y\\}$ such that

(18) $\displaystyle\begin{aligned} c(x\otimes x)&=x\otimes x,&c(y\otimes
x)&=x\otimes y,\\\ c(x\otimes y)&=(y+x)\otimes x,&c(y\otimes y)&=(y+x)\otimes
y.\end{aligned}$

The Nichols algebra $\mathscr{B}(\mathcal{V}(1,2))$ is called the _restricted
Jordan plane_.

######  Lemma 4.3.

[11] The restricted Jordan plane is generated by $x$, $y$ with relations

(19) $\displaystyle yx-xy+\tfrac{1}{2}x^{2},$ $\displaystyle x^{p},$
$\displaystyle y^{p}.$

The set $\\{x^{a}y^{b}:0\leq a,b<p\\}$ is a basis of
$\mathscr{B}(\mathcal{V}(1,2))$, so $\dim\mathscr{B}(\mathcal{V}(1,2))=p^{2}$.
∎

The minimal bosonization of $\mathcal{V}(1,2)$ arises as follows. Let
$\Gamma=\mathbb{Z}/p=\langle g\rangle$. We realize $\mathcal{V}(1,2)$ in
${}^{\Bbbk\Gamma}_{\Bbbk\Gamma}\mathcal{YD}$ by

$\displaystyle g\cdot x$ $\displaystyle=x,$ $\displaystyle g\cdot y$
$\displaystyle=y+x,$ $\displaystyle\deg x=\deg y$ $\displaystyle=g.$

Thus the Hopf algebra $H=\mathscr{B}(\mathcal{V}(1,2))\\#\Bbbk\Gamma$ has
dimension $p^{3}$. We get a presentation of $H$ by generators $x,y,g$, where
we identify $x=x\otimes 1,y=y\otimes 1,g=1\otimes g\in H$, with defining
relations

$\displaystyle x^{p}=y^{p}$ $\displaystyle=0,$ $\displaystyle g^{p}$
$\displaystyle=1,$ $\displaystyle gx$ $\displaystyle=xg,$ $\displaystyle gy$
$\displaystyle=yg+xg,$ $\displaystyle yx$
$\displaystyle=xy-\tfrac{1}{2}x^{2}.$

The comultiplication of $H$ is determined by

$\displaystyle\Delta(g)$ $\displaystyle=g\otimes g,$ $\displaystyle\Delta(x)$
$\displaystyle=x\otimes 1+g\otimes x,$ $\displaystyle\Delta(y)$
$\displaystyle=y\otimes 1+g\otimes y.$

In addition, the monomials $g^{i}x^{j}y^{\ell}$, $0\leq i,j,\ell\leq p-1$,
form a basis of $H$.

Let $K=\Bbbk\langle x,g\rangle\subset H$ and $L=\Bbbk[\zeta]/(\zeta^{p})$ with
$\zeta$ primitive.

######  Lemma 4.4.

The Hopf algebra $H$ fits into a split abelian extension $\Bbbk\to
K\overset{\iota}{\rightarrow}H\overset{\pi}{\rightarrow}L\to\Bbbk$, where
$\iota$ is the inclusion and $\pi$ is defined by $\pi(x)=0$, $\pi(g)=1$ and
$\pi(y)=\zeta$.

###### Proof.

The defining relations of $H$ imply that $K$ is commutative. Furthermore $L$
is generated by the primitive element $\zeta$, whence cocommutative. Clearly
$\pi$ is well-defined and $\ker\pi=HK^{+}$. We thus obtain an abelian exact
sequence $\Bbbk\to
K\overset{\iota}{\rightarrow}H\overset{\pi}{\rightarrow}L\to\Bbbk$.

Since $y^{p}=0=-\mathcal{S}(y)^{p}$, there exists a unique algebra map
$\mathpzc{s}:L\to H$ such that $\mathpzc{s}(\zeta)=-\mathcal{S}(y)=g^{-1}y$.
Clearly, $\pi\mathpzc{s}=\operatorname{id}_{L}$. Being an algebra map,
$\mathpzc{s}$ is automatically invertible for the convolution product. The
$L$-colinearity of $\mathpzc{s}$ follows from the relation

$\displaystyle(\mathpzc{s}\otimes\operatorname{id})\Delta(\zeta)=g^{-1}y\otimes
1+1\otimes\zeta=(\operatorname{id}\otimes\pi)\Delta\mathpzc{s}(\zeta)=(\operatorname{id}\otimes\pi)(g^{-1}y\otimes
g^{-1}+1\otimes g^{-1}y).$

Dually, let $\mathpzc{r}:H\to K$ be the linear map defined by

$\displaystyle\mathpzc{r}(g^{i}{x}^{j}y^{\ell})$
$\displaystyle=g^{i}{x}^{j}\,\,\text{ if }\,\,\ell=0,$ and
$\displaystyle\mathpzc{r}(g^{i}{x}^{j}y^{\ell})$ $\displaystyle=0\text{ if
}\,\,\ell>0.$

It is clear that $\mathpzc{r}$ is $K$-linear,
$\mathpzc{r}\iota=\operatorname{id}_{K}$ and
$\mathpzc{r}\mathpzc{s}=\epsilon_{L}1_{K}$. We next show that $\mathpzc{r}$ is
a coalgebra map. Since $\mathpzc{r}|_{K}=\operatorname{id}_{K}$, we have
$\Delta\mathpzc{r}(x)=(\mathpzc{r}\otimes\mathpzc{r})\Delta(x)$, for all $x\in
K$. Let $I=\ker\mathpzc{r}\subseteq H$, in other words, $I$ is the linear span
of all monomials $g^{i}{x}^{j}y^{\ell}$ with $\ell>0$. The relation
$\Delta(y)=y\otimes 1+g\otimes y$ implies that $\Delta(I)\subseteq I\otimes
H+H\otimes I$. Therefore $(\mathpzc{r}\otimes\mathpzc{r})\Delta(I)=0$,
implying that $\mathpzc{r}$ is a coalgebra map. This shows that
$(\mathpzc{s},\mathpzc{r})$ is a splitting and finishes the proof of the
lemma. ∎

######  Remark 4.5.

[8] The Drinfeld double of $H$ fits into an abelian exact sequence

$\displaystyle\Bbbk\to\mathbf{R}\rightarrow
D(H)\rightarrow{\mathfrak{u}}(\mathfrak{sl}_{2}(\Bbbk))\to\Bbbk,$

where $\mathbf{R}$ is a local commutative Hopf algebra.

The following result appeared already in [36] with a different proof.

######  Proposition 4.6.

The Hopf algebra $\mathscr{B}(\mathcal{V}(1,2))\\#\Bbbk\Gamma$ and the Nichols
algebra $\mathscr{B}(\mathcal{V}(1,2))$ have fgc.

###### Proof.

By Lemma 4.4, we may apply Theorem 1.2 and Lemma 4.2. ∎

###  4.3. More bosonizations

To deal with different realizations of the Jordan plane, we recall the notions
of YD-pairs and YD-triples that are available in any characteristic. Let $A$
be a Hopf algebra.

  * $\circ$

A pair $(g,\chi)\in G(A)\times\operatorname{Alg}(A,\Bbbk)$ is called a _YD-
pair_ for $A$ if

(20) $\displaystyle\chi(h)\,g$
$\displaystyle=\chi(h_{\left(2\right)})h_{\left(1\right)}\,g\,\mathcal{S}(h_{\left(3\right)}),$
$\displaystyle h$ $\displaystyle\in A.$

A YD-pair $(g,\chi)$ gives rise to $\Bbbk_{g}^{\chi}\in{}^{A}_{A}\mathcal{YD}$
of dimension 1, with action and coaction given by $\chi$ and $g$,
respectively. Any one-dimensional object in ${}^{A}_{A}\mathcal{YD}$ is like
this.

  * $\circ$

A collection $(g,\chi,\eta)$ where $(g,\chi)$ is a YD-pair for $A$ and
$\eta\in\operatorname{Der}_{\chi,\chi}(A,\Bbbk)$ is called a _YD-triple_ for
$A$ if

(21) $\displaystyle\eta(h)g$
$\displaystyle=\eta(h_{\left(2\right)})h_{\left(1\right)}g\mathcal{S}(h_{\left(3\right)}),$
$\displaystyle h$ $\displaystyle\in A,$ (22) $\displaystyle\text{and }\chi(g)$
$\displaystyle=\eta(g)=1.$

Notice that the existence of a YD-triple for $A$ when $\dim A<\infty$ forces
that $\operatorname{char}\Bbbk>0$, since $\eta\in A^{*}$ is a non-zero
$(\chi,\chi)$\- primitive.

A YD-triple $(g,\chi,\eta)$ gives rise to
$\mathcal{V}_{g}(\chi,\eta)\in{}^{A}_{A}\mathcal{YD}$, defined as the vector
space with a basis $\\{x,y\\}$, whose $A$-action and $A$-coaction are given by

$\displaystyle h\cdot x$ $\displaystyle=\chi(h)x,$ $\displaystyle h\cdot y$
$\displaystyle=\chi(h)y+\eta(h)x,$ $\displaystyle h$ $\displaystyle\in A;$
$\displaystyle\delta(x)$ $\displaystyle=g\otimes x,$ $\displaystyle\delta(y)$
$\displaystyle=g\otimes y.$

By assumption (22), $\mathcal{V}_{g}(\chi,\eta)\simeq\mathcal{V}(1,2)$ as a
braided vector space.

We now turn back to the assumption that $\operatorname{char}\Bbbk=p>2$.

######  Question 4.7.

Let $(g,\chi,\eta)$ be a YD-triple for $A$ where $\dim A<\infty$. Does
$\mathscr{B}(\mathcal{V}_{g}(\chi,\eta))\\#A$ have fgc?

We point out that Theorem 4.1 does not apply in the present situation, by the
following observation.

######  Remark 4.8.

Let $A$ be a finite-dimensional Hopf algebra that admits a YD-triple
$(g,\chi,\eta)$. Then $A$ is not semisimple.

###### Proof.

Observe that the restriction $\eta:\langle g\rangle\to\Bbbk$ is a morphism of
abelian groups, hence $\eta(g^{p})=0$, which implies that $p$ divides the
order of $g$. Thus $\Bbbk\langle g\rangle$ is not semisimple, and a fortiori
$A$ is not semisimple by [37, 10.3.4]. ∎

There are examples fitting into abelian exact sequences that are not
necessarily split.

######  Remark 4.9.

Let $F$ be a finite group and let $(g,\chi,\eta)$ be a YD-triple for $\Bbbk
F$. It consists of $g\in Z(F)$,
$\chi\in\widehat{F}\coloneqq\operatorname{Hom}_{\rm gps}(F,\Bbbk^{\times})$
and $\eta\in\operatorname{Der}_{\chi,\chi}(\Bbbk F,\Bbbk)$ such that
$\chi(g)=\eta(g)=1$. Let $N\coloneqq\ker\chi\cap Z(F)\lhd F$. On one hand we
consider the subalgebra of $H=\mathscr{B}(\mathcal{V}_{g}(\chi,\eta))\\#\Bbbk
F$:

$\displaystyle K$ $\displaystyle\coloneqq\Bbbk\langle x,\gamma:\gamma\in
N\rangle\simeq\Bbbk\langle x\rangle\\#\Bbbk N,$

Here $K$ is commutative but not necessarily cocommutative. On the other hand,
$\chi$ induces a character $\overline{\chi}$ of $F/N$. Let
$\Bbbk\zeta\in{}^{\Bbbk(F/N)}_{\Bbbk(F/N)}\mathcal{YD}$ corresponding to the
YD-pair $(e,\overline{\chi})$. Then

$\displaystyle L$
$\displaystyle\coloneqq\mathscr{B}(\zeta)\\#\Bbbk(F/N)\simeq\Bbbk[\zeta]/(\zeta^{p})\\#\Bbbk(F/N);$

here $\zeta$ is primitive. Clearly $L$ is cocommutative but not necessarily
commutative.

Let $\iota:K\to H$ be the inclusion and let $\pi:H\to L$ be the map defined by
$\pi(x)=0$, $\pi(y)=\zeta$ and $\pi(\gamma)=$ the class of $\gamma$ in $F/N$.
Then $H$ fits into the abelian exact sequence $\Bbbk\to
K\overset{\iota}{\rightarrow}H\overset{\pi}{\rightarrow}L\to\Bbbk$, which is
not split, for instance, when the exact sequence of groups $1\to N\rightarrow
F\rightarrow F/N\to 1$ is not split.

In the setting of the previous remark, the Hopf algebra $H^{*}$ has fgc.

######  Proposition 4.10.

Let $H=\mathscr{B}(\mathcal{V}_{g}(\chi,\eta))\\#\Bbbk F$, where $F$ is a
finite group and $(g,\chi,\eta)$ is a YD-triple for $\Bbbk F$. Then $H^{*}$
has fgc.

###### Proof.

Arguing as in [8, Lemma 1.5], we see that
$H^{*}\simeq\mathcal{R}\\#\Bbbk^{F}$, where $\mathcal{R}\simeq\mathscr{B}(W)$
is isomorphic to $\mathscr{B}(\mathscr{V}(1,2))$ as algebras (although not as
braided Hopf algebras). Anyway, $\mathcal{R}$ has fgc by Proposition 4.6,
hence $H^{*}$ has fgc by Theorem 4.1. ∎

##  5\. The first Laestrygonian algebra
$\mathscr{B}(\mathfrak{L}_{q}(1,\mathscr{G}))$

###  5.1. The Nichols algebra $\mathscr{B}(\mathfrak{L}_{q}(1,\mathscr{G}))$

The next example of our interest depends on the data: $q\in\Bbbk^{\times}$ and
$a\in{\mathbb{F}_{p}}^{\times}$. Let

$\displaystyle\mathtt{r}$ $\displaystyle\in\\{1-p,2-p,\dots,-2,-1\\}$ such
that $\displaystyle\mathtt{r}$ $\displaystyle\equiv 2a\mod p.$

The _ghost_ is the integer
$\mathscr{G}\coloneqq-\mathtt{r}\in\mathbb{I}_{p-1}$; since $p$ is odd,
$\mathscr{G}$ determines $a$. To this data we attach the braided vector space
$\mathfrak{L}_{q}(1,\mathscr{G})$ with basis
$\mathpzc{b}=\\{x_{1},y_{1},x_{2}\\}$ and braiding given by

(23) $\displaystyle(c(b\otimes b^{\prime}))_{b,b^{\prime}\in\mathpzc{b}}$
$\displaystyle=\begin{pmatrix}x_{1}\otimes x_{1}&(y_{1}+x_{1})\otimes
x_{1}&q\,x_{2}\otimes x_{1}\\\ x_{1}\otimes y_{1}&(y_{1}+x_{1})\otimes
y_{1}&q\,x_{2}\otimes y_{1}\\\ q^{-1}x_{1}\otimes
x_{2}&q^{-1}(y_{1}+ax_{1})\otimes x_{2}&x_{2}\otimes x_{2}\end{pmatrix}.$

Thus $V_{1}\coloneqq\Bbbk x_{1}+\Bbbk y_{1}\simeq\mathcal{V}(1,2)$ and
$V_{2}\coloneqq\Bbbk x_{2}$ satisfy $c:V_{i}\otimes V_{j}=V_{j}\otimes V_{i}$,
$i,j\in\\{1,2\\}$; in particular $V_{1}$ and $V_{2}$ are braided subspaces of
$V$. We introduce

(24) $\displaystyle z_{0}$ $\displaystyle\coloneqq x_{2},$ $\displaystyle
z_{n+1}$ $\displaystyle\coloneqq y_{1}z_{n}-qz_{n}y_{1},$ $\displaystyle n$
$\displaystyle>0.$

######  Lemma 5.1.

[4, 4.3.1] The algebra $\mathscr{B}(\mathfrak{L}_{q}(1,\mathscr{G}))$ is
presented by generators $x_{1},y_{1},x_{2}$ and relations (19), and, in the
notation (24),

(25) $\displaystyle x_{1}x_{2}$ $\displaystyle=q\,x_{2}x_{1},$ (26)
$\displaystyle z_{1+\mathscr{G}}$ $\displaystyle=0,$ (27) $\displaystyle
z_{t}z_{t+1}$ $\displaystyle=q^{-1}\,z_{t+1}z_{t},$ $\displaystyle 0\leq$
$\displaystyle t<\mathscr{G},$ (28) $\displaystyle z_{t}^{p}$
$\displaystyle=0,$ $\displaystyle 0\leq$ $\displaystyle t\leq\mathscr{G}.$

The algebra $\mathscr{B}(\mathfrak{L}_{q}(1,\mathscr{G}))$ has a PBW-basis

(29) $B=\\{x_{1}^{m_{1}}y_{1}^{m_{2}}z_{\mathscr{G}}^{n_{\mathscr{G}}}\dots
z_{1}^{n_{1}}z_{0}^{n_{0}}:m_{i},n_{j}\in\mathbb{I}_{0,p}\\},$

hence $\dim\mathscr{B}(\mathfrak{L}_{q}(1,\mathscr{G}))=p^{\mathscr{G}+3}$. ∎

###  5.2.  A suitable realization

In order to realize $\mathfrak{L}_{q}(1,\mathscr{G})$ in
${}^{\Bbbk\Gamma}_{\Bbbk\Gamma}\mathcal{YD}$ for some finite group $\Gamma$,
we need to assume that $q$ is a root of 1. Set
$d\coloneqq\operatorname{ord}q$; then $d$ is coprime to
$p=\operatorname{char}\Bbbk$. Fix a positive integer $f$ which is a multiple
of $pd$. A suitable choice of group is

$\displaystyle\Gamma={\mathbb{Z}/f}\times{\mathbb{Z}/f}$
$\displaystyle=\langle g_{1}\rangle\oplus\langle g_{2}\rangle,$
$\displaystyle\text{where }\operatorname{ord}g_{1}$
$\displaystyle=\operatorname{ord}g_{2}=f.$

It is not difficult to see that $\mathfrak{L}_{q}(1,\mathscr{G})$ can be
realized in ${}^{\Bbbk\Gamma}_{\Bbbk\Gamma}\mathcal{YD}$ by

(30) $\displaystyle\begin{aligned} g_{1}\cdot x_{1}&=x_{1},&g_{1}\cdot
y_{1}&=y_{1}+x_{1},&g_{1}\cdot x_{2}&=qx_{2},\\\ g_{2}\cdot
x_{1}&=q^{-1}x_{1},&g_{2}\cdot y_{1}&=q^{-1}(y_{1}+ax_{1}),&g_{2}\cdot
x_{2}&=x_{2},\\\ \deg x_{1}&=g_{1},&\deg y_{1}&=g_{1},&\deg
x_{2}&=g_{2}.\end{aligned}$

Therefore the Hopf algebra
$\widetilde{H}\coloneqq\mathscr{B}(\mathfrak{L}_{q}(1,\mathscr{G}))\\#\Bbbk\Gamma$
is presented by generators $x_{1}$, $y_{1}$, $x_{2}$, $g_{1}$, $g_{2}$, with
relations (19), (25), (26), (27), (28), and

(31) $\displaystyle g_{1}^{f}$ $\displaystyle=1,$ $\displaystyle g_{2}^{f}$
$\displaystyle=1,$ $\displaystyle g_{1}g_{2}$ $\displaystyle=g_{2}g_{1},$ (32)
$\displaystyle g_{1}x_{1}$ $\displaystyle=x_{1}g_{1},$ $\displaystyle
g_{1}y_{1}$ $\displaystyle=y_{1}g_{1}+g_{1}x_{1},$ $\displaystyle g_{1}x_{2}$
$\displaystyle=qx_{2}g_{1},$ (33) $\displaystyle g_{2}x_{1}$
$\displaystyle=q^{-1}x_{1}g_{2},$ $\displaystyle g_{2}y_{1}$
$\displaystyle=q^{-1}(y_{1}g_{2}+ax_{1}g_{2}),$ $\displaystyle g_{2}x_{2}$
$\displaystyle=x_{2}g_{2}.$

The comultiplication of $\widetilde{H}$ is determined by
$\Delta(g_{i})=g_{i}\otimes g_{i}$, $i=1,2$, and

(34) $\Delta(x_{1})=x_{1}\otimes 1+g_{1}\otimes
x_{1},\quad\Delta(y_{1})=y_{1}\otimes 1+g_{1}\otimes
y_{1},\quad\Delta(x_{2})=x_{2}\otimes 1+g_{2}\otimes x_{2}.$

Clearly $\dim\widetilde{H}=p^{\mathscr{G}+3}f^{2}$.

###  5.3. The split case

In this subsection we deal with
$\mathscr{B}(\mathfrak{L}_{1}(1,\mathscr{G}))$; Let $\Gamma$ be as above with
$f$ divisible by $pd$ with $d=\operatorname{ord}q$. Let $K\subseteq
H\coloneqq\mathscr{B}(\mathfrak{L}_{1}(1,\mathscr{G}))\\#\Bbbk\Gamma$ be the
Hopf subalgebra

$\displaystyle K=\Bbbk\langle x_{1},g_{1},g_{2}\rangle.$

We shall consider the restricted enveloping algebra

$\displaystyle L=\mathfrak{u}(\mathfrak{l}),$

where $\mathfrak{l}$ is the restricted Lie algebra defined as follows. Let
$V(\mathscr{G})$ be the simple $\mathfrak{sl}(2)$-module of highest weight
$\mathscr{G}$. Let $E=(\begin{smallmatrix}0&1\\\ 0&0\end{smallmatrix})$. Pick
a basis $(v_{n})_{n\in\mathbb{I}_{0,\mathscr{G}}}$ of $V(\mathscr{G})$ such
that

$\displaystyle E\cdot v_{n}$ $\displaystyle=v_{n+1},$ $\displaystyle n$
$\displaystyle\in\mathbb{I}_{0,\mathscr{G}-1},$ $\displaystyle E\cdot
v_{\mathscr{G}}$ $\displaystyle=0.$

Let $\mathfrak{l}=V(\mathscr{G})\rtimes\Bbbk E$, a Lie subalgebra of the
motion Lie algebra $V(\mathscr{G})\rtimes\mathfrak{sl}(2)$; it follows from
Lemma 2.6 that $\mathfrak{l}$ is restricted with $p$-operation equal to $0$.
That is, $L$ is presented by generators $v_{0}$ and $E$ with defining
relations, in terms of $v_{n+1}=Ev_{n}-v_{n}E$,
$n\in\mathbb{I}_{0,\mathscr{G}-1}$,

$\displaystyle v_{m}v_{n}$ $\displaystyle=v_{n}v_{m},$ $\displaystyle
v_{n}^{p}$ $\displaystyle=0,$ $\displaystyle m,n$
$\displaystyle\in\mathbb{I}_{0,\mathscr{G}},$ $\displaystyle
Ev_{\mathscr{G}}-v_{\mathscr{G}}E$ $\displaystyle=0,$ $\displaystyle E^{p}$
$\displaystyle=0.$

######  Proposition 5.2.

The Hopf algebra
$H=\mathscr{B}(\mathfrak{L}_{1}(1,\mathscr{G}))\\#\Bbbk\Gamma$ fits into a
split abelian exact sequence $\Bbbk\to
K\overset{\iota}{\to}H\overset{\pi}{\to}L\to\Bbbk$, where $\iota$ is the
inclusion and $\pi$ is defined by

(35) $\displaystyle\pi(g_{1})$ $\displaystyle=1,$ $\displaystyle\pi(g_{2})$
$\displaystyle=1,$ $\displaystyle\pi(x_{1})$ $\displaystyle=0,$
$\displaystyle\pi(y_{1})$ $\displaystyle=E,$ $\displaystyle\pi(x_{2})$
$\displaystyle=v_{0}.$

###### Proof.

By assumption, the Hopf subalgebra $K$ is commutative. In addition $L$ is
cocommutative. Notice that $\dim K=pf^{2}$. We see by inspection that $K$ is
normal, i.e., $HK^{+}=K^{+}H$. Thus we have an abelian exact sequence

$\displaystyle\Bbbk\to K\overset{\iota}{\to}H\to H/HK^{+}\to\Bbbk.$

Since $\dim H=p^{\mathscr{G}+3}f^{2}$, $\dim H/HK^{+}=p^{\mathscr{G}+2}=\dim
L$. Now (35) determines an algebra map $\pi:H\to L$ by (24) and the defining
relations of $\mathscr{B}(\mathfrak{L}_{1}(1,\mathscr{G}))$. The map $\pi$ is
surjective, has $HK^{+}\subseteq\ker\pi$ and preserves the comultiplication
since the classes of $y_{1}$ and $x_{2}$ are primitive in $H/HK^{+}$. By
dimension counting, $\pi$ induces an isomorphism of Hopf algebras $H/HK^{+}\to
L$.

Observe that $\pi(z_{n})=v_{n}$, for all $n\geq 0$. To define a splitting, we
argue as in the proof of Lemma 4.4. The universal property of
$\mathfrak{u}(\mathfrak{l})$ implies the existence of a unique algebra map
$\mathpzc{s}:L=\mathfrak{u}(\mathfrak{l})\to H$ such that

$\mathpzc{s}(E)=-\mathcal{S}(y_{1}),\quad\mathpzc{s}(v_{n})=-\mathcal{S}(z_{n}),\quad\forall
n=0,1,\dots$

Clearly $\pi\mathpzc{s}=\operatorname{id}_{L}$. To see that $\mathpzc{s}$ is
$L$-colinear, it will be enough to verify that the maps
$(\mathpzc{s}\otimes\operatorname{id}_{L})\Delta$ and
$(\operatorname{id}\otimes\pi)\Delta\mathpzc{s}$ agree on $E$ and $v_{0}$,
because both $(\mathpzc{s}\otimes\operatorname{id}_{L})\Delta$ and
$(\operatorname{id}_{H}\otimes\pi)\Delta\mathpzc{s}$ are algebra maps and $E$
and $v_{0}$ generate $L$ as an algebra. This follows at once from (34).

Let now $\mathpzc{r}:H\to K$ be the linear map defined on the basis of $H$
arising from (29) by

$\mathpzc{r}(g_{1}^{c}g_{2}^{b}x_{1}^{m_{1}}y_{1}^{m_{2}}z_{\mathscr{G}}^{n_{\mathscr{G}}}\dots
z_{1}^{n_{1}}z_{0}^{n_{0}})=\begin{cases}g_{1}^{c}g_{2}^{b}x_{1}^{m_{1}},\;\textrm{if
}m_{2}+n_{\mathscr{G}}+\dots+n_{0}=0,\\\ 0,\quad\textrm{otherwise}\end{cases}$

As in the proof of Lemma 4.4, we see that $\mathpzc{r}$ is a $K$-linear
retraction of $\iota$ and that $\mathpzc{r}\mathpzc{s}=\epsilon_{L}1_{K}$.
Moreover, $\mathpzc{r}$ is a coalgebra map. To see this, we consider the
subspace $I=\ker\mathpzc{r}$ which coincides with the linear span of all
monomials
$g_{1}^{c}g_{2}^{b}x_{1}^{m_{1}}y_{1}^{m_{2}}z_{\mathscr{G}}^{n_{\mathscr{G}}}\dots
z_{1}^{n_{1}}z_{0}^{n_{0}}$ such that $m_{2}+n_{\mathscr{G}}+\dots+n_{0}>0$.
The defining relations of $H$ imply that $I$ is a left ideal. We claim that
$\Delta(I)\subseteq I\otimes H+H\otimes I$. Indeed, (34) implies that
$\Delta(y_{1}),\Delta(x_{2})\in I\otimes H+H\otimes I$. By [3, 4.2.10, pp. 28]
and the comultiplication formula of the bosonization, there exist
$\nu_{j,n}\in\Bbbk$ such that

$\displaystyle\Delta(z_{n})$ $\displaystyle=z_{n}\otimes
1+\sum_{j=0}^{n}\nu_{j,n}\;x_{1}^{n-j}g_{1}^{j}g_{2}\otimes z_{j},$
$\displaystyle\forall n$ $\displaystyle\in\mathbb{I}_{0,\mathscr{G}}.$

Thus $\Delta(z_{n})\in I\otimes H+H\otimes I$, for all $n\geq 0$. Then the
claim follows. Since $\mathpzc{r}|_{K}=\operatorname{id}_{K}$, we conclude
that $\mathpzc{r}$ is a coalgebra map. Thus $(\mathpzc{s},\mathpzc{r})$ is a
splitting and the proof of the Proposition is complete. ∎

######  Proposition 5.3.

The Hopf algebra
$H=\mathscr{B}(\mathfrak{L}_{1}(1,\mathscr{G}))\\#\Bbbk\Gamma$ and its double
$D(H)$ have fgc.

###### Proof.

This follows from Proposition 5.2 and Theorem 3.6. ∎

###  5.4. The general case

Recall that $q\in\Bbbk^{\times}$ has order $d$, that $f$ is a multiple of $pd$
and that $\Gamma=\langle
g_{1},g_{2}:\,g_{1}^{f}=g_{2}^{f}=1,\,g_{1}g_{2}=g_{2}g_{1}\rangle\simeq\mathbb{Z}/f\times\mathbb{Z}/f$.

######  Theorem 5.4.

The Hopf algebra
$\widetilde{H}=\mathscr{B}(\mathfrak{L}_{q}(1,\mathscr{G}))\\#\Bbbk\Gamma$ has
fgc.

###### Proof.

Consider the bilinear form $\sigma:\Gamma\times\Gamma\to\Bbbk^{\times}$
determined by

$\sigma(g_{i},g_{j})=\begin{cases}1,\quad i=j,\\\ q,\quad i=1,j=2,\\\ 1,\quad
i=2,j=1.\end{cases}$

Then $\sigma$ is a 2-cocycle in $\Gamma$. Let
$\vartheta:\Gamma\times\Gamma\to\Bbbk^{\times}$,
$\vartheta(g,h)=\sigma(g,h)\sigma(h,g)^{-1}$ be the associated antisymmetric
bilinear form. Thus,

$\vartheta(g_{i},g_{j})=\begin{cases}1,\quad i=j,\\\ q,\quad i=1,j=2,\\\
q^{-1},\quad i=2,j=1.\end{cases}$

As in Subsection A.3 of the Appendix, let
${\mathcal{F}}_{\sigma}:{}^{\Bbbk\Gamma}_{\Bbbk\Gamma}\mathcal{YD}\to{}^{\Bbbk\Gamma}_{\Bbbk\Gamma}\mathcal{YD}$
be the monoidal functor associated to $\sigma$. The image under
${\mathcal{F}}_{\sigma}$ of the braided vector space
$V=\mathfrak{L}_{1}(1,\mathscr{G})$ in 5.2 is ${\mathcal{F}}_{\sigma}(V)=V$
with the same grading as $V$ and $\Gamma$-action (64); that is,

(36) $\displaystyle\begin{aligned} g_{1}\cdot x_{1}&=x_{1},&g_{1}\cdot
y_{1}&=y_{1}+x_{1},&g_{1}\cdot x_{2}&=qx_{2},\\\ g_{2}\cdot
x_{1}&=q^{-1}x_{1},&g_{2}\cdot y_{1}&=q^{-1}(y_{1}+ax_{1}),&g_{2}\cdot
x_{2}&=x_{2},\\\ \deg x_{1}&=g_{1},&\deg y_{1}&=g_{1},&\deg
x_{2}&=g_{2}.\end{aligned}$

Hence ${\mathcal{F}}_{\sigma}(V)=\mathfrak{L}_{q}(1,\mathscr{G})$. By Lemma
A.4. the bosonization
$\widetilde{H}=\mathscr{B}(\mathfrak{L}_{q}(1,\mathscr{G}))\\#\Bbbk\Gamma$ is
a cocycle deformation of the bosonization
$H=\mathscr{B}(\mathfrak{L}_{1}(1,\mathscr{G}))\\#\Bbbk\Gamma$. By Proposition
5.3, the double $D\left(H\right)$ has fgc. Hence $\widetilde{H}$ also has fgc,
as claimed. ∎

######  Corollary 5.5.

Let $q\in\mathbb{G}_{\infty}$. The Nichols algebra
$\mathscr{B}(\mathfrak{L}_{q}(1,\mathscr{G}))$ has fgc. ∎

##  6\. Pointed Hopf algebras over abelian groups

###  6.1. A class of braided vector spaces

We now proceed with a family of Hopf algebras which are bosonizations of some
Nichols algebras introduced in [4], analogues in odd characteristic of Nichols
algebras appearing in [3]. We show that, up to a cocycle deformation, the Hopf
algebra $H=\mathscr{B}(\mathscr{V}(\mathfrak{q},\mathbf{a}))\\#\Bbbk\Gamma$
fits into a split abelian exact sequence $\Bbbk\to K\rightarrow
H\rightarrow\mathfrak{u}(\mathfrak{l})\to\Bbbk$, where the restricted Lie
algebra $\mathfrak{l}$ is determined explicitly.

#### Data

We shall consider braided vector spaces depending on:

  * $\circ$

two positive integers $t<\theta$;

  * $\circ$

a matrix $\mathfrak{q}=(q_{ij})_{i,j\in\mathbb{I}_{\theta}}$ such that

(37) $\displaystyle q_{ij}q_{ji}$ $\displaystyle=1,$ $\displaystyle q_{ii}$
$\displaystyle=1,$ $\displaystyle i,j$
$\displaystyle\in\mathbb{I}_{\theta},\,i\neq j.$

  * $\circ$

a family
$\mathbf{a}=(a_{ij})_{\begin{subarray}{c}i\in\mathbb{I}_{t+1,\theta},\\\
j\in\mathbb{I}_{t}\end{subarray}}$ with entries in ${\mathbb{F}}_{p}$. We lift
this family to $\mathbb{Z}$ as follows:

  * $\diamond$

when $a_{ij}\neq 0$, we take

(38) $\displaystyle\mathtt{r}_{ij}$
$\displaystyle\in\\{1-p,2-p,\dots,-2,-1\\}$ such that
$\displaystyle\mathtt{r}_{ij}$ $\displaystyle\equiv 2a_{ij}\mod p,$

and then we set
$\mathscr{G}_{i,j}\coloneqq-\mathtt{r}_{ij}\in\mathbb{I}_{p-1}$.

  * $\diamond$

when $a_{ij}=0$, we lift it to $\mathscr{G}_{i,j}=0$.

The family
$\mathscr{G}\coloneqq(\mathscr{G}_{i,j})_{\begin{subarray}{c}i\in\mathbb{I}_{t+1,\theta},\\\
j\in\mathbb{I}_{t}\end{subarray}}$, equivalent to $\mathbf{a}$, is the _ghost_
; both $\mathbf{a}$ and $\mathscr{G}$ are needed.

#### Definition

We define a braided vector space $(\mathscr{V}(\mathfrak{q},\mathbf{a}),c)$
from the data above. First, it has a decomposition
$\mathscr{V}(\mathfrak{q},\mathbf{a})=V_{1}\oplus\dots\oplus
V_{t}\oplus\dots\oplus V_{\theta}$ such that

$\displaystyle c(V_{i}\otimes V_{j})$ $\displaystyle=V_{j}\otimes V_{i},\quad
i,j\in\mathbb{I}_{\theta}.$

Then we assume:

  * •

If $j\in\mathbb{I}_{t}$, then $V_{j}\simeq\mathcal{V}(1,2)$ (the blocks). Let
$\\{x_{j},y_{j}\\}$ be a basis of $V_{j}$ realizing (18):

(39) $\displaystyle\begin{aligned} c(x_{j}\otimes x_{j})&=x_{j}\otimes
x_{j},&c(y_{j}\otimes x_{j})&=x_{j}\otimes y_{j},\\\ c(x_{j}\otimes
y_{j})&=(y_{j}+x_{j})\otimes x_{j},&c(y_{j}\otimes
y_{j})&=(y_{j}+x_{j})\otimes y_{j}.\end{aligned}$

  * •

If $i\in\mathbb{I}_{t+1,\theta}$, then $\dim V_{i}=1$; these are the points.
We fix a basis $\\{x_{i}\\}$ of $V_{i}$. Thus,
$\\{x_{i}:i\in\mathbb{I}_{\theta}\\}\amalg\\{y_{j}:j\in\mathbb{I}_{t}\\}$ is a
basis of $\mathscr{V}(\mathfrak{q},\mathbf{a})$.

The braidings $c_{ij}=c_{|V_{i}\otimes V_{j}}$, $i,j\in\mathbb{I}_{\theta}$
are given by the data $\mathfrak{q}$ and $\mathbf{a}$ as follows:

  * $\circ$

If $i,j\in\mathbb{I}_{\theta}$, then

(40) $\displaystyle c(x_{i}\otimes x_{j})$ $\displaystyle=q_{ij}x_{j}\otimes
x_{i}.$

  * $\circ$

If $i,j\in\mathbb{I}_{t}$ are blocks, then $c_{ij}=q_{ij}\tau$, $\tau$ being
the flip, for $i\neq j$ while $c_{jj}$ is given by (39).

  * $\circ$

If $j\in\mathbb{I}_{t}$ a block and $i\in\mathbb{I}_{t+1,\theta}$ is a point,
then $c_{|(V_{i}\oplus V_{j})\otimes(V_{i}\oplus V_{j})}$ is given by (40) and

(41) $\displaystyle c(y_{j}\otimes x_{i})$ $\displaystyle=q_{ji}x_{i}\otimes
y_{j},$ $\displaystyle c(x_{i}\otimes y_{j})$
$\displaystyle=q_{ij}(y_{j}+a_{ij}x_{j})\otimes x_{i}.$

As in [3], $\mathscr{V}(\mathfrak{q},\mathbf{a})$ is described by a diagram of
the following shape:

That is, there are $\mathpzc{r}$ blocks, $\theta-t$ points and a line
decorated by $\mathscr{G}_{k,\ell}$ when $\mathscr{G}_{k,\ell}\neq 0$, joining
the block $k$ with the point $\ell$; this graph is admissible in the sense of
[3, Definition 1.3.7].

######  Remark 6.1.

The data above is an $\mathfrak{ab}$-triple
$\mathscr{T}=(\mathbf{n},\mathfrak{q},\mathbf{a})$ of rank $\theta$, cf. §
A.1, as follows:

  * •

$\mathbf{n}=(n_{j})_{j\in\mathbb{I}_{\theta}}$ is given by $n_{j}=2$, if
$1\leq j\leq t$; and $n_{j}=1$, if $t<j\leq\theta$.

  * •

the matrix $\mathfrak{q}=(q_{ij})_{i,j\in\mathbb{I}_{\theta}}$ is as given
above with the constraint (37);

  * •

the family $\mathbf{t}=(\mathbf{t}_{ij})_{i,j\in\mathbb{I}_{\theta,}}$ where
$\mathbf{t}_{ij}\in\operatorname{End}\Bbbk^{n_{j}}$ is given by

$\displaystyle\mathbf{t}_{ij}$ $\displaystyle=0,$ $\displaystyle\text{when }t$
$\displaystyle<j\leq\theta\text{ or }1\leq i,j\leq t;$
$\displaystyle\mathbf{t}_{ij}(x_{j})$ $\displaystyle=0,$
$\displaystyle\mathbf{t}_{ij}(y_{j})$ $\displaystyle=a_{ij}x_{j},$
$\displaystyle\text{when }1$ $\displaystyle\leq j\leq t\text{ and
}t<i\leq\theta.$

If $t=\theta=1$, then we have the restricted Jordan plane as in Section 4. If
$t=1$ and $\theta=2$, then we recover $\mathfrak{L}_{q}(1,\mathscr{G})$ as in
Section 5.

#### Grading

The subspace $\mathscr{U}$ of
$\mathscr{V}=\mathscr{V}(\mathfrak{q},\mathbf{a})$ spanned by
$(x_{i})_{i\in\mathbb{I}_{\theta}}$ is a braided subspace and
$\mathscr{U}\subseteq\mathscr{V}$ is a filtration of braided vector spaces.
The associated graded vector space
$\operatorname{gr}\mathscr{V}=\mathscr{U}\oplus\mathscr{V}/\mathscr{U}$ is of
diagonal type; indeed, if $\overline{y}_{j}$ is the class of $y_{j}$ in
$\mathscr{V}/\mathscr{U}$, $j\in\mathbb{I}_{t}$, then
$(x_{i})_{i\in\mathbb{I}_{\theta}}\amalg(\overline{y}_{j})_{j\in\mathbb{I}_{t}}$
is a basis of $\operatorname{gr}\mathscr{V}$ and the braiding of
$\operatorname{gr}\mathscr{V}$ is given by

$\displaystyle\begin{aligned} c(x_{i}\otimes x_{k})&=q_{ik}\,x_{k}\otimes
x_{j},&c(\overline{y}_{j}\otimes
x_{i})&=q_{ji}\,x_{i}\otimes\overline{y}_{j},\\\
c(x_{i}\otimes\overline{y}_{j})&=q_{ij}\,\overline{y}_{j}\otimes
x_{i},&c(\overline{y}_{j}\otimes\overline{y}_{\ell})&=q_{j\ell}\,\overline{y}_{\ell}\otimes\overline{y}_{j},\end{aligned}$
$\displaystyle\text{for all }i,k$ $\displaystyle\in\mathbb{I}_{\theta},\
j,\ell\in\mathbb{I}_{t}.$

Precisely, let $Q\simeq\mathbb{Z}^{\theta+t}$ be the free abelian group with
basis
$(\alpha_{i})_{i\in\mathbb{I}_{\theta}}\amalg(\beta_{j})_{j\in\mathbb{I}_{t}}$;
then the braiding is given by the bilinear form $\mathbf{p}:Q\times
Q\to\Bbbk^{\times}$ defined by

$\displaystyle\mathbf{p}(\alpha_{i}\otimes\alpha_{j})=\mathbf{p}(\alpha_{i}\otimes\beta_{j})=\mathbf{p}(\beta_{i}\otimes\alpha_{j})=\mathbf{p}(\beta_{i}\otimes\beta_{j})$
$\displaystyle=q_{ij}$

whenever $i$ and $j$ make sense. By (37),
$\mathscr{B}(\operatorname{gr}\mathscr{V})$ is a quantum linear space and we
have

$\displaystyle\mathbf{p}(\gamma\otimes\delta)$
$\displaystyle=\mathbf{p}(\delta\otimes\gamma)$
$\displaystyle\forall\gamma,\delta$ $\displaystyle\in Q.$

Now consider the grading of $\mathscr{V}$ and its extension to
$T(\mathscr{V})$ given by $\deg x_{i}=\alpha_{i}$, $\deg y_{j}=\beta_{j}$.

Given points $h,\ell\in\mathbb{I}_{t+1,\theta}$ we set

(42) $\displaystyle\mbox{щ}_{h,\mathbf{n}}$
$\displaystyle\coloneqq(\operatorname{ad}_{c}y_{1})^{n_{1}}\dots(\operatorname{ad}_{c}y_{t})^{n_{t}}x_{h},\qquad\mathbf{n}=(n_{1},\dots,n_{t})\in\mathbb{N}^{t}_{0};$
(43) $\displaystyle\mathcal{A}_{h}$
$\displaystyle\coloneqq\\{\mathbf{n}\in\mathbb{N}^{t}_{0}:0\leq\mathbf{n}\leq\mathscr{G}_{h}=(\mathscr{G}_{h,1},\dots,\mathscr{G}_{h,t})\\},\text{
ordered lexicographically.}$

In terms of the bilinear form $\mathbf{p}$, we also set

$\displaystyle\mathbf{p}_{h,\ell;\mathbf{m},\mathbf{n}}$
$\displaystyle\coloneqq\mathbf{p}_{\,\deg\mbox{щ}_{h,\mathbf{m}},\deg\mbox{щ}_{\ell,\mathbf{n}}}=\big{(}\prod_{k,j\in\mathbb{I}_{t}}q_{kj}^{m_{k}n_{j}}\big{)}\big{(}\prod_{k\in\mathbb{I}_{t}}q_{k\ell}^{m_{k}}\big{)}\big{(}\prod_{j\in\mathbb{I}_{t}}q_{hj}^{n_{j}}\big{)}q_{h\ell},\quad\mathbf{m}\in\mathcal{A}_{h},\
\mathbf{n}\in\mathcal{A}_{\ell}.$

######  Remark 6.2.

In the definition (42), $\operatorname{ad}_{c}$ means the braided adjoint.
However, it could be replaced by a sequence of $q$-commutators (with various
$q$). More precisely, given $h\in\mathbb{I}_{t+1,\theta}$ and
$\mathbf{n}=(n_{1},\dots,n_{t})\in\mathbb{N}^{t}_{0}$, set
$\mbox{щ}=\mbox{щ}_{h,\mathbf{n}}$. Then

(44) щ
$\displaystyle=y_{j}\,\widetilde{\mbox{щ}}-q\,\widetilde{\mbox{щ}}\,y_{j},$

where $j=\min\\{i\in\mathbb{I}_{t}:n_{i}\neq 0\\}$,
$\widetilde{\mathbf{n}}=(0,\dots,0,n_{j}-1,n_{j+1},\dots,n_{t})$,
$\widetilde{\mbox{щ}}=\mbox{щ}_{h,\widetilde{\mathbf{n}}}$ and

$\displaystyle q$ $\displaystyle=q_{jh}\prod_{j\leq i\leq t}q_{ji}^{n_{i}}.$

See the proof of [3, Lemma 7.2.3]. In particular, if $\mathfrak{q}=\mathbf{1}$
is the matrix with all entries equal to $1$, then $q=1$ and we can replace
$\operatorname{ad}_{c}$ by the usual adjoint in (42).

#### The Nichols algebra $\mathscr{B}(\mathscr{V}(\mathfrak{q},\mathbf{a}))$

The following result is not included in [4] but its proof is similar to that
of [4, 4.3.1, 6.1].

######  Lemma 6.3.

The algebra $\mathscr{B}(\mathscr{V}(\mathfrak{q},\mathbf{a}))$ is presented
by generators $x_{i}$, $i\in\mathbb{I}_{\theta}$, $y_{j}$,
$j\in\mathbb{I}_{t}=\mathbb{I}_{t}$, and relations

(45) $\displaystyle x_{j}^{p}=0,\quad y_{j}^{p}=0,\quad
y_{j}x_{j}-x_{j}y_{j}+\tfrac{1}{2}x_{j}^{2}=0,$ $\displaystyle j$
$\displaystyle\in\mathbb{I}_{t};$ (46) $\displaystyle
x_{k}x_{j}=q_{kj}\,x_{j}x_{k},\,x_{k}y_{j}=q_{kj}\,y_{j}x_{k},\quad
y_{k}y_{j}=q_{kj}\,y_{j}y_{k}$ $\displaystyle k\neq j$
$\displaystyle\in\mathbb{I}_{t};$ (47) $\displaystyle
x_{j}x_{h}=q_{jh}\,x_{h}x_{j},$ $\displaystyle j$
$\displaystyle\in\mathbb{I}_{t},\,\,h\in\mathbb{I}_{t+1,\theta};$ (48)
$\displaystyle(\operatorname{ad}_{c}y_{j})^{1+\mathscr{G}_{jh}}(x_{h})=0,$
$\displaystyle j$
$\displaystyle\in\mathbb{I}_{t},\,\,h\in\mathbb{I}_{t+1,\theta};$ (49)
$\displaystyle\mbox{щ}_{h,\mathbf{n}}^{p}=0,$ $\displaystyle\mathbf{n}$
$\displaystyle\in\mathcal{A}_{h},\,\,h\in\mathbb{I}_{t+1,\theta}$ (50)
$\displaystyle\mbox{щ}_{h,\mathbf{m}}\mbox{щ}_{\ell,\mathbf{n}}=\mathbf{p}_{h,\ell;\mathbf{m},\mathbf{n}}\,\mbox{щ}_{\ell,\mathbf{n}}\mbox{щ}_{h,\mathbf{m}},$
$\displaystyle h,\ell\in\mathbb{I}_{t+1,\theta},\,$
$\displaystyle\mathbf{m}\in\mathcal{A}_{h},\,\mathbf{n}\in\mathcal{A}_{\ell}.$

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

(51) $\displaystyle B=\big{\\{}x_{1}^{m_{1}}y_{1}^{m_{2}}\dots
x_{t}^{m_{2t-1}}y_{t}^{m_{2t}}\prod_{\begin{subarray}{c}h\in\mathbb{I}_{t+1,\theta}\\\
\mathbf{n}\in\mathcal{A}_{h}\end{subarray}}\mbox{щ}_{h,\mathbf{n}}^{n_{h,\mathbf{n}}}:\,$
$\displaystyle 0\leq n_{h,\mathbf{n}},m_{j}<p\mbox{ if
}j\in\mathbb{I}_{2t},h\in\mathbb{I}_{t+1,\theta},\mathbf{n}\in\mathcal{A}_{h}\big{\\}}.$

Hence
$\dim\mathscr{B}(\mathscr{V}(\mathfrak{q},\mathbf{a}))=p^{2t+\sum_{h\in\mathbb{I}_{t+1,\theta}}|\mathcal{A}_{h}|}$.

###### Proof.

Argue as in the proofs of [4, 4.3.1, 6.1]; relation (50) follows as in the
discussion in [3, pp. 107]. ∎

###  6.2. A suitable realization

In order to realize $\mathscr{V}(\mathfrak{q},\mathbf{a})$ in
${}^{\Bbbk\Gamma}_{\Bbbk\Gamma}\mathcal{YD}$ for some finite group $\Gamma$,
we need to assume that the entries of the matrix $\mathfrak{q}$ are roots of
1. Set

$\displaystyle
d\coloneqq\operatorname{lcm}\\{\operatorname{ord}q_{ij}:i,j\in\mathbb{I}_{\theta}\\};$

then $d$ is coprime to $p=\operatorname{char}\Bbbk$. Fix a positive integer
$f$ which is a multiple of $pd$. A suitable choice of group is

$\displaystyle\Gamma$ $\displaystyle=({\mathbb{Z}/f})^{\theta}=\langle
g_{1}\rangle\oplus\cdots\oplus\langle g_{\theta}\rangle,$ where
$\displaystyle\operatorname{ord}g_{1}$
$\displaystyle=\operatorname{ord}g_{2}=\dots\operatorname{ord}g_{\theta}=f.$

In other words, $\Gamma$ is generated by $g_{1},\dots,g_{\theta}$ with
relations

(52) $\displaystyle g_{i}^{f}$ $\displaystyle=1,$ $\displaystyle g_{i}g_{j}$
$\displaystyle=g_{j}g_{i},$ $\displaystyle i,j$
$\displaystyle\in\mathbb{I}_{\theta}.$

It is not difficult to see that $\mathscr{V}(\mathfrak{q},\mathbf{a})$ can be
realized in ${}^{\Bbbk\Gamma}_{\Bbbk\Gamma}\mathcal{YD}$ by

(53) $\displaystyle\begin{aligned} g_{k}\cdot
V_{\ell}&=q_{k\ell}\operatorname{id}_{V_{\ell}},&k&\in\mathbb{I}_{t},\,\ell\in\mathbb{I}_{\theta},\,k\neq\ell;\text{
or }k,\ell\in\mathbb{I}_{t+1,\theta};\\\ g_{i}\cdot x_{j}&=q_{ij}x_{j},\quad
g_{i}\cdot
y_{j}=q_{ij}(y_{j}+a_{ij}x_{j}),&i&\in\mathbb{I}_{t+1,\theta},j\in\mathbb{I}_{t},\\\
g_{j}\cdot x_{j}&=x_{j},\quad g_{j}\cdot
y_{j}=y_{j}+x_{j},&j&\in\mathbb{I}_{t},\\\ \deg x_{\ell}&=g_{\ell},\quad\deg
y_{j}=g_{j},&\ell&\in\mathbb{I}_{\theta}.\,j\in\mathbb{I}_{t}.\end{aligned}$

Let $\widetilde{H}$ denote the Hopf algebra
$\mathscr{B}(\mathscr{V}(\mathfrak{q},\mathbf{a}))\\#\Bbbk\Gamma$; it has a
presentation by generators $x_{i}$, $y_{j}$, $g_{k}$,
$i,k\in\mathbb{I}_{\theta}$, $j\in\mathbb{I}_{t}$ with relations (45), (46),
(47), (48), (49), (50), (52) and those induced by (53). The comultiplication
is given by

(54) $\displaystyle\Delta(x_{i})$ $\displaystyle=x_{i}\otimes 1+g_{i}\otimes
x_{i},$ $\displaystyle\Delta(y_{j})$ $\displaystyle=y_{j}\otimes
1+g_{j}\otimes y_{j},$ $\displaystyle\Delta(g_{i})$
$\displaystyle=g_{i}\otimes
g_{i},\,i\in\mathbb{I}_{\theta},j\in\mathbb{I}_{t}.$

One has

$\displaystyle\dim\widetilde{H}=p^{2t+\sum_{h\in\mathbb{I}_{t+1,\theta}}|\mathcal{A}_{h}|}f^{\theta}.$

###  6.3. The split case

In this subsection we deal with
$\mathscr{B}(\mathscr{V}(\mathbf{1},\mathbf{a}))$, where
$\mathbf{1}\in\Bbbk^{\theta}$ has all entries equal to $1$. Let $\Gamma$ be as
above with $f$ divisible by $pd$ with $d=\operatorname{ord}q$. Consider the
Hopf subalgebra $K\subseteq
H=\mathscr{B}(\mathscr{V}(\mathbf{1},\mathbf{a}))\\#\Bbbk\Gamma$

$\displaystyle K=\Bbbk\langle
x_{1},\dots,x_{t},\,g_{1},\dots,g_{\theta}\rangle.$

We shall consider the restricted enveloping algebra

$\displaystyle L=\mathfrak{u}(\mathfrak{l}),$

where $\mathfrak{l}$ is the restricted Lie algebra defined by the following
steps.

  * $\circ$

$\mathfrak{g}\coloneqq\mathfrak{g}_{1}\oplus\cdots\oplus\mathfrak{g}_{t}$ is
the direct sum of $\mathpzc{r}$ copies
$\mathfrak{g}_{j}\simeq\mathfrak{sl}(2)$, $j\in\mathbb{I}_{t}$.

  * $\circ$

$E_{j}\in\mathfrak{g}_{j}$ is the element corresponding to
$(\begin{smallmatrix}0&1\\\ 0&0\end{smallmatrix})$;
$\mathfrak{n}\coloneqq\Bbbk E_{1}\oplus\cdots\oplus\Bbbk
E_{t}\hookrightarrow\mathfrak{g}$.

  * $\circ$

For any point $h\in\mathbb{I}_{t+1,\theta}$, let $V(\mathscr{G}_{h,j})$ be the
simple $\mathfrak{g}_{j}$-module of highest weight $\mathscr{G}_{h,j}$. Pick a
basis $(v_{h,j;n})_{n\in\mathbb{I}_{0,\mathscr{G}_{h,j}}}$ of
$V(\mathscr{G}_{h,j})$ such that

$\displaystyle E_{j}\cdot v_{h,j;m}$ $\displaystyle=v_{h,j;m+1},$
$\displaystyle 0\leq m$ $\displaystyle<\mathscr{G}_{h,j},$ $\displaystyle
E_{j}\cdot v_{h,\mathscr{G}_{hij}}$ $\displaystyle=0.$

  * $\circ$

For any point $h\in\mathbb{I}_{t+1,\theta}$, let $V(\mathscr{G}_{h})$ be the
simple $\mathfrak{g}$-module

$\displaystyle V(\mathscr{G}_{h})\coloneqq
V(\mathscr{G}_{h,1})\otimes\cdots\otimes V(\mathscr{G}_{h,t}).$

Recalling the notation (43), a basis of $V(\mathscr{G}_{h})$ is formed by the
elements

$\displaystyle v_{h,\mathbf{m}}$ $\displaystyle\coloneqq
v_{h,1,[m_{1}]}\otimes\cdots\otimes v_{h,t,[m_{t}]},$
$\displaystyle\mathbf{m}$
$\displaystyle=(m_{1},\dots,m_{t})\in\mathcal{A}_{h}.$

For $h\in\mathbb{I}_{t+1,\theta}$, set $v_{h}\coloneqq
v_{h,1,[0]}\otimes\cdots\otimes v_{h,t,[0]}$. Then for any
$\mathbf{m}\in\mathcal{A}_{h}$, we have

$\displaystyle v_{h,\mathbf{m}}$ $\displaystyle=E_{1}^{m_{1}}\cdots
E_{t}^{m_{t}}\cdot v_{h}.$

Indeed, $V(\mathscr{G}_{h})$ is the simple $\mathfrak{g}$-module of highest
weight $\mathscr{G}_{h}=(\mathscr{G}_{h,1},\dots,\mathscr{G}_{h,t})$.

  * $\circ$

Finally, let $\mathfrak{l}=V(\mathscr{G})\rtimes\mathfrak{n}$, where

$\displaystyle V(\mathscr{G})\coloneqq V(\mathscr{G}_{t+1})\oplus\cdots\oplus
V(\mathscr{G}_{\theta}).$

By Lemma 2.6, $\mathfrak{l}$ is a restricted Lie algebra with $p$-operation
equal to $0$. The restricted enveloping algebra $L=\mathfrak{u}(\mathfrak{l})$
is presented by generators $E_{j}$, $j\in\mathbb{I}_{t}$, and $v_{h}$,
$h\in\mathbb{I}_{t+1,\theta}$; set

$\displaystyle v_{h,\mathbf{m}}$
$\displaystyle=(\operatorname{ad}E_{1})^{m_{1}}\cdots(\operatorname{ad}{E_{t}})^{m_{t}}(v_{h}).$

Then the defining relations are

(55) $\displaystyle E_{j}E_{k}$ $\displaystyle=E_{k}E_{j},$ $\displaystyle
j,k$ $\displaystyle\in\mathbb{I}_{t};$ (56)
$\displaystyle(\operatorname{ad}E_{j})^{1+\mathscr{G}_{jh}}(v_{h})$
$\displaystyle=0,$ $\displaystyle j$
$\displaystyle\in\mathbb{I}_{t},\,\,h\in\mathbb{I}_{t+1,\theta};$ (57)
$\displaystyle v_{h,\mathbf{m}}v_{i,\mathbf{n}}$
$\displaystyle=v_{i,\mathbf{n}}v_{h,\mathbf{m}},$ $\displaystyle i,h$
$\displaystyle\in\mathbb{I}_{t+1,\theta},\,\,\mathbf{m}\in\mathcal{A}_{h},\,\,\mathbf{n}\in\mathcal{A}_{i};$
(58) $\displaystyle E_{j}^{p}$ $\displaystyle=0,$ $\displaystyle j$
$\displaystyle\in\mathbb{I}_{t};$ (59) $\displaystyle v_{h,\mathbf{m}}^{p}$
$\displaystyle=0,$ $\displaystyle h$
$\displaystyle\in\mathbb{I}_{t+1,\theta},\,\,\mathbf{m}\in\mathcal{A}_{h}.$

One has

$\displaystyle\dim
L=p^{t+\sum_{h\in\mathbb{I}_{t+1,\theta}}|\mathcal{A}_{h}|}.$

######  Remark 6.4.

Clearly $\mathfrak{l}$ is a Lie subalgebra of
$V(\mathscr{G})\rtimes\mathfrak{g}$; we do not need the reference to
$\mathfrak{g}$ here but it will be necessary in further developments.

######  Proposition 6.5.

The Hopf algebra
$H=\mathscr{B}(\mathscr{V}(\mathbf{1},\mathbf{a}))\\#\Bbbk\Gamma$ fits into a
split abelian exact sequence $\Bbbk\to
K\overset{\iota}{\rightarrow}H\overset{\pi}{\rightarrow}L\to\Bbbk$, where
$\iota$ is the inclusion and $\pi:H\to L$ is determined by

(60) $\displaystyle\pi(g_{i})$ $\displaystyle=1,$ $\displaystyle
i\in\mathbb{I}_{\theta},$ $\displaystyle\pi(x_{j})$ $\displaystyle=0,$
$\displaystyle\pi(y_{j})$ $\displaystyle=E_{j},$ $\displaystyle
j\in\mathbb{I}_{t},$ $\displaystyle\pi(x_{\ell})$ $\displaystyle=v_{\ell},$
$\displaystyle\ell\in\mathbb{I}_{t+1,\theta}$ .

The proof has the same pattern as the proof of Proposition 5.2.

###### Proof.

Evidently $K$ is commutative, $L$ is cocommutative and $\dim
K=p^{t}f^{\theta}$. By inspection, $K$ is normal, i.e., $HK^{+}=K^{+}H$. Thus
we have an abelian exact sequence

$\displaystyle\Bbbk\to K\overset{\iota}{\to}H\to H/HK^{+}\to\Bbbk.$

We have $\dim H/HK^{+}=\dim L$. By the defining relations of
$\mathscr{B}(\mathscr{V}(\mathbf{1},\mathbf{a}))$, see Remark 6.2, the
assignment (60) determines a Hopf algebra map $\pi:H\to L$, which is
surjective and has $HK^{+}\subseteq\ker\pi$. Thus $\pi$ induces an isomorphism
of Hopf algebras $H/HK^{+}\to L$.

Observe that $\pi(\mbox{щ}_{h,\mathbf{n}})=v_{h,\mathbf{n}}$, for all
$h,\mathbf{n}$. The section $\mathpzc{s}:L=\mathfrak{u}(\mathfrak{l})\to H$ is
the unique algebra map such that

$\displaystyle\mathpzc{s}(E_{j})$
$\displaystyle=-\mathcal{S}(y_{j})=-g_{j}^{-1}y_{j},$
$\displaystyle\mathpzc{s}(v_{h})$
$\displaystyle=-\mathcal{S}(x_{h})=-g_{h}^{-1}x_{h},$ $\displaystyle\forall j$
$\displaystyle\in\mathbb{I}_{t},\,h\in\mathbb{I}_{t+1,\theta}.$

We appeal again to Remark 6.2 to see that $\mathpzc{s}$ is indeed well-
defined. As in the proof of Proposition 5.2, we get that $\mathpzc{s}$ is an
$L$-colinear section of $\pi$.

Observe that the generic element of the basis of $H$ arising from (51) can be
expressed as

$\displaystyle u=g_{1}^{b_{1}}\dots g_{\theta}^{b_{\theta}}x_{1}^{m_{1}}\dots
x_{t}^{m_{t}}y_{1}^{n_{1}}\dots
y_{t}^{n_{t}}\prod_{\begin{subarray}{c}h\in\mathbb{I}_{t+1,\theta}\\\
\mathbf{n}\in\mathcal{A}_{h}\end{subarray}}\mbox{щ}_{h,\mathbf{n}}^{n_{h,\mathbf{n}}}.$

Let $\mathpzc{r}:H\to K$ be the linear map defined on $u$ by the formula

$\mathpzc{r}(u)=\begin{cases}g_{1}^{b_{1}}\dots
g_{\theta}^{b_{\theta}}x_{1}^{m_{1}}\dots x_{t}^{m_{t}},\;\textrm{if
}n_{k}=n_{h,\mathbf{n}}=0,\,\forall k,h,\mathbf{n};\\\
0,\quad\textrm{otherwise.}\end{cases}$

Clearly, $\mathpzc{r}$ is a $K$-linear retraction of $\iota$ and
$\mathpzc{r}\mathpzc{s}=\epsilon_{L}1_{K}$. We claim that $\mathpzc{r}$ is a
coalgebra map. To see this, we consider the subspace $I=\ker\mathpzc{r}$ which
is the linear span of the monomials

$\displaystyle g_{1}^{b_{1}}\dots g_{\theta}^{b_{\theta}}x_{1}^{m_{1}}\dots
x_{t}^{m_{t}}y_{1}^{n_{1}}\dots
y_{t}^{n_{t}}\prod_{\begin{subarray}{c}h\in\mathbb{I}_{t+1,\theta}\\\
\mathbf{n}\in\mathcal{A}_{h}\end{subarray}}\mbox{щ}_{h,\mathbf{n}}^{n_{h,\mathbf{n}}}:\qquad
n_{1}+\dots+n_{h}+\sum_{\begin{subarray}{c}h\in\mathbb{I}_{t+1,\theta}\\\
\mathbf{n}\in\mathcal{A}_{h}\end{subarray}}n_{h,\mathbf{n}}>0.$

By the defining relations of $H$, $I$ is a left ideal.

We claim that $\Delta(I)\subseteq I\otimes H+H\otimes I$. First,
$\Delta(y_{j}),\Delta(x_{h})\in I\otimes H+H\otimes I$ for all
$j\in\mathbb{I}_{t}$, $h\in\mathbb{I}_{t+1,\theta}$ by (54). Next, by [3,
Lemmas 7.2.3 (b) and 7.2.4] and the comultiplication formula of the
bosonization, we have

$\displaystyle\Delta(\mbox{щ}_{h,\mathbf{n}})$
$\displaystyle\in\mbox{щ}_{h,\mathbf{n}}\otimes
1+\sum_{0\leq\mathbf{k}\leq\mathbf{n}}H\otimes\mbox{щ}_{h,\mathbf{k}}\subset
I\otimes H+H\otimes I,$ $\displaystyle\forall h$ $\displaystyle,\mathbf{n}.$

Let $u$, $v$ be monomials in $I$ such that $\Delta(u),\Delta(v)\in I\otimes
H+H\otimes I$. Then $\Delta(uv)\in I\otimes H+H\otimes I$. By a recursive
argument, the second claim follows. Since
$\mathpzc{r}|_{K}=\operatorname{id}_{K}$, we get the first claim, i.e., that
$\mathpzc{r}$ is a coalgebra map. Thus $(\mathpzc{s},\mathpzc{r})$ is a
splitting and the proof of the Proposition is complete. ∎

######  Proposition 6.6.

The Hopf algebra
$H=\mathscr{B}(\mathscr{V}(\mathbf{1},\mathbf{a}))\\#\Bbbk\Gamma$ and its
double $D(H)$ have fgc.

###### Proof.

This follows from Proposition 6.5 and Theorem 3.6. ∎

###  6.4. The general case

######  Theorem 6.7.

The Hopf algebra
$\widetilde{H}=\mathscr{B}(\mathscr{V}(\mathfrak{q},\mathbf{a}))\\#\Bbbk\Gamma$
has fgc.

###### Proof.

By Corollary A.8, the bosonization
$\widetilde{H}=\mathscr{B}(\mathscr{V}(\mathfrak{q},\mathbf{a}))\\#\Bbbk\Gamma$
is a cocycle deformation of the bosonization
$H=\mathscr{B}(\mathscr{V}(\mathbf{1},\mathbf{a}))\\#\Bbbk\Gamma$. By
Proposition 6.6, the double $D\left(H\right)$ has fgc. Hence $\widetilde{H}$
also has fgc, as claimed. ∎

Since $H$ is free over $\mathscr{B}(\mathscr{V}(\mathfrak{q},\mathbf{a}))$,
Theorem 6.7 together with [7, Theorem 3.2.1] implies:

######  Corollary 6.8.

The Nichols algebra $\mathscr{B}(\mathscr{V}(\mathfrak{q},\mathbf{a}))$ has
fgc. ∎

##  Appendix A Cocycle-equivalence of Nichols algebras

In this Appendix we describe braided vector spaces over abelian groups and
spell out conditions for their Nichols algebras being twist-equivalent. Except
in Corollary A.8, $\operatorname{char}\Bbbk$ is arbitrary.

###  A.1. Braided vector spaces over abelian groups

######  Definition A.1.

Let $\theta\in\mathbb{N}$. An $\mathfrak{ab}$-triple (of rank $\theta$) is a
collection $\mathscr{T}=(\mathbf{n},\mathfrak{q},\mathbf{t})$ where

  * •

$\mathbf{n}=(n_{j})_{j\in\mathbb{I}_{\theta}}$ is a family of positive
integers, normalized by $n_{1}\geq n_{2}\geq\dots\geq n_{\theta}$;

  * •

$\mathfrak{q}=(q_{ij})_{i,j\in\mathbb{I}_{\theta}}$ is a matrix with
invertible entries and

  * •

$\mathbf{t}=(\mathbf{t}_{ij})_{i,j\in\mathbb{I}_{\theta,}}$ is a family where
$\mathbf{t}_{ij}\in\operatorname{End}\Bbbk^{n_{j}}$ satisfies
$\mathbf{t}_{ij}=0$ when $\dim V_{j}=1$ and
$\mathbf{t}_{ik}\mathbf{t}_{jk}=\mathbf{t}_{jk}\mathbf{t}_{ik}$ for all
$i,j,k$.

An $\mathfrak{ab}$-triple is _nilpotent_ if every $\mathbf{t}_{ij}$ is
nilpotent.

We attach a braided vector space to an $\mathfrak{ab}$-triple
$(\mathbf{n},\mathfrak{q},\mathbf{t})$ by the following recipe. Let
$V=\oplus_{j\in\mathbb{I}_{\theta}}V_{j}$ be a vector space with a
decomposition such that $\dim V_{j}=n_{j}$ for all $j\in\mathbb{I}_{\theta}$;
pick a basis of $V_{j}$, pull back $\mathbf{t}_{ij}$ to
$\mathbf{t}_{ij}\in\operatorname{End}V_{j}$ and define $c\in GL(V\otimes V)$
by

(61) $\displaystyle c(x\otimes y)$
$\displaystyle=q_{ij}\left(y+\mathbf{t}_{ij}(y)\right)\otimes x,$
$\displaystyle x\in V_{i},\quad y\in V_{j}.$

The proof of the following result is left to the reader.

######  Lemma A.2.

The pair $(V,c)$ is a braided vector space that can be realized over
${}^{\Bbbk\Lambda}_{\Bbbk\Lambda}\mathcal{YD}$, where
$\Lambda\simeq\mathbb{Z}^{\theta}$ with canonical basis
$\alpha_{1},\dots,\alpha_{\theta}$, by

$\displaystyle V_{\alpha_{i}}$ $\displaystyle=V_{i},$
$\displaystyle\alpha_{i}\rightharpoonup x$
$\displaystyle=q_{ij}\left(x+\mathbf{t}_{ij}(x)\right),$ $\displaystyle x\in
V_{j},\ i,j\in\mathbb{I}_{\theta}.\qed$

###  A.2. Cocycle equivalence

We start by discussing the relationship between cocycle deformation and
bosonization, as it appears in [30]. Let $H$ be a Hopf algebra; let
$\sigma:H\otimes H\to\Bbbk$ be an invertible 2-cocycle; let $H_{\sigma}$ be
the Hopf algebra which is $H$ as coalgebra and has multiplication

(62) $\displaystyle x\cdot_{\sigma}y$
$\displaystyle=\sigma(x_{\left(1\right)},y_{\left(1\right)})x_{\left(2\right)}y_{\left(2\right)}\sigma^{-1}(x_{\left(3\right)},y_{\left(3\right)}),$
$\displaystyle x,y$ $\displaystyle\in H,$

Let now $R$ be a Hopf algebra in ${}^{H}_{H}\mathcal{YD}$ and $A\coloneqq
R\\#H$ the bosonization with canonical projection and injection $\pi:A\to H$
and $\iota:H\to A$. Let $\sigma^{\pi}:A\otimes A\to\Bbbk$ be given by
$\sigma^{\pi}\coloneqq\sigma(\pi\otimes\pi)$; this is an invertible 2-cocycle
on $A$. Then $\pi:A_{\sigma^{\pi}}\to H_{\sigma}$ and $\iota:H_{\sigma}\to
A_{\sigma^{\pi}}$ are still Hopf algebra maps. Hence $A_{\sigma^{\pi}}\simeq
R_{\sigma}\\#H_{\sigma}$ where $R_{\sigma}$ is a Hopf algebra in
${}^{H_{\sigma}}_{H_{\sigma}}\mathcal{YD}$ that coincides with $R$ as vector
subspace of $A$, with multiplication

(63) $\displaystyle x\cdot_{\sigma}y$
$\displaystyle=\sigma(x_{\left(0\right)},y_{\left(0\right)})x_{\left(1\right)}y_{\left(1\right)},$
$\displaystyle x,y$ $\displaystyle\in R_{\sigma}.$

######  Lemma A.3.

[9, 2.13] If $R=\oplus_{n\in\mathbb{N}_{0}}R(n)$ is a graded Hopf algebra in
${}^{H}_{H}\mathcal{YD}$, then $R_{\sigma}$ is a graded Hopf algebra in
${}^{H_{\sigma}}_{H_{\sigma}}\mathcal{YD}$ with $R(n)=R_{\sigma}(n)$ as vector
spaces for all $n\geq 0$. Also, $R$ is a Nichols algebra if and only if
$R_{\sigma}$ is. ∎

###  A.3. Cocycles over an abelian group

We fix an abelian group $\Gamma$. Let
$\sigma:\Gamma\times\Gamma\to\Bbbk^{\times}$ be a group 2-cocycle:
$\sigma(gh,k)\sigma(g,h)=\sigma(g,hk)\sigma(h,k)$, for all $g,h,k\in\Gamma$.
Then the map $\vartheta:\Gamma\times\Gamma\to\Bbbk^{\times}$ given by

$\displaystyle\vartheta(g,h)$ $\displaystyle=\sigma(g,h)\sigma^{-1}(h,g),$
$\displaystyle g,h$ $\displaystyle\in\Gamma,$

is bilinear and antisymmetric. Let
${\mathcal{F}}_{\sigma}:{}^{\Bbbk\Gamma}_{\Bbbk\Gamma}\mathcal{YD}\to{}^{\Bbbk\Gamma}_{\Bbbk\Gamma}\mathcal{YD}$
be the monoidal functor that assigns $V\mapsto{\mathcal{F}}_{\sigma}(V)=V$
with the same grading and action

(64) $\displaystyle g\rightharpoonup_{\sigma}v$
$\displaystyle=\vartheta(g,h)g\rightharpoonup v,$ $\displaystyle
g,h\in\Gamma,v\in V_{h},$

where $\rightharpoonup$ is the action on $V$. The braiding $c_{\sigma}$ in
${\mathcal{F}}_{\sigma}(V)\otimes{\mathcal{F}}_{\sigma}(W)$ is given by

(65) $\displaystyle c_{\sigma}(v\otimes w)$
$\displaystyle=\vartheta(g,h)c(v\otimes w),$ $\displaystyle v\in V_{g},w$
$\displaystyle\in W_{h},\ g,h\in\Gamma.$

Fix $V\in{}^{\Gamma}_{\Gamma}\mathcal{YD}$ and let
$A=\mathscr{B}(V)\\#\Bbbk\Gamma$; as before $\pi:A\to\Bbbk\Gamma$ and
$\iota:\Bbbk\Gamma\to A$ are the canonical projection and the inclusion.
Clearly, the linear extension $\sigma:\Bbbk\Gamma\otimes\Bbbk\Gamma\to\Bbbk$
of $\sigma$ is a 2-cocycle as in the previous subsection. Let
$\sigma^{\pi}=\sigma(\pi\otimes\pi)$ be as above.

######  Lemma A.4.

The Hopf algebra $A_{\sigma^{\pi}}$ is isomorphic to
$\mathscr{B}(V)_{\sigma}\\#\Bbbk\Gamma$ with the same comultiplication of
$\mathscr{B}(V)$ and the multiplication given by

(66) $\displaystyle x._{\sigma^{\pi}}y$ $\displaystyle=\sigma(g,h)xy,$
$\displaystyle x$ $\displaystyle\in\mathscr{B}(V)_{g},\
y\in\mathscr{B}(V)_{h},$ $\displaystyle g,h$ $\displaystyle\in\Gamma.$

Furthermore
$\mathscr{B}(V)_{\sigma}\simeq\mathscr{B}({\mathcal{F}}_{\sigma}(V))$.

_Proof._ The first part follows from the discussion above; (66) is particular
instance of (63). For the second part we show that the degree one homogenous
component of $\mathscr{B}(V)_{\sigma}$ is ${\mathcal{F}}_{\sigma}(V)$ and
apply Lemma A.3. If $y\in V_{h}$, then

$\displaystyle g._{\sigma^{\pi}}y$
$\displaystyle=\sigma(g,\pi(y))g\sigma^{-1}(g,1)+\sigma(g,h)\,gy\sigma^{-1}(g,1)+\sigma(g,h)\,gh\sigma^{-1}(g,\pi(y))$
$\displaystyle=\sigma(g,h)gy;$

Similarly,
$(gy)._{\sigma^{\pi}}g^{-1}=\sigma(gh,g^{-1})gyg^{-1}\sigma^{-1}(g,g^{-1})$ so
the action is given by

$\displaystyle(g._{\sigma^{\pi}}y)._{\sigma^{\pi}}g^{-1}$
$\displaystyle=\sigma(g,h)(gy._{\sigma^{\pi}}g^{-1})=\sigma(g,h)\sigma(gh,g^{-1})gyg^{-1}\sigma^{-1}(g,g^{-1})$
$\displaystyle=\sigma(g,h)\sigma(h,g)^{-1}g\rightharpoonup
y=g\rightharpoonup_{\sigma^{\pi}}y.\hskip 90.0pt\qed$

######  Definition A.5.

[7, 9, §2.4]. Two braided Hopf algebras $R$ and $S$ are _cocycle-equivalent_
if there exist a Hopf algebra $H$ and an invertible 2-cocycle $\sigma:H\otimes
H\to\Bbbk$ such that

  * •

$R$ is realizable in ${}^{H}_{H}\mathcal{YD}$;

  * •

$S$ is isomorphic to $R_{\sigma}$ as a braided Hopf algebra.

The following definition extends [7, Lemma 4.3] and [9, §2.4].

######  Definition A.6.

Two braided vector spaces $(V,c)$ and $(V^{\prime},c^{\prime})$ arising from
$\mathfrak{ab}$-triples $(\mathbf{n},\mathfrak{q},\mathbf{t})$ and
$(\mathbf{n}^{\prime},\mathfrak{q}^{\prime},\mathbf{t}^{\prime})$ are twist-
equivalent if

(67) $\displaystyle\begin{aligned}
\mathbf{n}&=\mathbf{n}^{\prime},&\mathbf{t}&=\mathbf{t}^{\prime},&q_{ii}&=q^{\prime}_{ii},&q_{ij}q_{ji}&=q^{\prime}_{ij}q^{\prime}_{ji},&i,j&\in\mathbb{I}_{\theta}.\end{aligned}$

######  Lemma A.7.

If the braided vector spaces $(V,c)$ and $(V^{\prime},c^{\prime})$ arising
from the $\mathfrak{ab}$-triples $(\mathbf{n},\mathfrak{q},\mathbf{t})$ and
$(\mathbf{n}^{\prime},\mathfrak{q}^{\prime},\mathbf{t}^{\prime})$ are twist-
equivalent, then the Nichols algebras $\mathscr{B}(V)$ and
$\mathscr{B}(V^{\prime})$ are cocycle-equivalent.

###### Proof.

By (67) there exists a linear isomorphism $\psi:V\to V^{\prime}$ preserving
the decompositions $V=\oplus_{i\in\mathbb{I}_{\theta}}V_{i}$ and
$V^{\prime}=\oplus_{i\in\mathbb{I}_{\theta}}V^{\prime}_{i}$ and intertwining
the endomorphisms $\mathbf{t}_{ij}$ and $\mathbf{t}^{\prime}_{ij}$. We realize
$(V,c)$ as in Lemma A.2. We consider the unique bilinear form
$\sigma:\Lambda\times\Lambda\to\Bbbk^{\times}$, hence a group 2-cocycle, given
by

(68)
$\sigma(\alpha_{i},\alpha_{j})=\begin{cases}&q^{\prime}_{ij}q_{ij}^{-1},\qquad
i\leq j,\\\ &1,\qquad\qquad i>j.\end{cases}$

We claim that $\psi:V_{\sigma}\to V^{\prime}$ is an isomorphism in
${}^{\Bbbk\Lambda}_{\Bbbk\Lambda}\mathcal{YD}$. Clearly $\psi$ preserves the
grading. Assume $i\leq j$ and let $x\in V_{i}$, $y\in V_{j}$. Then

$\displaystyle\alpha_{j}\rightharpoonup_{\sigma}x$
$\displaystyle\overset{\eqref{eq:twistingaction}}{=}\sigma(\alpha_{j},\alpha_{i})\sigma^{-1}(\alpha_{i},\alpha_{j})q_{ji}\left(x+\mathbf{t}_{ji}(x)\right)$
$\displaystyle=(q^{\prime}_{ij})^{-1}q_{ij}q_{ji}\left(x+\mathbf{t}_{ji}(x)\right)\overset{\eqref{eq:twistequiv}}{=}q^{\prime}_{ji}\left(x+\mathbf{t}_{ji}(x)\right);$
$\displaystyle\alpha_{i}\rightharpoonup_{\sigma}y$
$\displaystyle\overset{\eqref{eq:twistingaction}}{=}\sigma(\alpha_{i},\alpha_{j})\sigma^{-1}(\alpha_{j},\alpha_{i})q_{ij}\left(y+\mathbf{t}_{ij}(i)\right)=q^{\prime}_{ij}\left(y+\mathbf{t}_{ij}(i)\right).$

Thus $\psi$ preserves the action of $\Lambda$, hence it extends to an
isomorphism $\Psi:\mathscr{B}(V_{\sigma})\to\mathscr{B}(V^{\prime})$ of Hopf
algebras in ${}^{\Bbbk\Lambda}_{\Bbbk\Lambda}\mathcal{YD}$. Now
$\mathscr{B}({\mathcal{F}}_{\sigma}(V))\simeq\mathscr{B}(V)_{\sigma}$ by Lemma
A.4. ∎

The following statement is needed in the paper. Assume that
$\operatorname{char}\Bbbk=p$ is odd. Let
$\mathscr{V}(\mathfrak{q},\mathbf{a})$ and
$\mathscr{V}(\mathfrak{q}^{\prime},\mathbf{a})$ be two braided vector spaces
as in Section 6 with the same $\theta$. By hypothesis,
$q_{ii}=q^{\prime}_{ii}=1$, and
$q_{ij}q_{ji}=q^{\prime}_{ij}q^{\prime}_{ji}=1$, for all
$i,j\in\mathbb{I}_{\theta}$.

######  Corollary A.8.

Assume that there exists a positive integer $f$ such that

(69) $\displaystyle p$ $\displaystyle\text{ divides }f,$ (70)
$\displaystyle\operatorname{ord}q_{ij}$ $\displaystyle\text{ divides }f,$
$\displaystyle\operatorname{ord}q^{\prime}_{ij}$ $\displaystyle\text{ divides
}f,$ $\displaystyle\forall i,j$ $\displaystyle\in\mathbb{I}_{\theta}.$

Let $\Gamma=(\mathbb{Z}/f)^{\theta}$. Then
$\mathscr{V}(\mathfrak{q},\mathbf{a})$ and
$\mathscr{V}(\mathfrak{q}^{\prime},\mathbf{a})$ are realizable in
${}^{\Bbbk\Gamma}_{\Bbbk\Gamma}\mathcal{YD}$ and there exists an invertible
2-cocycle $\sigma:\Bbbk\Gamma\otimes\Bbbk\Gamma\to\Bbbk$ such that
$\mathscr{B}(\mathscr{V}(\mathfrak{q}^{\prime},\mathbf{a}))$ is isomorphic to
$\mathscr{B}(\mathscr{V}(\mathfrak{q},\mathbf{a}))_{\sigma}$ as Hopf algebras
in ${}^{\Bbbk\Gamma}_{\Bbbk\Gamma}\mathcal{YD}$.

###### Proof.

The proof of the first claim on the realizations is straightforward using the
hypotheses (69) and (70). By (70), there is a unique bilinear form
$\sigma:\Gamma\times\Gamma\to\Bbbk^{\times}$ given by (68). Then the second
claim follows as in the proof of Lemma A.7. ∎

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