Pointed Hopf algebras over non abelian groups with decomposable braidings, I

TL;DR

This paper classifies certain finite-dimensional pointed Hopf algebras with specific decomposable braidings over non-abelian groups, providing presentations and confirming a key conjecture in the field.

Contribution

It offers a complete description and presentation of these Hopf algebras and verifies the Andruskiewitsch-Schneider Conjecture for this class.

Findings

Classification of pointed Hopf algebras with decomposable braidings

Explicit generators and relations for the Nichols algebra

Verification of the Andruskiewitsch-Schneider Conjecture

Abstract

We describe all finite-dimensional pointed Hopf algebras whose infinitesimal braiding is a fixed Yetter-Drinfeld module decomposed as the sum of two simple objects: a point and the one of transpositions of the symmetric group in three letters. We give a presentation by generators and relations of the corresponding Nichols algebra and show that Andruskiewitsch-Schneider Conjecture holds for this kind of pointed Hopf algebras.

Equations

$$(c\otimes \mathrm{id})(\mathrm{id}\otimes c)(c\otimes \mathrm{id})=(\mathrm{id}\otimes c)(c\otimes \mathrm{id})(\mathrm{id}\otimes c).$$

$$\delta (h\cdot v)=h_{(1)}{v}_{(-1)}\mathcal{S}(h_{(3)})\otimes h_{(2)}\cdot v_{(0)}$$

$$f_{(-1)}{f}_{(0)}(v)$$

$$c_{V,U}(v\otimes u)$$

$${\mu }_{A\otimes B}=({\mu }_A\otimes {\mu }_B)({\mathrm{id}}_A\otimes c_{A,B}\otimes {\mathrm{id}}_B).$$

$$\delta (g\cdot v)=gh{g}^{-1}\otimes g\cdot v$$

$$\mathrm{supp}V:=\{g\in \Gamma :V_g\ne 0\}\subset \Gamma .$$

$$V^{\chi }:=\{v\in V:g\cdot v=\chi (g)v\text{ for all }g\in \Gamma \}.$$

$$({\mathrm{ad}}_{c}x)y$$

$$x⊳(y⊳z)$$

$$q_{x,y⊳z}{q}_{y,z}$$

$$c(v_x\otimes v_y)$$

$$\delta (v_i)$$

$$x_{i_1\dots i_n}$$

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

$$\mathcal{B}$$

$$\begin{array}{rl}\sigma (x,1)=\sigma (1,x)& =\epsilon (x),\\ \sigma (x_{(1)},y_{(1)})\sigma (x_{(2)}{y}_{(2)},z)& =\sigma (y_{(1)},z_{(1)})\sigma (x,y_{(2)}{z}_{(2)}),\end{array}$$

$$x{\cdot }_{\sigma }y$$

$$c(x_i\otimes x_j)$$

$$\begin{array}{rlrl}{x}_i^2& =0,& & i\in {\mathbb{I}}_3,\\ x_1{x}_2+x_3{x}_1+x_2{x}_3& =0,& & x_2{x}_1+x_1{x}_3+x_3{x}_2=0.\end{array}$$

$$\{x_1^{n_1}{x}_2^{n_2}{x}_3^{n_3}:0\le n_1,n_2,n_3\le 1\}\cup \{x_1^{n_1}{x}_2{x}_1{x}_3^{n_3}:0\le n_1,n_3\le 1\}.$$

$$x_1^2$$

$$\begin{array}{rl}{\lambda }_1& =0\text{ if either }{\chi }_i^2\ne \epsilon \text{ or }{g}_i^2=1\text{ for some }i\in {\mathbb{I}}_3,\\ {\lambda }_2& =0\text{ if either }{\chi }_i{\chi }_j\ne \epsilon \text{ or }{g}_i{g}_j=1\text{ for some }i\ne j\in {\mathbb{I}}_3.\end{array}$$

$$x_i^2$$

$$\omega :=-q_1{q}_2\in {\mathbb{G}}_3^{\prime }.$$

$$c(x_i\otimes x_4)$$

$$\begin{array}{rlrlrlrlrl}{g}_i^{\Gamma }& =\{g_1,g_2,g_3\},& g_4^{\Gamma }& =\{g_4\}& g_i^2& =g_j^2,& g_i{g}_j& =g_{2i-j}{g}_i,& i,& j\in {\mathbb{I}}_3;\end{array}$$

$$\begin{array}{rlrlrl}{\chi }_i(g_j)& =\{\begin{array}{ll}-1,& j\in {\mathbb{I}}_3,\\ q_2,& j=4\end{array},& i& \in {\mathbb{I}}_3,& {\chi }_4(g_j)& =\{\begin{array}{ll}{q}_1,& j\in {\mathbb{I}}_3,\\ -{\omega }^2,& j=4\end{array}.\end{array}$$

$$\gamma \cdot x_i$$

$$\gamma \nu$$

Full text

2010 Mathematics Subject Classification. 16T20, 17B37. The work of I. A. and G. S. was partially supported by CONICET, Secyt (UNC), the MathAmSud project GR2HOPF

1. Introduction

The theory of Hopf algebras has grown recently, focused especially in the finite dimensional case. The idea of classifying these algebras leads to separate into families that share certain invariants, mainly associated with their coalgebra structure. The first invariant to take in care is the coradical (the sum of all simple subcoalgebras). The first distinguished family is that of cosemisimple Hopf algebras, those which coincide with the coradical. A second important family is constituted by pointed Hopf algebras (those whose coradical coincides with the subalgebra generated by group-like elements) and was intensively studied after the introduction of quantized enveloping algebras by Drinfeld and Jimbo.

A turning point in the classification of pointed Hopf algebras was given by the introduction of the Lifting Method by Andruskiewitsch and Schneider [AS1]: a sequence of steps proposed to systematically describe all the deformations of a graded Hopf algebra attached to a fixed finite group $\Gamma$ (giving the coradical) and a Yetter-Drinfeld module $V$ over $\Gamma$ (describing the infinitesimal braiding). At this point an algebra $\mathcal{B}(V)$ universally contrusted from $V$ appears: the so-called Nichols algebra of $V$, cf. §2.4. The steps of this method are the following:

1. (a)

Classify all Yetter-Drinfeld modules $V$ over $\Gamma$ such that $\dim\mathcal{B}(V)<\infty$. 2. (b)

Give a presentation by generators and relations of $\mathcal{B}(V)$. 3. (c)

Check if, given $H$ with $H_{0}\simeq\Bbbk\Gamma$, the associated graded Hopf algebra satisfies $\operatorname{gr}H\simeq\mathcal{B}(V)\#\Bbbk\Gamma$ (generation in degree one problem). 4. (d)

Compute all Hopf algebras obtained as deformations of $\mathcal{B}(V)\#\Bbbk\Gamma$.

The Lifting Method was succefully applied to classify finite-dimensional pointed Hopf algebras over abelian groups $\Gamma$ of order not divisible by small primes [AS2]. After the appeareance of this result, Masouka [Ma] was able to prove that all these pointed Hopf algebras are cocycle deformations of their associated graded Hopf algebras.

For general finite abelian groups $\Gamma$, Heckenberger [H] gave a complete answer of (a) and Angiono [An1] finished (b) and (c). The final step was recently completed in [AG], based on a strategy to construct Hopf algebras using cocycle deformations developed in [A+, AAG].

If we turn towards non-abelian groups, the first step (a) was almost completed [HV2, HV3]. The authors gave all non-simple Yetter-Drinfeld modules over non-abelian groups such that the associated Nichols algebras are finite-dimensional. They describe their root systems and compute the dimensions. At the moment there is no general answers for the remaining steps of the Lifting Method. Anyway the method was applied successfully on certain pairs of non-abelian groups and infinitesimal braidings whose Nichols algebras are finite dimensional, see e.g. [GV, GV2]

The present work starts the computation of all liftings of Nichols algebras of Yetter-Drinfeld modules classified in [HV2, HV3]. We fix a brainding (that we call ${\mathtt{HV}_{1}}$) decomposable as the sum of two simple Yetter-Drinfeld modules, one related with the so-called Fomin-Kirillov algebra [MS, AG] over the symmetric group in three letters and another of dimension one. We complete the answer for steps (b), (c) and (d). Indeed, we give a presentation by generators and relations of $\mathcal{B}({\mathtt{HV}_{1}})$ in §3. Next we give a positive answer to the generation-in-degree-one problem in §4: the unique finite-dimensional post-Nichols algebra of ${\mathtt{HV}_{1}}$ is the Nichols algebra $\mathcal{B}({\mathtt{HV}_{1}})$ itself. We also introduce a distinguished pre-Nichols algebra of ${\mathtt{HV}_{1}}$, whose behavior is analogous to those pre-Nichols algebras in [An2] for braidings of diagonal type. Finally, the main result of this paper is contained in §5 and gives a complete list of all finite-dimensional pointed Hopf algebras whose group of group-like elements is a group $\Gamma$ such that ${\mathtt{HV}_{1}}$ admits a principal realization over $\Gamma$. We follow the strategy of [A+, AAG] and prove at the same time that all of them are cocycle deformations of $\mathcal{B}({\mathtt{HV}_{1}})\#\Bbbk\Gamma$.

Acknowledgements

We thank Nicolás Andruskiewitsch for posing us the problem and contributing with ideas for its solution. Many interesting discussions with him helped us to improve the paper.

2. Preliminaries

2.1. Notation

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

Throughout this work $\Bbbk$ denotes an algebraically closed field of characteristic zero. All vector spaces, algebras and tensor products are over $\Bbbk$. 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 $\mathbb{G}^{\prime}_{\infty}=\mathbb{G}_{\infty}-\{1\}$.

Let $A$ be an algebra. The ideal generated by a subset $(a_{i})_{i\in I}$ is denoted by $\langle a_{i}:i\in I\rangle$ while the subalgebra generated by this set is denoted by $\Bbbk\langle a_{i}:i\in I\rangle$.

Let $H$ be a Hopf algebra (or a Hopf algebra in a braided tensor category). We will use the Sweedler notation for the coalgebra structure and for comodules; e.g. the coproduct of $h\in H$ is denoted $\Delta(h)=h_{(1)}\otimes h_{(2)}$. Let $G(H)=\{x\in H^{\times}:\Delta(x)=x\otimes x\}$ be the group of group-like elements. For each $g,h\in G(H)$, $\mathcal{P}_{g,h}(H)$ is the subspace of $(g,h)$-primitive elements; in particular $\mathcal{P}(H):=\mathcal{P}_{1,1}(H)$ is the set of primitive elements.

2.2. Braided vector spaces, Yetter-Drinfeld modules and braided Hopf algebras

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

[TABLE]

Let $H$ be a Hopf algebra with bijective antipode. Attached to $H$ there is a braided tensor category ${}^{H}_{H}\mathcal{YD}$. An object of ${}^{H}_{H}\mathcal{YD}$, called a Yetter-Drinfeld module over $H$, is a left $H$-module and left $H$-comodule $V$ such that

[TABLE]

The monoidal structure on ${}^{H}_{H}\mathcal{YD}$ comes from the usual tensor product of modules and comodules over a Hopf algebra. The left dual of a finite dimensional $V\in{}^{H}_{H}\mathcal{YD}$ is the vector space $V^{*}$, where $H$ acts via the antipode $(h\cdot f)(v)=f(\mathcal{S}(h)\cdot v)$ and the coaction $\delta(f)=f_{(-1)}\otimes f_{(0)}$ is determined by

[TABLE]

For $V,U\in{}^{H}_{H}\mathcal{YD}$, the braiding $c_{V,U}\colon V\otimes U\to U\otimes V$ is given by

[TABLE]

The pair $(V,c_{V,V})$ is a braided vector space.

Reciprocally, let $(V,c)$ be a braided vector space. A realization of $V$ over $H$ is an structure on $V$ of Yetter-Drinfeld module over $H$ such that the braiding $c$ is the categorical braiding $c_{V,V}$. The realization is principal if there exists a basis $\{v_{i}\}_{i\in I}$ of $V$ and $\{g_{i}\}_{i\in I}\in G(H)$ such that $\delta(v_{i})=g_{i}\otimes v_{i}$, $i\in I$.

There is a monoidal structure in the category of algebras in ${}^{H}_{H}\mathcal{YD}$. Namely, if $(A,\mu_{A})$ and $(B,\mu_{B})$ are algebras in ${}^{H}_{H}\mathcal{YD}$, then $A\underline{\otimes}B=(A\otimes B,\mu_{A\otimes B})$ also is, where the multiplication is given by

[TABLE]

We are mainly interested in Yetter-Drinfeld modules over a group algebra $\Bbbk\Gamma$. Let $V$ be simultaneously a left $\Bbbk\Gamma$-comodule and a left $\Bbbk\Gamma$-module. Then $V=\oplus_{g\in\Gamma}V_{g}$, where $V_{g}=\{v\in V\colon\delta(v)=v\otimes g\}$. In this setting the Yetter-Drinfeld compatibility (2.1) is read as

[TABLE]

Thus $V\in{}^{\Bbbk\Gamma}_{\Bbbk\Gamma}\mathcal{YD}$ means that $g\cdot V_{h}\subset V_{ghg^{-1}}$ for all $g,h\in\Gamma$.

We recall that the support of $V$ is the subset

[TABLE]

We denote by $\widehat{\Gamma}$ the group of characters of $\Gamma$. For each $\chi\in\widehat{\Gamma}$,

[TABLE]

We also use the following notation: $V_{g}^{\chi}=V_{g}\cap V^{\chi}$.

We recall that a Hopf algebra in ${}^{H}_{H}\mathcal{YD}$ is a collection $(R,\mu,\Delta,\mathcal{S})$, where $R$ is an object ${}^{H}_{H}\mathcal{YD}$ with structures $(R,\mu)$ of algebra in ${}^{H}_{H}\mathcal{YD}$ and $(R,\Delta)$ of coalgebra in ${}^{H}_{H}\mathcal{YD}$ that are compatible in the sense that $\Delta\colon R\to R\underline{\otimes}R$ and $\varepsilon\colon R\to\Bbbk$ are algebra maps, and such that $\mathcal{S}$ is a convolution inverse of the identity of $R$. In this case, the left adjoint action of $R$ on itself is the linear map $\operatorname{ad}_{c}\colon R\to\operatorname{End}(R)$,

[TABLE]

so the action of a primitive element $x\in R$ is

[TABLE]

2.3. Racks

We recall now a prototypical example of Yetter-Drinfeld modules over groups. A rack $X=(X,\rhd)$ is a pair where $X$ is a non-empty set and $\rhd:X\times X\rightarrow X$ is an operation such that

[TABLE]

and the maps $\phi_{x}\colon X\longmapsto X$, $\phi_{x}(y)=x\rhd y$, $y\in X$, are bijective for all $x\in X$. A quandle is a rack such that $x\rhd x=x$ for all $x\in X$.

Let $\Gamma$ be a group. An example of a rack (which is moreover a quandle) is given by $X$ a union of conjugacy classes inside $\Gamma$ and the operation given by $g\rhd h=ghg^{-1}$, $g,h\in\mathcal{O}$.

A 2-cocycle on $X$ is a function $\mathfrak{q}:X\times X\rightarrow\Bbbk^{\times}$, $(x,y)\mapsto q_{x,y}$, such that

[TABLE]

Let $(X,\rhd)$ be a rack and $\mathfrak{q}$ a 2-cocycle on $X$. Let $V$ be the vector space with basis $\{v_{x}\colon x\in X\}$. Let $c:V\otimes V\to V\otimes V$ be the linear map

[TABLE]

The pair $(V,c)$ is a braided vector space denoted by $V(X,\mathfrak{q})$.

Next we recall the definition of a principal realization of a braided vector space $V(X,\mathfrak{q})$ (here we restrict to the case of group algebras as in [AG, Definition 3.2], see [GV2, §4] for the case of an arbitrary Hopf algebra). It is given by the data $(\cdot,(g_{i})_{i\in X},(\chi_{i})_{i\in X})$ consisting of:

• •

an action $\cdot:G\times X\to X$ of $G$ on $X$;

• •

a family of elements $g_{i}\in G$, $i\in X$, such that $g_{h\cdot i}=hg_{i}h^{-1}$ and $g_{i}\cdot j=i\rhd j$, for all $i,j\in X$, $h\in G$;

• •

a family of functions $\chi_{i}:G\to\Bbbk^{\times}$, $i\in X$, called $1$-cocycles, such that $\chi_{i}(g(j))=q_{ji}$, $i,j\in X$, $\chi_{i}(ht)=\chi_{i}(t)\chi_{t\cdot i}(h)$, for all $i\in X$, $h,t\in G$.

This data defines a Yetter-Drinfeld structure on $V(X,q)$ by

[TABLE]

2.4. Nichols algebras

Let $V\in{}^{H}_{H}\mathcal{YD}$. The Nichols algebra of $V$ is the unique quotient of the tensor algebra $\mathcal{B}(V)=T(V)/{\mathcal{J}}(V)$ in ${}^{H}_{H}\mathcal{YD}$ that is a graded braided Hopf algebra and such that $V$ is the space of primitive elements. Among the several descriptions of the universally defined ideal ${\mathcal{J}}(V)=\oplus_{n\geq 2}{\mathcal{J}}_{n}(V)$, we recall the most useful for our purposes: ${\mathcal{J}}(V)$ is the unique maximal element in the class of graded Hopf ideals and Yetter-Drinfeld submodules of $T(V)$ with zero components on degree [math] and $1$.

We recall that a pre-Nichols algebra $\widetilde{\mathcal{B}}$ of $V$ is a graded braided Hopf algebra $\widetilde{\mathcal{B}}=\oplus_{n\geq 0}\widetilde{\mathcal{B}}^{n}$ such that $\widetilde{\mathcal{B}}^{0}=\Bbbk 1$, $\widetilde{\mathcal{B}}^{1}=V$ and $\widetilde{\mathcal{B}}$ is generated as an algebra by $V$; that is, a graded braided Hopf algebra, intermediate between $T(V)$ and $\mathcal{B}(V)$. By definition, the inclusion $V\hookrightarrow\widetilde{\mathcal{B}}$ induces graded Hopf algebra epimorphisms $T(V)\twoheadrightarrow\widetilde{\mathcal{B}}$ and $\widetilde{\mathcal{B}}\twoheadrightarrow\mathcal{B}(V)$.

Fix $(x_{i})_{i\in I}$ a basis of $V$, then we use the following notation:

[TABLE]

Dually, a post-Nichols algebra $\mathcal{L}$ of $V$ is a graded braided Hopf algebra, intermediate between $T^{c}(V)$, the free associative coalgebra of $V$, and $\mathcal{B}(V)$; that is, a graded braided Hopf algebra $\mathcal{L}=\oplus_{n\geq 0}\mathcal{L}^{n}$ such that $\mathcal{L}^{0}=\Bbbk 1$, $\mathcal{L}^{1}=V$ and $\mathcal{P}(\mathcal{L})=\mathcal{L}^{1}$.

Suppose $V$ has a basis $(x_{i})_{i\in\mathbb{I}}$ where the coaction is given by $\delta(x_{i})=g_{i}\otimes x_{i}$ for some $(g_{i})_{i\in\mathbb{I}}\subset G(H)$. Let $(f_{i})_{i\in\mathbb{I}}$ be the dual basis of $(x_{i})_{i\in\mathbb{I}}$. Define a skew derivation $\partial_{i}\colon T(V)\to T(V)$ by $\partial_{i}(1)=0$, $\partial_{i}|_{V}=f_{i}$ and recursively

[TABLE]

It descends to every pre-Nichols algebra $\mathcal{B}$ and gives a criterion to characterize Nichols algebras:

[TABLE]

In particular every $x\in\mathcal{B}$ such that $\partial_{i}(x)=0$ for all $i\in\mathbb{I}$ projects to 0 onto the Nichols algebra $\mathcal{B}(V)$.

2.5. Cocycle deformations, cleft objects

A normalized $2$-cocycle on a Hopf algebra $H$ is a convolution invertible map $\sigma\colon H\otimes H\to\Bbbk$ such that

[TABLE]

This allows to modify the multiplication of $H$ to a new associative multiplication $\cdot_{\sigma}\colon H\otimes H\to\Bbbk$ given by

[TABLE]

The antipode $\mathcal{S}$ of $H$ can also be modified via $\sigma$ to produce certain $\mathcal{S}_{\sigma}$ in such a way that $H_{\sigma}:=(H,\cdot_{\sigma},1,\Delta,\varepsilon,\mathcal{S}_{\sigma})$ is a Hopf algebra. We denote the set of $2-$cocycles on $H$ by $Z^{2}(H,\Bbbk)$.

A cocycle deformation of $H$ is a Hopf algebra isomorphic to some $H_{\sigma}$.

A right cleft object for $H$ is a right $H$-comodule algebra $C$ that satisfies $C^{\operatorname{co}H}=\Bbbk$ and admits a convolution invertible comodule map $\gamma\colon H\to C$. In this case, $\gamma$ can be normalized to $\gamma(1)=1$ and is called a section. We denote by $\operatorname{Cleft}(H)$ the set of iso-classes of cleft objects.

There exists an analogous notion of left cleft object. Also, given $K,H$ two Hopf algebras, $A$ is a $(K,H)$-bicleft object if $A$ is simultaneously a right $H$-cleft object and a left $K$-object, and the two coactions commute.

These two notions are equivalent. Indeed each cocycle $\sigma$ gives raise to a right cleft object $A$ such that $A$ is a $(H_{\sigma},H)$bicleft object. Reciprocally for each right $H$-cleft object $A$ there exists a Hopf algebra $L:=L(A,H)$, called the Schauenburg algebra, such that $A$ is $(L,H)$-bicleft and $L(A,H)\simeq H_{\sigma}$ for some $\sigma$, see [S] for details; $L$ is unique up to isomorphism.

2.6. Liftings of the Fomin-Kirillov algebra $\mathtt{FK}_{3}$

Let $\mathcal{O}_{2}^{3}$ be the rack of transpositions in $\mathbb{S}_{3}$. Let $(V,c)$ be the braided vector space determined by $\mathcal{O}_{2}^{3}$ with cocycle $\mathfrak{q}\equiv-1$. Thus $V$ has a basis $(x_{i})_{i\in\mathbb{I}_{3}}$ such that

[TABLE]

where $2i-j$ means the class modulo 3.

By [FK, MS], the Nichols algebra ${\mathtt{FK}}_{3}:=\mathcal{B}(V)$ is presented by generators $x_{i}$, $i\in\mathbb{I}_{3}$, with relations

[TABLE]

Also, $\dim\mathtt{FK}_{3}=12$. Indeed, the following set is a basis of $\mathtt{FK}_{3}$:

[TABLE]

Let $\Gamma$ be a group such that $V$ admits a principal realization $(g_{i},\chi_{i})_{i\in\mathbb{I}_{3}}$ over $H=\Bbbk\Gamma$, see §2.3. For example we may take $\Gamma=\mathbb{S}_{3}$. By [GV2, Theorem 1.2] $L$ is a lifting of $\mathtt{FK}_{3}$ over $H$ if and only if $L$ is (isomorphic to) the quotient of $T(V)\#H$ by the relations

[TABLE]

where the deformation parameters $\lambda_{1},\lambda_{2}\in\Bbbk$ satisfy the constraints

[TABLE]

Notice that (2.12) implies that the following relations also hold:

[TABLE]

3. The Nichols algebra $\mathcal{B}({\mathtt{HV}_{1}})$

In this section we introduce the Nichols algebra $\mathcal{B}({\mathtt{HV}_{1}})$. We recall the definition of principal realization for the associated braided vector space, describe the root system of ${\mathtt{HV}_{1}}$ and give a (minimal) presentation by generators and relations of this Nichols algebra.

3.1. The problem

Our aim is to study the Nichols algebra $\mathcal{B}({\mathtt{HV}_{1}})$, cf. [HV2, Example 1.10]. Here ${\mathtt{HV}_{1}}=V_{1}\oplus V_{2}$, where $V_{1}$ is as in §2.6 and $V_{2}$ has dimension one: we fix $x_{4}\in V-0$. Let $q_{1},q_{2}\in\Bbbk^{\times}$ be such that

[TABLE]

The braiding of ${\mathtt{HV}_{1}}$ is determined by

• •

$(V_{1},c)$ is a braided vector subspace;

• •

$V_{2}$ is a point with label $-\omega^{2}$, so $c(x_{4}\otimes x_{4})=-\omega^{2}x_{4}\otimes x_{4}$;

• •

the braiding between them is

[TABLE]

The braided vector space ${\mathtt{HV}_{1}}$ is associated to the rack $\mathcal{O}_{2}^{3}\times\{4\}$ and the matrix $\mathfrak{q}:=\begin{pmatrix}-1&q_{1}\\ q_{2}&-\omega^{2}\end{pmatrix}$, cf. [A, Example 38].

The root system of ${\mathtt{HV}_{1}}$ is of standard type $B_{2}$, cf. [HS2, HV2]. We fix the following notation for the set of positive roots: $\varDelta_{+}=\{1,112,12,2\}$

3.2. Principal realizations

Let $\Gamma$ be a group. We assume that there exist $g_{i}\in\Gamma$, $i\in\mathbb{I}_{4}$, and $1$-cocycles $\chi_{i}:\Gamma\to\Bbbk$, such that

[TABLE]

Thus the conjugation induces an action $\triangleright\colon\Gamma\times\mathbb{I}_{4}\to\mathbb{I}_{4}$ given by $g_{g\triangleright i}=gg_{i}g^{-1}$. Following [AG, Definition 3.2] we obtain a principal YD-realization of $(\mathcal{O}_{2}^{3}\times{4},\mathfrak{q})$ over $\Gamma$: ${\mathtt{HV}_{1}}$ is realized in ${}^{\Bbbk\Gamma}_{\Bbbk\Gamma}\mathcal{YD}$ by defining

[TABLE]

Example 3.1**.**

Let $\Gamma_{3}$ be the group generated by $\nu,\gamma,\zeta$ with relations

[TABLE]

It is isomorphic to the enveloping group of the quandle $\mathcal{O}_{2}^{3}$ [HV1, Section 2]. The elements $g_{i}:=\gamma\nu^{i-1}$, $i\in\mathbb{I}_{3}$, and $g_{4}=\zeta$ generate $\Gamma_{3}$ and satisfy (3.3). There exist 1-cocycles $\chi_{i}\colon\Gamma_{3}\to\Bbbk^{\times}$, $i\in\mathcal{O}_{2}^{3}\times{4}$, determined by

[TABLE]

Hence $\Gamma_{3}$ provides a principal YD-realization of our quandle.

We restrict now to a group $\Gamma$ as above and consider ${\mathtt{HV}_{1}}\in{}^{\Bbbk\Gamma}_{\Bbbk\Gamma}\mathcal{YD}$. For a shorter notation, we set

[TABLE]

Theorem 3.2**.**

[HV2, Theorem 8.2]** The multiplication map

[TABLE]

is an isomorphism of $\mathbb{N}_{0}^{2}$-graded objects in ${}^{\Bbbk\Gamma}_{\Bbbk\Gamma}\mathcal{YD}$. Hence

[TABLE]

Remark 3.3*.*

The algebra $\mathcal{B}(V_{2})$ is presented by a single generator $x_{4}$ and the relation $x_{4}^{6}=0$. Since $V_{1}$ and $V_{2}$ are Yetter-Drinfeld submodules of ${\mathtt{HV}_{1}}$, we have inclusions of Hopf algebras in ${}^{\Bbbk\Gamma}_{\Bbbk\Gamma}\mathcal{YD}$

[TABLE]

thus the relations (2.12) and $x_{4}^{6}=0$ hold in $\mathcal{B}({\mathtt{HV}_{1}})$.

3.3. Structure of $V_{12}=(\operatorname{ad}_{c}V_{1})V_{2}$

Let

[TABLE]

Some general features of the Yetter-Drinfeld module $V_{12}$ are given in [HV2, Lemma 6.1]. Next we describe explicitly this structure for our realization.

Lemma 3.4**.**

1. (a)

The subspace $V_{12}$ is a Yetter-Drinfeld submodule of $\mathcal{B}({\mathtt{HV}_{1}})$ with basis $(x_{i4})_{i\in\mathbb{I}_{3}}$. The action, coaction and braiding satisfy

[TABLE] 2. (b)

The inclusion induces an algebra map $\mathcal{B}(V_{12})\to\mathcal{B}({\mathtt{HV}_{1}})$.

Proof.

(a) Since $g_{4}$ in central in $G$ and $g_{i}\cdot x_{4}=q_{1}x_{4}$ for $i\in\mathbb{I}_{3}$, we have that

[TABLE]

is homogeneous of degree $g_{i}g_{4}$. The set $(x_{i4})_{i\in\mathbb{I}_{3}}$ spans $V_{12}$, hence it is enough to show that they are linearly independent. This is achieved computing their skew derivations:

[TABLE]

for $i,j\in\mathbb{I}_{3}$, so they are linearly independent. The action of $\Gamma$ is given by

[TABLE]

and the formula for the braiding of $V_{12}$ follows from the action of $\Gamma$.

Now (b) follows by [HS1, Theorem 2.6, Corollary 2.7(2)]. ∎

Remark 3.5*.*

By [FK, MS] the following set is a basis of $\mathcal{B}(V_{12})$:

[TABLE]

3.4. Structure of $V_{112}=(\operatorname{ad}_{c}V_{1})^{2}V_{2}$

We set

[TABLE]

Lemma 3.6**.**

1. (a)

The subspace $V_{112}$ is a Yetter-Drinfeld submodule of $\mathcal{B}({\mathtt{HV}_{1}})$ with basis $\{x_{124},x_{134}\}$. The action and the coaction satisfy

[TABLE]

The braiding is of diagonal type, with matrix and Dynkin diagram

[TABLE]

Hence $V_{112}$ is of super type ${\mathbf{A}}(\omega|\mathbb{I}_{2})$ **[AA, Section 5.1]**. 2. (b)

The inclusion induces an algebra map $\mathcal{B}(V_{112})\to\mathcal{B}({\mathtt{HV}_{1}})$. Hence the following relations hold:

[TABLE]

Proof.

(a) Since the $x_{i}$’s are primitive in $\mathcal{B}({\mathtt{HV}_{1}})$, we have

[TABLE]

As $x_{i}^{2}=0$, we have that $x_{ii4}=(\operatorname{ad}_{c}x_{i})^{2}x_{4}=0$. Hence $V_{112}$ is spanned by $x_{ij4}$, $i\neq j\in\mathbb{I}_{3}$.

Claim**.**

If $i\neq j\in\mathbb{I}_{3}$, then

[TABLE]

Hence $V_{112}$ is spanned by $\{x_{124},x_{134}\}$.

We have to compute the skew derivations $\partial_{h}$ of both sides. If $h\neq i\in\mathbb{I}_{3}$, then $\partial_{h}(x_{ij4})=0$ by (3.10). For $h=i,4$,

[TABLE]

Hence (3.13) follows using (2.12). Thus $V_{112}$ is spanned by $\{x_{124},x_{134}\}$. Now $x_{124}$ and $x_{134}$ are nonzero because $\partial_{j}\partial_{4}(x_{1i4})=-\omega^{2}x_{i}\neq 0$, and linearly independent since $g_{1}g_{i}g_{4}\neq g_{1}g_{j}g_{4}$ if $i\neq j$. The action of $G$ and the braiding are obtained by direct computation.

Now (b) is a consequence of [HS1, Theorem 2.6, Corollary 2.7(2)] and [AA, Section 5.1]. ∎

Remark 3.7*.*

By [AA, Section 5.1.11], $\mathcal{B}(V_{112})$ has a PBW basis:

[TABLE]

3.5. Defining relations

Recall that

[TABLE]

The aim of this subsection is to give a minimal presentation of the Nichols algebra $\mathcal{B}({\mathtt{HV}_{1}})$. We start by stating the main result.

Theorem 3.8**.**

The Nichols algebra $\mathcal{B}({\mathtt{HV}_{1}})$ is presented by generators $x_{i}$, $i\in\mathbb{I}_{4}$, and relations

[TABLE]

Let ${\mathcal{G}}$ be the set of generators (3.14)–(3.19) of the ideal ${\mathcal{J}}({\mathtt{HV}_{1}})$. Then ${\mathcal{G}}$ is minimal. The set

[TABLE]

with $0\leq n_{4}<6$, $0\leq n_{14},n_{24},n_{34},n_{124},n_{134},n_{1},n_{2},n_{3}<2$, $0\leq n_{124134}<3$ is a basis of $\mathcal{B}({\mathtt{HV}_{1}})$, where $x_{124134}=x_{124}x_{134}+\omega^{2}x_{134}x_{124}$.

Here the brackets mean that we choose either an element of the first line or else one of the second, with the restrictions for the $n_{\alpha}$.

3.5.1. A first presentation of $\mathcal{B}({\mathtt{HV}_{1}})$

We start with a set of defining relations of the Nichols algebra $\mathcal{B}({\mathtt{HV}_{1}})$. This set is far from being minimal but will help to prove Theorem 3.8.

By Remark 3.3 and Lemmas 3.4, 3.6, the following relations hold in $\mathcal{B}({\mathtt{HV}_{1}})$: (3.14), (3.15), (3.16), (3.17), (3.18),

[TABLE]

Lemma 3.9**.**

The algebra $\mathcal{B}({\mathtt{HV}_{1}})$ is presented by generators $x_{i}$, $i\in\mathbb{I}_{4}$, with relations (3.14)–(3.25).

Proof.

Let $\widetilde{\mathcal{B}}$ be the algebra presented by generators $\widetilde{x}_{i}$, $i\in\mathbb{I}_{4}$, with relations (3.14)–(3.25), replacing $x_{h4}$ and $x_{ij4}$ by

[TABLE]

Claim 1**.**

There exists a surjective algebra map $\phi\colon\widetilde{\mathcal{B}}\to\mathcal{B}({\mathtt{HV}_{1}})$.

Proof.

We already know that relations (3.14)–(3.18), and (3.21)–(3.25) hold in $\mathcal{B}({\mathtt{HV}_{1}})$. Hence it suffices to show that (3.19) also hold in $\mathcal{B}({\mathtt{HV}_{1}})$. Notice that $\partial_{i}$, $i\in\mathbb{I}_{3}$, vanish both sides of (3.19). Now

[TABLE]

Hence

[TABLE]

so (3.19) holds in $\mathcal{B}({\mathtt{HV}_{1}})$. ∎

Let $S_{2}=\{\widetilde{x}_{4}\}$, $S_{12}=\{\widetilde{x}_{14},\widetilde{x}_{24},\widetilde{x}_{34}\}$, $S_{112}=\{\widetilde{x}_{124},\widetilde{x}_{134}\}$, $S_{1}=\{\widetilde{x}_{1},\widetilde{x}_{2},\widetilde{x}_{3}\}$; let $\widetilde{\mathcal{B}}_{\alpha}$ be the subalgebra of $\widetilde{\mathcal{B}}$ generated by $S_{\alpha}$ and denote by $\phi_{\alpha}$ the restriction of $\phi$ to $\widetilde{\mathcal{B}}_{\alpha}$, $\alpha\in\varDelta_{+}$.

Claim 2**.**

$\phi_{\alpha}\colon\widetilde{\mathcal{B}}_{\alpha}\to\mathcal{B}(V_{\alpha})$* is an algebra isomorphism, $\alpha\in\varDelta_{+}$.*

Proof.

Since $S_{\alpha}$ generates $\widetilde{\mathcal{B}}_{\alpha}$ and is mapped by $\phi$ onto the set of generators of $\mathcal{B}(V_{\alpha})$, it follows that $\phi(\widetilde{\mathcal{B}}_{\alpha})=\mathcal{B}(V_{\alpha})$. Moreover, since (3.14)–(3.25) hold in $\widetilde{\mathcal{B}}$, the defining relations of $\mathcal{B}(V_{\alpha})$ are satisfied by the corresponding elements of $S_{\alpha}$. This yields an algebra map $\mathcal{B}(V_{\alpha})\to\widetilde{\mathcal{B}}_{\alpha}$ that is a section of $\phi_{\alpha}$. ∎

Claim 3**.**

The multiplication $\widetilde{\mathcal{B}}_{2}\otimes\widetilde{\mathcal{B}}_{12}\otimes\widetilde{\mathcal{B}}_{112}\otimes\widetilde{\mathcal{B}}_{1}\to\widetilde{\mathcal{B}}$ is surjective.

Proof.

We start with an auxiliar relation, that holds either in $\widetilde{\mathcal{B}}$ or $\mathcal{B}({\mathtt{HV}_{1}})$:

[TABLE]

In fact, by (3.24) we may rewrite $x_{1j4}=\omega^{m}x_{ih4}$, where $i=h-j+1$ and $m\in\mathbb{I}_{0,2}$ depends on $h$ and $j$. Using (3.21) twice,

[TABLE]

Let $\mathcal{D}_{2}=\widetilde{\mathcal{B}}_{2}$, $\mathcal{D}_{12}=\widetilde{\mathcal{B}}_{2}\widetilde{\mathcal{B}}_{12}$, $\mathcal{D}_{112}=\widetilde{\mathcal{B}}_{2}\widetilde{\mathcal{B}}_{12}\widetilde{\mathcal{B}}_{112}$, $\mathcal{D}_{1}=\widetilde{\mathcal{B}}_{2}\widetilde{\mathcal{B}}_{12}\widetilde{\mathcal{B}}_{112}\widetilde{\mathcal{B}}_{1}$. Our goal is to prove that $\widetilde{\mathcal{B}}=\mathcal{D}_{1}$. Since $\mathcal{D}_{1}$ contains $1$, this can be achieved showing that $\mathcal{D}_{1}$ is a left ideal of $\widetilde{\mathcal{B}}$, which reduces to prove that $\widetilde{x}_{i}\mathcal{D}_{1}\subset\mathcal{D}_{1}$ for all $i\in\mathbb{I}_{4}$: The case $i=4$ is straightforward. We proceed by steps.

Step 1**.**

$\mathcal{D}_{112}S_{12}\subset\mathcal{D}_{112}$, $\mathcal{D}_{1}S_{112}\subset\mathcal{D}_{1}$.

We start with the first inclusion. We have to prove that $\mathcal{D}_{112}\widetilde{x}_{h4}\subset\mathcal{D}_{112}$ for all $h\in\mathbb{I}_{3}$. The subspace $\mathcal{D}_{112}$ is spanned by monomials

[TABLE]

Fix $h\in\mathbb{I}_{3}$. If $l=0$, then $y\widetilde{x}_{h4}\in\mathcal{D}_{112}$. If $l>0$, then

[TABLE]

by repeated applications of (3.26). Thus $y\widetilde{x}_{h4}\in\mathcal{D}_{112}$.

Now we notice that $\mathcal{D}_{1}$ is spanned by elements

[TABLE]

If $m=0$, then $y\widetilde{x}_{1j4}\in\mathcal{D}_{1}$ by definition. If $m>0$, then

[TABLE]

for some $b\in\mathbb{N}_{0}$, $j^{\prime}\in\mathbb{I}_{2,3}$, by (3.25). Hence $y\widetilde{x}_{1j4}\in\mathcal{D}_{1}$.

Step 2**.**

For all $i\in\mathbb{I}_{3}$,

[TABLE]

For the first inclusion, we only need to show that $\widetilde{x}_{i}\widetilde{\mathcal{B}}_{2}\subset\mathcal{D}_{12}\,\Bbbk\langle\widetilde{x}_{i}\rangle$ since $\widetilde{x}_{i}^{2}=0$ and $\widetilde{\mathcal{B}}_{2}\subset\mathcal{D}_{12}$. The claim holds by the following formula, which follows inductively from (3.19):

[TABLE]

For the second, it suffices to prove that $\widetilde{x}_{i}\widetilde{\mathcal{B}}_{12}\subset\mathcal{D}_{112}\,\Bbbk\langle\widetilde{x}_{i}\rangle$. We prove by induction on $k$ that $\widetilde{x}_{i}\widetilde{x}_{h_{1}4}\cdots\widetilde{x}_{h_{k}4}\in\mathcal{D}_{112}\,\Bbbk\langle\widetilde{x}_{i}\rangle$. The case $k=0$ is trivial, while $k=1$ is just the definition of the $\widetilde{x}_{ij4}$’s. We assume it holds for $k$. Then, for any $h\in\mathbb{I}_{3}$,

[TABLE]

where we use inductive hypothesis, Step 1 and the following facts:

[TABLE]

The last inclusion follows from (3.25).

Step 3**.**

$\mathcal{D}_{2}\mathcal{D}_{12}\subset\mathcal{D}_{12}$, $\mathcal{D}_{12}\mathcal{D}_{112}\subset\mathcal{D}_{112}$.

The first inclusion can be written as $\widetilde{\mathcal{B}}_{2}\widetilde{\mathcal{B}}_{2}\widetilde{\mathcal{B}}_{12}\subset\widetilde{\mathcal{B}}_{2}\widetilde{\mathcal{B}}_{12}$, and the proof follows since $\widetilde{\mathcal{B}}_{2}$ is a subalgebra. The proof of the second inclusion is similar since $\widetilde{\mathcal{B}}_{12}\widetilde{\mathcal{B}}_{2}=\widetilde{\mathcal{B}}_{2}\widetilde{\mathcal{B}}_{12}$, and $\widetilde{\mathcal{B}}_{12}$, $\widetilde{\mathcal{B}}_{2}$ are subalgebras.

Step 4**.**

$\widetilde{x}_{i}\mathcal{D}_{112}\subset\mathcal{D}_{112}\,\Bbbk\langle\widetilde{x}_{i}\rangle$, $i\in\mathbb{I}_{3}$.

By (3.25), $\Bbbk\langle\widetilde{x}_{i}\rangle\widetilde{\mathcal{B}}_{112}=\widetilde{\mathcal{B}}_{112}\Bbbk\langle\widetilde{x}_{i}\rangle$. Now we have

[TABLE]

where the first two inclusions follow from Step 2, the third from Step 3 and $\Bbbk\langle\widetilde{x}_{i}\rangle\widetilde{\mathcal{B}}_{2}\subset\widetilde{\mathcal{B}}_{2}\Bbbk\langle\widetilde{x}_{i}\rangle$, and the last equality from $\mathcal{D}_{112}\widetilde{\mathcal{B}}_{112}=\mathcal{D}_{112}$.

Step 5**.**

$\widetilde{x}_{i}\mathcal{D}_{1}\subset\mathcal{D}_{1}$, $i\in\mathbb{I}_{3}$.

Indeed, by Step 4 and the fact $\Bbbk\langle\widetilde{x}_{i}\rangle\widetilde{\mathcal{B}}_{1}=\widetilde{\mathcal{B}}_{1}$ it follows

[TABLE]

This concludes the proof of Claim 3. ∎

By Claim 3 in $\ast$, Claim 2 and Theorem 3.2 in $\star$ and Claim 1 in $\dagger$, we have

[TABLE]

Since $\phi$ is surjective it must be an isomorphism. ∎

3.5.2. Proof of Theorem 3.8

The statement about the basis follows from Theorem 3.2 and Lemmas 3.4, 3.6.

We seek now for a minimal set of generators of the ideal ${\mathcal{J}}({\mathtt{HV}_{1}})$. Hence we need to obtain relations in Lemma 3.9 from relations in Theorem 3.8.

Lemma 3.10**.**

Let $S$ be a quotient algebra of $T({\mathtt{HV}_{1}})$ such that (3.14), (3.15), (3.18) and (3.19) hold in $S$. Then (3.21), (3.22), (3.24), (3.25) and (3.23) also hold.

Proof.

Let $h,j\in\mathbb{I}_{3}$. As we assume that (3.19) holds, untwining (3.19) we get $x_{4}(x_{h}x_{4}-q_{1}x_{4}x_{h})=q_{2}(x_{h}x_{4}-q_{1}x_{4}x_{h})x_{4}$, which can be rewritten as

[TABLE]

Using this equality we compute

[TABLE]

Now (3.21) follows from (3.27) for $j=h$ and (3.14).

For (3.22), let $i\in\mathbb{I}_{2,3}$. Using (3.27) again and (3.15),

[TABLE]

Next we prove (3.24). Let $i\in\mathbb{I}_{2,3}$. By (3.15), $x_{1i4}+x_{5-i\,14}+x_{i\,5-i\,4}=0$, so

[TABLE]

Now we turn to (3.25). Let $i,j\in\mathbb{I}_{3}$. Since $x_{i}^{2}=0$ we have that

[TABLE]

We now split the proof of the six relations of (3.25) in different cases. Assume first $i=1$. Then (3.25) means $x_{1}x_{1j4}=q_{1}x_{1\,2-j\,4}x_{1}$ for any $j\in\mathbb{I}_{3}$, which is (3.28). Now assume $i\in\mathbb{I}_{2,3}$. By (3.24) and (3.28) we get

[TABLE]

Finally we consider (3.23). By (3.14), $x_{114}=(\operatorname{ad}_{c}x_{1}^{2})x_{4}=0$. Hence

[TABLE]

for all $j\in\mathbb{I}_{2,3}$. ∎

We proceed with the proof of the statement about the presentation of ${\mathcal{J}}({\mathtt{HV}_{1}})$. By Lemmas 3.9 and 3.10 ${\mathcal{G}}$ generates ${\mathcal{J}}({\mathtt{HV}_{1}})$ as an ideal. Hence it remains to show the minimality: It is enough to show that if $r\in{\mathcal{G}}$, then the ideal ${\mathcal{I}}_{r}$ generated by ${\mathcal{G}}\setminus r$ is not the whole ${\mathcal{J}}({\mathtt{HV}_{1}})$. The tensor algebra $T({\mathtt{HV}_{1}})$ has a $\mathbb{N}_{0}^{2}$-grading with $x_{1},x_{2},x_{3}$ sitting on degree $(1,0)$ and $x_{4}$ on degree $(0,1)$. The relations (3.14)–(3.19) are homogeneous and the $\Bbbk\Gamma$-coaction is of the shape $\delta(r)=g_{r}\otimes r$ for some $g_{r}\in\Gamma$. The next table contains $\mathbb{N}_{0}^{2}$-degrees and $g_{r}$ for all $r\in{\mathcal{G}}$.

[TABLE]

Consider on $\mathbb{N}_{0}^{2}$ the partial order $(a,b)\preceq(a^{\prime},b^{\prime})$ if and only if $a\leq a^{\prime}$ and $b\leq b^{\prime}$. This induces an order on the set of homogeneous elements of $T({\mathtt{HV}_{1}})$.

If $r$ is (3.16) then it is minimal in ${\mathcal{G}}$, so $r\notin{\mathcal{I}}_{r}$ since its $\mathbb{N}_{0}^{2}$-degree is minimal. Now fix $r$ one of the words with degree $(2,0)$. Since no element in ${\mathcal{G}}$ has degree less than $(2,0)$, the only way to have $r\in{\mathcal{I}}_{r}$ is to write it as a linear combination of the other elements in ${\mathcal{G}}$ with degree $(2,0)$. This is impossible since the $g_{r}$ are pairwise different so they are linearly independent. The same argument holds if $r$ is one of (3.19).

Now we fix a relation $r_{i}=x_{i14}-\omega x_{1\,5-i\,4}$ in (3.18). Suppose that $r_{i}\in{\mathcal{I}}_{r}$: By $\mathbb{N}_{0}^{2}$-degree restrictions we must have

[TABLE]

Now we look at the $\Bbbk\Gamma$-coaction. Since $g_{r_{i}}=g_{1}g_{5-i}g_{4}$, it follows that $\nu=0$ and $\mu_{r}=\lambda_{r}=0$ if $g_{r}\neq g_{1}g_{5-i}$. So our sum becomes

[TABLE]

But untwining the definition we get

[TABLE]

and the term of the form $x_{1}x_{4}x_{5-i}$ does not appear on the right-hand side of (3.29). This contradiction shows that $r\notin{\mathcal{I}}_{r}$.

Finally, we show using GAP that (3.17) is not in the ideal generated by the other relations. ∎

4. Pre-Nichols algebras of ${\mathtt{HV}_{1}}$

We have two purposes in this §. On the one hand, we study finite-dimensional pre-Nichols algebras of ${\mathtt{HV}_{1}}$ to conclude that any pointed Hopf algebra with infinitesimal braiding ${\mathtt{HV}_{1}}$ is generated by skew-primitive and group-like elements. On the other hand, we introduce a pre-Nichols algebra $\widetilde{\mathcal{B}}({\mathtt{HV}_{1}})$ of ${\mathtt{HV}_{1}}$ which plays the role of distinguished pre-Nichols algebras for braidings of diagonal type [An2]: the Gelfand Kirillov dimension of $\widetilde{\mathcal{B}}({\mathtt{HV}_{1}})$ is finite and $\widetilde{\mathcal{B}}({\mathtt{HV}_{1}})$ has a skew central Hopf subalgebra ${\mathcal{Z}}({\mathtt{HV}_{1}})$ such that $\mathcal{B}({\mathtt{HV}_{1}})$ is the quotient of $\widetilde{\mathcal{B}}({\mathtt{HV}_{1}})$ by the ideal generated by ${\mathcal{Z}}({\mathtt{HV}_{1}})$.

4.1. Generation in degree one

Here we study finite-dimensional pre-Nichols algebras $\widetilde{\mathcal{B}}$ of ${\mathtt{HV}_{1}}$. We recall that there exists a homomorphism $T({\mathtt{HV}_{1}})\twoheadrightarrow\widetilde{\mathcal{B}}$ of graded Hopf algebras in ${}^{\Bbbk\Gamma}_{\Bbbk\Gamma}\mathcal{YD}$. In the following lemmas we proceed as in [AS2, Lemma 5.4] to show that this map factors through some relations of the defining ideal ${\mathcal{J}}({\mathtt{HV}_{1}})$. We start by looking at those relations which are primitive in $T({\mathtt{HV}_{1}})$.

Remark 4.1*.*

By direct computation,

[TABLE]

For (3.19) and (3.18), the comultiplication in $T({\mathtt{HV}_{1}})$ is

[TABLE]

Indeed, these formulas follow from the following ones:

[TABLE]

Lemma 4.2**.**

Let $\widetilde{\mathcal{B}}$ be a finite-dimensional pre-Nichols algebra of ${\mathtt{HV}_{1}}$. Then (3.14), (3.15), (3.16) and (3.19) hold in $\widetilde{\mathcal{B}}$.

Proof.

First we note that the Nichols algebra of the primitive elements $\mathcal{P}(\widetilde{\mathcal{B}})$ is finite dimensional. Indeed, $\mathcal{P}(\widetilde{\mathcal{B}})\#\Bbbk\Gamma$ is contained in the first term of the coradical filtration of $\widetilde{\mathcal{B}}\#\Bbbk\Gamma$, see [AS2, Lemma 5.4].

Since $T({\mathtt{HV}_{1}})\to\widetilde{\mathcal{B}}$ is a homomorphism of braided Hopf algebras, the elements (3.14), (3.15), (3.16) and (3.19) are primitive in $\widetilde{\mathcal{B}}$. The strategy now is to build braided subspaces of $\mathcal{P}(\widetilde{\mathcal{B}})$ which are either zero or generate an infinite-dimensional Nichols algebra. By direct computation, if $r$ is one of the relations in (3.14), (3.15) or (3.16), then $c(r\otimes r)=r\otimes r$. Hence $r=0$.

Now we turn to (3.19). Fix $h\in\mathbb{I}_{3}$ and set $r_{h}=x_{4}x_{h4}-q_{2}x_{h4}x_{4}\in\mathcal{P}(\widetilde{\mathcal{B}})$. Since $g_{4}$ is central, $r_{h}$ is homogeneous of degree $g_{h}g_{4}^{2}$. Suppose that $r_{h}$ is non-zero, we may consider the $2$-dimensional subspace $W_{h}=\Bbbk x_{h}\oplus\Bbbk r_{h}\subset\mathcal{P}(\widetilde{\mathcal{B}})$. The braiding of $W_{h}$ is diagonal, with braiding matrix and Dynkin diagram

[TABLE]

This diagram does not appear in [H, Table 1]. Hence $\dim\mathcal{B}(W_{h})=\infty$ and we have a contradiction since $\mathcal{B}(W_{h})\hookrightarrow\mathcal{B}(\mathcal{P}(\widetilde{\mathcal{B}}))$. Thus $r_{h}=0$. ∎

Lemma 4.3**.**

Let $\widetilde{\mathcal{B}}$ be as above. Then (3.18) holds in $\widetilde{\mathcal{B}}$.

Proof.

Set $y_{1}=x_{314}-\omega x_{124}$, $y_{2}=x_{214}-\omega x_{134}$ and $W=\Bbbk y_{1}+\Bbbk y_{2}$; we have $W\subset\mathcal{P}(\widetilde{\mathcal{B}})$ by Lemma 4.2. The $\Bbbk\Gamma$-coaction is given by $g_{1}g_{2}g_{4}$ and $g_{1}g_{3}g_{4}$ respectively. Let us compute the $\Gamma$-module structure of $W$. As (3.15) holds in $\widetilde{\mathcal{B}}$ by the previous step, $x_{1i4}+x_{2-i\,14}+x_{i\,2-i\,4}=0$, $i\in\mathbb{I}_{2,3}$. Then

[TABLE]

and similarly

[TABLE]

Assume $W\neq 0$. Since $g_{1}$ permutes the generators, we have $y_{1},y_{2}\neq 0$; moreover they are linearly independent since they have different $\Gamma$-degrees. A straightforward computation shows that $W$ has not $\Gamma$-stable $1$-dimensional subspace, so $W$ is a simple Yetter-Drinfeld module over $\Gamma$.

As $\operatorname{supp}V\oplus W$ generates the subgroup $\Gamma^{\prime}$ generated by $g_{i}$, $i\in\mathbb{I}_{4}$, we may realize our Nichols and pre-Nichols algebras over $\Gamma^{\prime}$. Hence we may (and will) assume that $\Gamma=\Gamma^{\prime}$. We may now evoke the classification theorem on [HV2]. Indeed, consider the pair $(V,W)$ of simple objects in ${}^{\Bbbk\Gamma}_{\Bbbk\Gamma}\mathcal{YD}$. As $W\subset\widetilde{\mathcal{B}}^{2}$, we have $V\cap W=0$. By Remark 4.1 and Lemma 4.2, we have $V\oplus W\subset\mathcal{P}(\widetilde{\mathcal{B}})$, so $\dim\mathcal{B}(V\oplus W)<\infty$. Thus $V\oplus W$ is one of the five-dimensional braided vector spaces in [HV2, Theorem 2.1]; moreover it is one of [HV2, Examples 1.9–1.11] since $V$ is the quandle of transpositions of $\mathbb{S}_{3}$. Next we compute the action of $\Gamma$ on $V\oplus W$:

[TABLE]

Hence $c_{W,V}c_{V,W}\neq\operatorname{id}_{V\otimes W}$ and the braiding of $W$ is diagonal, with matrix $\begin{pmatrix}-\omega^{2}&-1\\ -1&-\omega^{2}\end{pmatrix}$, $-\omega^{2}\in\mathbb{G}^{\prime}_{6}$. But all the $W$’s in [HV2, Examples 1.9 - 1.11] have vertices labeled with $-1$, which is a contradiction. Thus $W=0$. ∎

We can now prove the main result of this section, which states that $\mathcal{B}({\mathtt{HV}_{1}})$ is the unique finite dimensional pre-Nichols, respectively post-Nichols, algebra of ${\mathtt{HV}_{1}}$. In particular, we will show that the top degree relation (3.17) holds in any finite dimensional pre-Nichols algebra.

Theorem 4.4**.**

1. (a)

Let $\widetilde{\mathcal{B}}=\oplus_{n\geq 0}\widetilde{\mathcal{B}}^{n}$ be a finite-dimensional pre-Nichols algebra of ${\mathtt{HV}_{1}}$. Then $\widetilde{\mathcal{B}}\simeq\mathcal{B}({\mathtt{HV}_{1}})$. 2. (b)

Let $\mathcal{L}=\oplus_{n\geq 0}\mathcal{L}^{n}$ be a finite-dimensional post-Nichols algebra of ${\mathtt{HV}_{1}}$. Then $\mathcal{L}\simeq\mathcal{B}({\mathtt{HV}_{1}})$.

Proof.

For (a) we proceed as in [An1, Theorem 4.1]. By definition, we may identify $\widetilde{\mathcal{B}}=T({\mathtt{HV}_{1}})/I$, where $I$ is a graded Yetter-Drinfeld submodule and Hopf ideal of $T({\mathtt{HV}_{1}})$ generated by homogeneous elements of degree $\geq 2$, and $I\subseteq{\mathcal{J}}({\mathtt{HV}_{1}})$. Let $\pi\colon\widetilde{\mathcal{B}}\twoheadrightarrow\mathcal{B}({\mathtt{HV}_{1}})$ be the canonical projection of graded braided Hopf algebras. Assume ${\mathcal{J}}({\mathtt{HV}_{1}})\supsetneq I$, hence one of the generators in Theorem 3.8 does not belong to $I$. By Lemmas 4.2 and 4.3, it must be $r=(x_{124}x_{134}+\omega^{2}x_{134}x_{124})^{3}\notin I$. As ${\mathcal{G}}$ is a minimal set of defining relations of ${\mathcal{J}}({\mathtt{HV}_{1}})$ and this ideal is graded, $r$ has minimal degree $\geq 2$ among non-trivial elements in $\ker\pi$. Then $r$ is primitive in $\widetilde{\mathcal{B}}$ by [An1, Lemma 3.2]. For the braiding on $r$, we claim that $c(r\otimes r)=r\otimes r$. Indeed $r$ is homogeneous of degree $g_{1}^{12}g_{4}^{6}$ and

[TABLE]

so we have $g_{1}^{12}g_{4}^{6}\cdot r=r$. Hence $\Bbbk r$ is a braided vector subspace corresponding to an infinite dimensional Nichols algebra, and $r\in\mathcal{B}(\mathcal{P}(\widetilde{\mathcal{B}}))$, a contradiction. Thus $I={\mathcal{J}}({\mathtt{HV}_{1}})$ and $\widetilde{\mathcal{B}}\simeq\mathcal{B}({\mathtt{HV}_{1}})$.

Now we prove (b). Let us compute ${\mathtt{HV}_{1}}^{*}\in{}^{\Bbbk\Gamma}_{\Bbbk\Gamma}\mathcal{YD}$. Denote by $(f_{i})_{i\in\mathbb{I}_{4}}$ the basis of ${\mathtt{HV}_{1}}^{*}$ dual to $(x_{i})_{i\in\mathbb{I}_{4}}$. Then $\delta(f_{i})=g_{i}^{-1}\otimes f_{i}$, so $c(f_{i}\otimes f_{j})=g_{i}^{-1}\cdot f_{j}\otimes f_{i}$. Straightforward computations shows that

[TABLE]

Hence ${\mathtt{HV}_{1}}\to{\mathtt{HV}_{1}}^{*}$, $x_{i}\mapsto f_{i}$, $i\in\mathbb{I}_{4}$, is an isomorphism of braided vector spaces.

Let $\mathcal{L}=\oplus_{n\geq 0}\mathcal{L}^{n}$ be a finite-dimensional post-Nichols algebra of ${\mathtt{HV}_{1}}$. By [AS1, Lemma 5.5], $\mathcal{L}^{*}$ is a finite-dimensional pre-Nichols algebra of ${\mathtt{HV}_{1}}^{*}\simeq{\mathtt{HV}_{1}}$. By (a), $\mathcal{L}^{*}\simeq\mathcal{B}({\mathtt{HV}_{1}}^{*})$, hence $\mathcal{L}\simeq\mathcal{B}({\mathtt{HV}_{1}})$. ∎

Theorem 4.5**.**

Let $H$ be a finite-dimensional pointed Hopf algebra over $\Gamma$ with infinitesimal braiding ${\mathtt{HV}_{1}}$. Then $H$ is generated by its group-like and skew-primitive elements.

Proof.

Working as in [AS2, Theorem 5.5], we reduce to prove that the unique finite-dimensional post-Nichols algebra of ${\mathtt{HV}_{1}}$ is the Nichols algebra, which follows by Theorem 4.4 (b). ∎

4.2. The dintinguished pre-Nichols algebra

Now we introduce a pre-Nichols algebra of ${\mathtt{HV}_{1}}$ which mimics those given in [An2] for braidings of diagonal type.

Definition 4.6**.**

The distinguished pre-Nichols algebra $\widetilde{\mathcal{B}}({\mathtt{HV}_{1}})$ of ${\mathtt{HV}_{1}}$ is the quotient of $T({\mathtt{HV}_{1}})$ by the ideal $\widetilde{{\mathcal{J}}}({\mathtt{HV}_{1}})$ generated by the elements (3.14), (3.15), (3.18) and (3.19).

Remark 4.7*.*

$\widetilde{{\mathcal{J}}}({\mathtt{HV}_{1}})$ is a Hopf ideal by Remark 4.1, and there exists a canonical projection $\pi:\widetilde{\mathcal{B}}({\mathtt{HV}_{1}})\twoheadrightarrow\mathcal{B}({\mathtt{HV}_{1}})$. Also, (3.21)–(3.25) hold in $\widetilde{\mathcal{B}}({\mathtt{HV}_{1}})$ by Lemma 3.10. Let ${\mathcal{Z}}({\mathtt{HV}_{1}})$ be the subalgebra of $\widetilde{\mathcal{B}}({\mathtt{HV}_{1}})$ generated by

[TABLE]

Then $\mathcal{B}({\mathtt{HV}_{1}})$ is the quotient of $\widetilde{\mathcal{B}}({\mathtt{HV}_{1}})$ by the ideal generated by ${\mathcal{Z}}({\mathtt{HV}_{1}})$.

Next we will prove that ${\mathcal{Z}}({\mathtt{HV}_{1}})$ is a normal Hopf subalgebra of $\widetilde{\mathcal{B}}({\mathtt{HV}_{1}})$. In order to do so, we need an auxiliar computation.

Lemma 4.8**.**

Let $j\in\mathbb{I}_{2,3}$. The following relations hold in $\widetilde{\mathcal{B}}({\mathtt{HV}_{1}})$:

[TABLE]

Proof.

Fix $j\in\mathbb{I}_{2,3}$. For (4.4), we compute

[TABLE]

Now (4.5) follows from (4.4) by multiplying both sides by $q_{1}^{2}\omega^{2}$. ∎

Lemma 4.9**.**

1. (a)

${\mathcal{Z}}({\mathtt{HV}_{1}})$* is a normal Hopf subalgebra of $\widetilde{\mathcal{B}}({\mathtt{HV}_{1}})$.* 2. (b)

${\mathcal{Z}}({\mathtt{HV}_{1}})=^{\operatorname{co}\pi}\widetilde{\mathcal{B}}({\mathtt{HV}_{1}})=\widetilde{\mathcal{B}}({\mathtt{HV}_{1}})^{\operatorname{co}\pi}$. 3. (c)

${\mathcal{Z}}({\mathtt{HV}_{1}})$* is presented by generators $z_{4}$, $z_{124134}$, which satisfy the relation*

[TABLE]

The set $\widetilde{B}=\{z_{4}^{m}z_{124134}^{n}\colon m,n\in\mathbb{N}_{0}\}$ is a basis of ${\mathcal{Z}}({\mathtt{HV}_{1}})$.

Proof.

(a) We show that the generators of ${\mathcal{Z}}({\mathtt{HV}_{1}})$ are vanished by the braided adjoint action of $\widetilde{\mathcal{B}}({\mathtt{HV}_{1}})$.

We start with $z_{4}$. Clearly $(\operatorname{ad}_{c}x_{4})z_{4}=x_{4}x_{4}^{6}-(-\omega^{2})^{6}x_{4}^{6}x_{4}=0$. Let $i\in\mathbb{I}_{3}$. By [AS1, Equation (A.8)] we have

[TABLE]

On the other hand, $(\operatorname{ad}_{c}x_{4})^{2}x_{i}=0$ by (3.19), hence $(\operatorname{ad}_{c}x_{i})z_{4}=0$.

Now turn to $z_{124134}$. By Lemma 3.10, we have $x_{1j4}^{2}=0$ for $j\in\mathbb{I}_{2,3}$, so

[TABLE]

Let $i\in\mathbb{I}_{3}$. By (3.25), $(\operatorname{ad}_{c}x_{i})x_{1j4}=0$ for $j\in\mathbb{I}_{2,3}$. Since $\operatorname{ad}_{c}x_{i}$ is a braided derivation, it follows that $(\operatorname{ad}_{c}x_{i})z_{124134}=0$.

By Lemma 3.10, (3.26) holds in $\widetilde{\mathcal{B}}({\mathtt{HV}_{1}})$ since this relation follows from (3.21) and (3.24). By repeated applications of (4.5) and (3.26) we get

[TABLE]

By (3.22) we have that

[TABLE]

Hence ${\mathcal{Z}}({\mathtt{HV}_{1}})$ is a normal subalgebra. Also ${\mathcal{Z}}({\mathtt{HV}_{1}})$ is a Hopf subalgebra since $z_{4},z_{124134}\in\mathcal{P}(\widetilde{\mathcal{B}}({\mathtt{HV}_{1}}))$ by Remark 4.1 and [An1, Lemma 3.2]

(b) This fact follows from [A+, Proposition 3.6 (c)].

(c) Note that $z_{4},z_{124134}\neq 0$ since (3.14)-(3.19) minimally generate ${\mathcal{J}}({\mathtt{HV}_{1}})$. Also, $z_{4}$ and $z_{124134}$ $q$-commute by (4.6), so ${\mathcal{Z}}({\mathtt{HV}_{1}})$ is spanned by $\widetilde{B}$.

Let $K$ be the subalgebra of $\widetilde{\mathcal{B}}({\mathtt{HV}_{1}})\#\Bbbk\Gamma$ generated by $z_{4},z_{124134}$ and $\Gamma$: $K$ is a Hopf subalgebra, which is a pointed Hopf algebra since $z_{4}$ is $(1,g_{4}^{6})$-primitive and $z_{124134}$ is $(1,g_{1}^{12}g_{4}^{6})$-primitive. As $z_{4},z_{124134}$ are linearly independent, the infinitesimal braiding contains the braided vector space generated by them, which is of diagonal type with matrix $\begin{pmatrix}1&q_{1}^{72}\\ q_{2}^{72}&1\end{pmatrix}$ so the set $\{z_{4}^{m}z_{124134}^{n}\gamma\colon m,n\in\mathbb{N}_{0},\gamma\in\Gamma\}$ is linearly independent. Thus $\widetilde{B}$ is linearly independent. ∎

Proposition 4.10**.**

1. (a)

We have an extension of braided Hopf algebras:

[TABLE] 2. (b)

$\operatorname{GK-dim}\widetilde{\mathcal{B}}({\mathtt{HV}_{1}})=2$. 3. (c)

The following set is a basis of $\widetilde{\mathcal{B}}({\mathtt{HV}_{1}})$:

[TABLE]

Proof.

(a) This fact follows by Lemma 4.9 and Remark 4.7, cf. [AN, §2.5].

(b) By [A+, Proposition 3.6 (d)] there exists a right ${\mathcal{Z}}({\mathtt{HV}_{1}})$-linear isomorphism $\mathcal{B}({\mathtt{HV}_{1}})\otimes{\mathcal{Z}}({\mathtt{HV}_{1}})\simeq\widetilde{\mathcal{B}}({\mathtt{HV}_{1}})$, hence $\widetilde{\mathcal{B}}({\mathtt{HV}_{1}})$ is finitely generated as right ${\mathcal{Z}}({\mathtt{HV}_{1}})$-module. By [KL, Proposition 5.5],

[TABLE]

(c) First, $\widetilde{\mathcal{B}}({\mathtt{HV}_{1}})$ is spanned by the set (4.7): this follows using the defining relations and arguing as in Lemma 3.9.

Let $\{b_{i}\}_{i\in\mathbb{I}_{10368}}$ be an enumeration of (3.20). Let $\widetilde{b}_{i}$ be the corresponding element viewed in $\widetilde{\mathcal{B}}({\mathtt{HV}_{1}})$; thus $\pi(\widetilde{b}_{i})=b_{i}$. Since the elements of ${\mathcal{Z}}({\mathtt{HV}_{1}})$ $q$-commute with all the elements of $\widetilde{\mathcal{B}}({\mathtt{HV}_{1}})$, the set (4.7) coincides with

[TABLE]

up to non-zero scalars. Hence it suffices to prove that $B$ is linearly independent. Suppose that $0=\sum_{m,n\in\mathbb{N}_{0}}\sum_{i\in\mathbb{I}_{10368}}a_{mni}z_{4}^{m}z_{124134}^{n}\widetilde{b}_{i}$, where not all the scalars $a_{mni}$’s are zero. Let $i_{0}\in\mathbb{I}_{10368}$ be such that $a_{mni_{0}}\neq 0$ for some $m,n\in\mathbb{N}_{0}$ and $b_{i}$ is of maximal degree $N$. Let $f\in\mathcal{B}({\mathtt{HV}_{1}})^{*}$, $f(b_{i})=\delta_{ii_{0}}$, $i\in\mathbb{I}_{10368}$. Since $\Delta$ and $\pi$ are $\mathbb{N}_{0}$-graded, $(\operatorname{id}\otimes f)(\operatorname{id}\otimes\pi)\Delta(\widetilde{b}_{i})=0$ for all $\widetilde{b}_{i}$ of degree less than $N$; for those $\widetilde{b}_{i}$ of degree $N$, $(\operatorname{id}\otimes f)(\operatorname{id}\otimes\pi)\Delta(\widetilde{b}_{i})=\delta_{ii_{0}}1$ since $\Delta(\widetilde{b}_{i})\in 1\otimes\widetilde{b}_{i}+\sum_{j>0}\widetilde{\mathcal{B}}({\mathtt{HV}_{1}})^{j}\otimes\widetilde{\mathcal{B}}({\mathtt{HV}_{1}})^{N-j}$. Hence

[TABLE]

which is a contradiction by Lemma 4.9 (c). ∎

5. Liftings of ${\mathtt{HV}_{1}}$

We fix a group $\Gamma$ and a principal realization of ${\mathtt{HV}_{1}}$ as in §3.2. In order to compute the liftings of $\mathcal{B}({\mathtt{HV}_{1}})$ over $\Gamma$ we follow the strategy developed in [A+, AAG]. The procedure gives rise to a family of liftings which are cocycle deformations of $\mathcal{B}({\mathtt{HV}_{1}})\#\Bbbk\Gamma$. Moreover, it provides a criterion to check if we have an exhaustive list of liftings of $\mathcal{B}({\mathtt{HV}_{1}})$ over $\Gamma$.

The starting point of the strategy is a suitable chosen chain of subsequent quotients of pre-Nichols algebras of ${\mathtt{HV}_{1}}$:

[TABLE]

After bosonization with $\Bbbk\Gamma$ we obtain graded Hopf algebras $\mathcal{H}_{k}:=\mathcal{B}_{k}\#\Bbbk\Gamma$, $k\in\mathbb{I}_{l+1}$, related by a corresponding chain of quotients. If $\mathcal{A}\in\operatorname{Cleft}(\mathcal{H}_{k})$, then the left Schauenburg Hopf algebra $L(\mathcal{A},\mathcal{H}_{k})$ is (isomorphic to) a quotient of $\mathcal{T}:=\mathcal{H}_{0}=T({\mathtt{HV}_{1}})\#\Bbbk\Gamma$ by [A+, Proposition 5.10]; this yields a filtration $\mathfrak{F}$ on $L(\mathcal{A},\mathcal{H}_{k})$, induced by the filtration coming from the $\mathbb{N}_{0}$-graduation of $T({\mathtt{HV}_{1}})$. We denote by $\operatorname{gr}_{\mathfrak{F}}$ the associated graded object. Let

[TABLE]

Recursively, the strategy begins with $\operatorname{Cleft}^{\prime}(\mathcal{T})=\{\mathcal{T}\}$ and compute the set $\operatorname{Cleft}^{\prime}(\mathcal{H}_{k+1})$ from $\operatorname{Cleft}^{\prime}(\mathcal{H}_{k})$ using the ideas in [Gu].

Since $\mathcal{B}({\mathtt{HV}_{1}})$ is coradically graded, if $L$ is a lifting of ${\mathtt{HV}_{1}}$ then $\mathfrak{F}$ above coincides with the coradical filtration and we have

[TABLE]

so the last step of the recursion leaves us with a family of liftings that are cocycle deformations of $\mathcal{B}({\mathtt{HV}_{1}})$.

We mention particular features of the strategy that will clarify the upcoming computations. Fix $k\in\mathbb{I}_{l+1}$.

• •

All cleft objects of $\mathcal{H}_{k+1}$ arise as quotients of cleft objects of $\mathcal{H}_{k}$ [Gu].

• •

If $\mathcal{A}\in\operatorname{Cleft}^{\prime}(\mathcal{H}_{k})$, there is an algebra map $\mathcal{T}\twoheadrightarrow\mathcal{A}$. The $\mathcal{H}_{k}$-colinear section $\gamma_{k}\colon\mathcal{H}_{k}\to\mathcal{A}$ restricts to an algebra map $(\gamma_{k})_{|\Bbbk\Gamma}\in\operatorname{Alg}(\Bbbk\Gamma,\mathcal{A})$.

• •

Each $\mathcal{A}\in\operatorname{Cleft}^{\prime}(\mathcal{H}_{k})$ can be obtained as $\mathcal{A}={\mathcal{E}}\#\Bbbk\Gamma$ where ${\mathcal{E}}$ is a $\Bbbk\Gamma$-module algebra [A+, Proposition 5.8]. Moreover, the section $\gamma_{k}$ restricts to a braided $\mathcal{B}_{k}$-comodule isomorphism $\gamma_{k}\colon\mathcal{B}_{k}\to{\mathcal{E}}$ [AnG2].

• •

As algebras, each ${\mathcal{E}}$ is a quotient of some ${\mathcal{E}}^{\prime}$ of the previous step.

• •

$\mathcal{L}_{0}=\mathcal{T}$ and, if $\mathcal{A}\twoheadrightarrow\mathcal{A}^{\prime}$, then $L(\mathcal{A},\mathcal{H}_{k})\twoheadrightarrow L(\mathcal{A}^{\prime},\mathcal{H}_{k+1})$.

5.1. Stratification

The first task proposed by the strategy is to obtain a convenient stratification ${\mathcal{G}}={\mathcal{G}}_{0}\sqcup{\mathcal{G}}_{1}\sqcup\cdots\sqcup{\mathcal{G}}_{l}$ of the minimal set of generators of the ideal ${\mathcal{J}}({\mathtt{HV}_{1}})$ found in Theorem 3.8. We need that the elements of ${\mathcal{G}}_{k}$ are primitive in the braided Hopf algebra $\mathcal{B}_{k}:=T({\mathtt{HV}_{1}})/\langle\cup_{j=0}^{k-1}{\mathcal{G}}_{j}\rangle$, $k\in\mathbb{I}_{l+1}$. In our setting, we also require that the vector space spanned by ${\mathcal{G}}_{k}$ is a Yetter-Drinfeld submodule of $T({\mathtt{HV}_{1}})$ over $\Bbbk\Gamma$.

By Remark 4.1 we may consider the following stratification:

[TABLE]

5.1.1. Realization of the strata

To describe the liftings we need to determine the braided vector space structure of each step of the stratification. To this end, it is enough to describe the action of $g_{i}\in\Gamma$, $i\in\mathbb{I}_{4}$.

Recall (cf. §3.2) that we have elements $g_{i}\in\Gamma$, 1-cocycles $\chi_{i}:\Gamma\to\Bbbk$, $i\in\mathbb{I}_{4}$, and an action $\triangleright\colon\Gamma\times\mathbb{I}_{4}\to\mathbb{I}_{4}$ such that

[TABLE]

We summarize the structure of the submodules $\Bbbk{\mathcal{G}}_{k}\subset T({\mathtt{HV}_{1}})\in{}^{\Bbbk\Gamma}_{\Bbbk\Gamma}\mathcal{YD}$, $k\in\mathbb{I}_{0,4}$, for later reference.

Let $\widetilde{\triangleright}\colon\mathbb{I}_{4}\times\mathbb{I}_{2,3}\to\mathbb{I}_{2,3}$ such that $j\,\widetilde{\triangleright}\,i=\begin{cases}5-i&\text{ if }j\in\mathbb{I}_{3},\\ i&\text{ if }j=4.\end{cases}$

$\heartsuit$ ${\mathcal{G}}_{0}$: structure determined by $x_{4}x_{h4}-q_{2}x_{h4}x_{4}\in T({\mathtt{HV}_{1}})_{g_{h}g_{4}^{2}}$, $h\in\mathbb{I}_{3}$,

[TABLE]

$\heartsuit$ ${\mathcal{G}}_{1}$: Here $x_{h}^{2}\in T({\mathtt{HV}_{1}})_{g_{h}^{2}}$ and the action satisfies

[TABLE]

$\heartsuit$ ${\mathcal{G}}_{2}$. Put $r_{i}:=x_{1}x_{i}+x_{5-i}x_{1}+x_{i}x_{5-i}$, $i\in\mathbb{I}_{2,3}$. In this case $r_{i}\in T({\mathtt{HV}_{1}})_{g_{1}g_{i}}$ and the action satisfies

[TABLE]

$\heartsuit$ ${\mathcal{G}}_{3}$. Let $p_{i}:=x_{i14}-\omega x_{15-i\,4}$. Then $p_{i}\in T({\mathtt{HV}_{1}})_{g_{1}g_{5-i}g_{4}}$, $i\in\mathbb{I}_{2,3}$. By (4.1), there are maps $\eta_{i}\colon\mathbb{I}_{4}\to\Bbbk^{\times}$, $i\in\mathbb{I}_{2,3}$ such that

[TABLE]

$\heartsuit$ ${\mathcal{G}}_{4}$. Here $x_{4}^{6}\in T({\mathtt{HV}_{1}})_{g_{4}^{6}}^{\chi_{4}^{6}}$, and $(x_{124}x_{134}+\omega^{2}x_{134}x_{124})^{3}\in T({\mathtt{HV}_{1}})_{g_{1}^{12}g_{4}^{6}}^{\chi_{1}^{12}\chi_{4}^{6}}$.

For $r\in{\mathcal{G}}$, we denote by $g_{r}$ the element of $\Gamma$ such that $r\in T({\mathtt{HV}_{1}})_{g_{r}}$.

Lemma 5.1**.**

If $r\in{\mathcal{G}}$, then $g_{r}\neq g_{i}$ for all $i\in\mathbb{I}_{4}$.

Proof.

Suppose first $r\in{\mathcal{G}}_{0}$, so $g_{r}=g_{h}g_{4}^{2}$ for some $h\in\mathbb{I}_{3}$. Since $g_{4}$ is central and $g_{h}$ is not, we have $g_{h}g_{4}^{2}\neq g_{4}$. Also $g_{h}g_{4}^{2}\neq g_{h}$, because $g_{4}^{2}\cdot x_{4}=\omega x_{4}$ and $\omega\neq 1$. If $g_{h}g_{4}^{2}=g_{i}$ for some $i\in\mathbb{I}_{3}$, $i\neq h$, then $-x_{i}=g_{i}\cdot x_{i}=g_{h}g_{4}^{2}\cdot x_{i}=-q_{2}^{2}x_{2h-i}$, a contradiction.

Next, let $r\in{\mathcal{G}}_{1}$, so $g_{r}=g_{h}^{2}$ for some $h\in\mathbb{I}_{3}$. Since $g_{h}^{2}=g_{i}^{2}$ for all $i\in\mathbb{I}_{3}$, it follows $g_{h}^{2}\neq g_{i}$, $i\in\mathbb{I}_{3}$. If $g_{h}^{2}=g_{4}$ then

[TABLE]

hence $\omega=(q_{1}q_{2})^{10}=-\omega$, contradicting $\omega\in\mathbb{G}^{\prime}_{3}$.

Suppose $r\in{\mathcal{G}}_{2}$, so $g_{r}=g_{1}g_{i}$ for some $i\in\mathbb{I}_{2,3}$. Clearly $g_{1}g_{i}\neq g_{1},g_{i}$; since $g_{5-i}g_{1}=g_{1}g_{i}$, it also follows $g_{1}g_{i}\neq g_{5-i}$. If $g_{1}g_{i}=g_{4}$ then we have $x_{i}=g_{1}g_{i}\cdot x_{1}=g_{4}\cdot x_{1}=q_{2}x_{1}$, a contradiction.

Assume now $r\in{\mathcal{G}}_{3}$ and let $i\in\mathbb{I}_{2,3}$ such that $g_{r}=g_{1}g_{5-i}g_{4}$. Since $g_{1}g_{5-i}=g_{i}g_{1}=g_{5-i}g_{i}$, we have $g_{1}g_{5-i}g_{4}\neq g_{1},g_{i},g_{5-i}$, because $g_{5-i},g_{1},g_{i}$, respectively, are non-central. We also have $g_{1}g_{5-i}g_{4}\neq g_{4}$, since $g_{1}g_{5-i}$ acts non-trivially on $x_{1}$.

Turn now to ${\mathcal{G}}_{4}$. We have $g_{4}^{6}\neq g_{h}$ for $h\in\mathbb{I}_{3}$ because $g_{h}$ is non-central; also $g_{4}^{4}\neq g_{4}$, because $g_{4}^{5}\cdot x_{4}=-\omega x_{4}$ and $-\omega\neq 1$. Finally, assume $g_{r}=g_{1}^{12}g_{4}^{6}$. If $g_{1}^{12}g_{4}^{6}=g_{h}$ for some $h\in\mathbb{I}_{3}$, then for any $j\in\mathbb{I}_{3}$ different from $h$ we have $-x_{2h-j}=g_{h}\cdot x_{j}=g_{1}^{12}g_{4}^{6}\cdot x_{j}=g_{j}^{12}g_{4}^{6}\cdot x_{j}\in\Bbbk x_{j}$, a contradiction. If $g_{1}^{12}g_{4}^{6}=g_{4}$, we compute

[TABLE]

so $q_{2}^{5}=1$, $q_{1}^{12}=-\omega^{2}$. Then $1=(q_{1}q_{2})^{60}=(-\omega^{2})^{5}=-\omega$, a contradiction. ∎

5.2. Computing cleft objects

The second task of the strategy is the introduction of a suitable family of cleft objects of $\mathcal{H}$. To this end we define a family of $\Bbbk\Gamma$-module algebras such that, after bosonization with $\Bbbk\Gamma$, give the desired cleft extensions.

The set of deformation parameters $\mathcal{R}_{{\mathtt{HV}_{1}}}$ consists of 4-uples of scalars $\boldsymbol{\lambda}=(\lambda_{i})_{i\in\mathbb{I}_{4}}\in\Bbbk^{4}$ such that

[TABLE]

Let $\boldsymbol{\lambda}\in\mathcal{R}_{{\mathtt{HV}_{1}}}$. We define ${\mathcal{E}}_{0}(\boldsymbol{\lambda})=\mathcal{B}_{0}=T({\mathtt{HV}_{1}})$, ${\mathcal{E}}_{1}(\boldsymbol{\lambda})=\mathcal{B}_{1}$, but we change the labels of the generators to $(y_{i})_{i\in\mathbb{I}_{4}}$ in order to differentiate with generators $(x_{i})_{i\in\mathbb{I}_{4}}$ of the pre-Nichols algebras $\mathcal{B}_{k}$. Let

[TABLE]

Remark 5.2*.*

Each ${\mathcal{E}}_{i}(\boldsymbol{\lambda})$ is a $\Bbbk\Gamma$-module algebra since the ideal is stable by the $\Gamma$-action by (5.2). Thus we may introduce $\mathcal{A}_{i}(\boldsymbol{\lambda}):={\mathcal{E}}_{i}(\boldsymbol{\lambda})\#\Bbbk\Gamma$.

Lemma 5.3**.**

Let $k\in\mathbb{I}_{4}$. Then ${\mathcal{E}}_{k}(\boldsymbol{\lambda})\neq 0$ and each $\mathcal{A}_{k}(\boldsymbol{\lambda})$ is a $\mathcal{H}_{k}$-cleft object. There exists an $\mathcal{H}_{k}$-colinear section $\gamma_{k}\colon\mathcal{H}_{k}\to\mathcal{A}_{k}$ which restricts to an algebra map $(\gamma_{k})_{|\Bbbk\Gamma}\in\operatorname{Alg}(\Bbbk\Gamma,\mathcal{A}_{k})$.

Proof.

Fix $\boldsymbol{\lambda}\in\mathcal{R}({\mathtt{HV}_{1}})$; we prove the claim recursively on $k$. For the sake of simplicity of the notation, we call ${\mathcal{E}}_{k}={\mathcal{E}}_{k}(\boldsymbol{\lambda})$, $\mathcal{A}_{k}=\mathcal{A}_{k}(\boldsymbol{\lambda})$. For $k=1$ the claim is clear since ${\mathcal{E}}_{1}=\mathcal{B}_{1}$, $\mathcal{A}_{1}=\mathcal{H}_{1}$ so we take $\gamma_{1}=\operatorname{id}_{\mathcal{H}_{1}}$.

$\heartsuit$ For $k=2$, we notice that ${\mathcal{E}}_{2}\neq 0$ (and a fortiori $\mathcal{A}_{2}\neq 0$) since there exists an algebra map ${\mathcal{E}}_{2}\twoheadrightarrow{\mathcal{E}}(\lambda_{1},\lambda_{2})$, where ${\mathcal{E}}(\lambda_{1},\lambda_{2})$ is the corresponding non-trivial algebra given in [GV2, Proposition 7.2]: the map identifies the generators $y_{i}$ for $i\in\mathbb{I}_{3}$ and annihilates $y_{4}$. As

[TABLE]

we have that $\langle y_{i}^{2}-\lambda_{1}:i\in\mathbb{I}_{3}\rangle=\langle y_{1}^{2}-\lambda_{1}\rangle$. Using [A+, Proposition 5.8] and this equality of ideals, we have that

[TABLE]

Let $Y_{1}^{\prime}$ be the subalgebra of $\mathcal{H}_{1}$ generated by $x_{1}^{2}$. Then $Y_{1}^{\prime}$ is isomorphic to a polynomial ring in one variable since $x_{1}^{2}\in\mathcal{P}(\mathcal{B}_{1})_{g_{1}^{2}}-0$ and $g_{1}^{2}\cdot x_{1}^{2}=x_{1}^{2}$. As in [AAG] we set $Y_{1}=\mathcal{S}(Y_{1}^{\prime})$. Notice that

[TABLE]

Since $Y_{1}$ is also a polynomial algebra generated by $x_{1}^{2}g_{1}^{-2}$, there exists an algebra map $\phi:Y_{1}\to\mathcal{A}_{1}$ such that $\phi(x_{1}^{2}g_{1}^{-2})=y_{1}^{2}g_{1}^{-2}-\lambda_{1}g_{1}^{-2}$, which is $\mathcal{H}_{1}$-colinear. We notice that

[TABLE]

Hence $\mathcal{A}_{2}$ is a $\mathcal{H}_{2}$-cleft object by [Gu, Theorem 8]. The claim about the section $\gamma_{2}$ follows from [A+, Proposition 5.8].

$\heartsuit$ For $k=3$, ${\mathcal{E}}_{3}\neq 0$ (and a fortiori $\mathcal{A}_{3}\neq 0$) since the algebra map ${\mathcal{E}}_{2}\twoheadrightarrow{\mathcal{E}}(\lambda_{1},\lambda_{2})$ descends to an algebra map ${\mathcal{E}}_{3}\twoheadrightarrow{\mathcal{E}}(\lambda_{1},\lambda_{2})$.

Let $\mathtt{r}_{i}=y_{1}y_{i}+y_{5-i}y_{1}+y_{i}y_{5-i}$, $i\in\mathbb{I}_{2,3}$. As $g_{1}(\mathtt{r}_{2}-\lambda_{2})g_{1}^{-1}=\mathtt{r}_{3}-\lambda_{2}$, we have that $\langle\mathtt{r}_{i}-\lambda_{2}:i\in\mathbb{I}_{2,3}\rangle=\langle\mathtt{r}_{2}-\lambda_{2}\rangle$. Hence

[TABLE]

Let $Y_{2}^{\prime}$ be the subalgebra of $\mathcal{H}_{2}$ generated by $r_{2}=x_{1}x_{2}+x_{3}x_{1}+x_{2}x_{3}$. Then $Y_{2}^{\prime}$ is isomorphic to a polynomial ring in one variable since $r_{2}\in\mathcal{P}(\mathcal{B}_{2})_{g_{1}g_{2}}-0$ and $g_{1}g_{2}\cdot r_{2}=r_{2}$. As in [AAG] we set $Y_{2}=\mathcal{S}(Y_{2}^{\prime})$. Notice that $\mathcal{H}_{2}/\langle Y_{2}^{+}\rangle\simeq\mathcal{H}_{3}$. Since $Y_{2}$ is also a polynomial algebra generated by $r_{2}g_{2}^{-1}g_{1}^{-1}$, there exists an algebra map $\phi:Y_{2}\to\mathcal{A}_{2}$ such that $\phi(r_{2}g_{2}^{-1}g_{1}^{-1})=\mathtt{r}_{2}g_{2}^{-1}g_{1}^{-1}-\lambda_{2}g_{2}^{-1}g_{1}^{-1}$, which is $\mathcal{H}_{2}$-colinear. Hence $\mathcal{A}_{3}$ is a $\mathcal{H}_{3}$-cleft object by [Gu, Theorem 8], since $\mathcal{A}_{2}/\langle\phi(Y_{2}^{+})\rangle=\mathcal{A}_{2}/\langle\mathtt{r}_{2}-\lambda_{2}\rangle\simeq\mathcal{A}_{3}$. The claim about the section $\gamma_{3}$ again follows from [A+, Proposition 5.8].

$\heartsuit$ For $k=4$, we check that ${\mathcal{E}}_{4}\neq 0$ (and a fortiori $\mathcal{A}_{4}\neq 0$) using GAP. Let $\mathtt{p}_{i}=y_{5-i\,14}-\omega y_{1i4}-\lambda_{2}y_{4}$, $i\in\mathbb{I}_{2,3}$. As $g_{1}\mathtt{p}_{2}g_{1}^{-1}=q_{1}\mathtt{p}_{3}$, we have that $\langle\mathtt{p}_{i}:i\in\mathbb{I}_{2,3}\rangle=\langle\mathtt{p}_{2}\rangle$. Hence $\mathcal{A}_{4}\simeq\mathcal{A}_{3}/\langle\mathtt{p}_{2}\rangle$.

Let $Y_{3}^{\prime}$ be the subalgebra of $\mathcal{H}_{3}$ generated by $p_{2}=x_{314}-\omega x_{124}$. Using GAP we check that $p_{2}^{6}\neq 0$. Thus, by [A+, Lemma 5.13], $Y_{3}^{\prime}$ is isomorphic to a polynomial ring in one variable since $p_{2}\in\mathcal{P}(\mathcal{B}_{3})_{g_{1}g_{2}g_{4}}-0$ and $g_{1}g_{2}g_{4}\cdot p_{2}=-\omega^{2}p_{2}$. As in [AAG] we set $Y_{3}=\mathcal{S}(Y_{3}^{\prime})$. Notice that $\mathcal{H}_{3}/\langle Y_{3}^{+}\rangle\simeq\mathcal{H}_{4}$. Since $Y_{3}$ is also a polynomial algebra generated by $p_{2}g_{4}^{-1}g_{2}^{-1}g_{1}^{-1}$, there exists an algebra map $\phi:Y_{3}\to\mathcal{A}_{3}$ such that $\phi(p_{2}g_{4}^{-1}g_{2}^{-1}g_{1}^{-1})=\mathtt{p}_{2}g_{4}^{-1}g_{2}^{-1}g_{1}^{-1}$, which is $\mathcal{H}_{3}$-colinear. Hence $\mathcal{A}_{4}$ is a $\mathcal{H}_{4}$-cleft object by [Gu, Theorem 8], since $\mathcal{A}_{3}/\langle\phi(Y_{3}^{+})\rangle=\mathcal{A}_{3}/\langle\mathtt{p}_{2}\rangle\simeq\mathcal{A}_{4}$. The claim about the section $\gamma_{4}$ again follows from [A+, Proposition 5.8]. ∎

Next we define

[TABLE]

Again, ${\mathcal{E}}(\boldsymbol{\lambda})$ is a $\Bbbk\Gamma$-module algebra, so we may consider $\mathcal{A}(\boldsymbol{\lambda}):={\mathcal{E}}(\boldsymbol{\lambda})\#\Bbbk\Gamma$.

Lemma 5.4**.**

$\mathcal{A}(\boldsymbol{\lambda})$* is a $\mathcal{H}$-cleft object. There exists an $\mathcal{H}$-colinear section $\gamma\colon\mathcal{H}\to\mathcal{A}$ which restricts to an algebra map $(\gamma)_{|\Bbbk\Gamma}\in\operatorname{Alg}(\Bbbk\Gamma,\mathcal{A})$.*

Proof.

Notice that $\mathcal{B}_{4}=\widetilde{\mathcal{B}}({\mathtt{HV}_{1}})$. We keep the notation from Remark 4.7: $z_{4}=x_{4}^{6}$, $z_{124134}=(x_{124}x_{134}+\omega^{2}x_{134}x_{124})^{3}$. Let $\mathcal{B}_{5}=\mathcal{B}_{4}/\langle z_{4}\rangle$, hence $\mathcal{B}({\mathtt{HV}_{1}})=\mathcal{B}_{5}/\langle z_{124134}\rangle$. We denote the canonical projections as follows:

[TABLE]

We proceed in two steps, one for each relation.

$\heartsuit$ By Lemma 4.9 and [A+, Proposition 3.6 (c)], $\Bbbk[z_{4}]=\mathcal{B}_{4}^{\operatorname{co}\pi_{1}}$, so $X^{\prime}_{4}:=\mathcal{H}_{4}^{\operatorname{co}\pi_{1}\#\operatorname{id}}=\Bbbk[z_{4}]\#1$; set $X_{4}=\mathcal{S}(X^{\prime}_{4})$. There exists an algebra map $\phi\colon X_{4}\to\mathcal{A}_{4}$, $\phi(z_{4}g_{4}^{-6})=y_{4}^{6}g_{4}^{-6}-\lambda_{3}g_{4}^{-6}$, which is $\mathcal{H}_{4}$-colinear. Working as in [AAG, Theorems 4.7 & 5.15] we check that this map is also $\mathcal{H}_{4}$-linear. Indeed, $y_{i}y_{4}^{6}=q_{1}^{6}y_{4}^{6}y_{i}$ for all $i\in\mathbb{I}_{3}$ since (3.19) is not deformed; hence, by (5.2),

[TABLE]

Let $\mathcal{H}_{5}:=\mathcal{B}_{5}\#\Bbbk\Gamma$, ${\mathcal{E}}_{5}:={\mathcal{E}}_{4}/\langle y_{4}^{6}-\lambda_{3}\rangle$, $\mathcal{A}_{5}:={\mathcal{E}}_{5}\#\Bbbk\Gamma$. By [Gu, Theorem 4] $\mathcal{A}_{5}$ is a $\mathcal{H}_{5}$-cleft object.

$\heartsuit$ By Lemma 4.9 and [A+, Proposition 3.6 (c)], $\Bbbk[z_{124134}]=\mathcal{B}_{5}^{\operatorname{co}\pi_{2}}$, so $X^{\prime}_{5}:=\mathcal{H}^{\operatorname{co}\pi_{2}\#\operatorname{id}}=\Bbbk[z_{124134}]\#1$; set $X_{5}=\mathcal{S}(X^{\prime}_{5})$. There exists an algebra map $\phi\colon X_{5}\to\mathcal{A}_{5}$, $\phi(z_{124134}g_{1}^{-12}g_{4}^{-6})=\gamma_{4}(z_{124134})g_{1}^{-12}g_{4}^{-6}-\lambda_{4}g_{1}^{-12}g_{4}^{-6}$, which is $\mathcal{H}_{5}$-colinear. We claim that this map is also $\mathcal{H}_{5}$-linear. Indeed, $\phi(g_{i}\cdot z_{124134}g_{1}^{-12}g_{4}^{-6})=g_{i}\cdot\phi(z_{124134}g_{1}^{-12}g_{4}^{-6})$ by (5.2). Set $\mathtt{y}_{124134}=\gamma_{4}(z_{124134})$. By Lemma 4.9 (a), $\phi(x_{i}\cdot z_{124134}g_{1}^{-12}g_{4}^{-6})=0$. Notice that

[TABLE]

By direct computation,

[TABLE]

hence $y_{i}\mathtt{y}_{124134}-\chi_{1}^{12}\chi_{4}^{6}(g_{i})\mathtt{y}_{124134}y_{i}=\mu$ for some $\mu_{i}\in\Bbbk$, $i\in\mathbb{I}_{4}$.

On the other hand, $\gamma_{4}$ is $\Gamma$-linear by [A+, Proposition 5.8 (c)]. If $i\in\mathbb{I}_{3}$,

[TABLE]

If $\mu_{i}\neq 0$ we get $-q_{1}^{2}=1=q_{2}^{5}\omega$, thus $\omega=\omega^{10}=(q_{1}q_{2})^{10}=-\omega$, a contradiction. Hence $\mu_{i}=0$ for $i\in\mathbb{I}_{3}$. Analogously, if $\mu_{4}\neq 0$, then $-q_{2}^{4}=1=q_{1}^{3}$, a contradiction since $q_{1}q_{2}\in\mathbb{G}^{\prime}_{6}$. Thus $\phi$ is $\mathcal{H}_{5}$-linear.

By [Gu, Theorem 4] $\mathcal{A}$ is a $\mathcal{H}$-cleft object. The claim about the section $\gamma$ follows from [A+, Proposition 5.8]. ∎

5.3. Computing the liftings

Let $\boldsymbol{\lambda}\in\mathcal{R}_{{\mathtt{HV}_{1}}}$ and define $\mathcal{L}_{1}=\mathcal{H}_{1}$, but we change the labels of the generators to $(a_{i})_{i\in\mathbb{I}_{4}}$,

[TABLE]

Notice that, in $\mathcal{L}_{2}(\boldsymbol{\lambda})$,

[TABLE]

The same happens for the other sets of relations in $\mathcal{L}_{3}(\boldsymbol{\lambda})$ and $\mathcal{L}_{4}(\boldsymbol{\lambda})$.

Lemma 5.5**.**

Let $\boldsymbol{\lambda}\in\mathcal{R}_{{\mathtt{HV}_{1}}}$, $i\in\mathbb{I}_{4}$. Then $\mathcal{A}_{i}(\boldsymbol{\lambda})$ is a $(\mathcal{L}_{i}(\boldsymbol{\lambda}),\mathcal{H}_{i})$-biGalois object.

Proof.

For $i=1,2,3$, $\mathcal{L}_{i}(\boldsymbol{\lambda})\simeq L(\mathcal{A}_{i}(\boldsymbol{\lambda}),\mathcal{H}_{i})$ by [A+, Corollary 5.12], since all the involved relations are skew-primitive elements of $\mathcal{T}$.

For $i=4$, we use Remark 4.1 to compute

[TABLE]

Hence $\mathcal{L}_{4}(\boldsymbol{\lambda})\simeq L(\mathcal{A}_{4}(\boldsymbol{\lambda}),\mathcal{H}_{4})$, again by [A+, Corollary 5.12]. ∎

Thanks to the previous Lemma and [A+, Corollary 5.12], there exists a $(g_{1}^{12}g_{4}^{6},1)$-primitive element $\mathtt{a}_{124134}\in\mathcal{L}_{4}$ 111We performed an algorithm in GAP to compute explicitly $\mathtt{a}_{124134}$ and $\gamma_{4}(z_{124134})$ but the program did not finish the computation. such that

[TABLE]

Next we define

[TABLE]

Theorem 5.6**.**

Let $\Gamma$ be a group with a principal realization of ${\mathtt{HV}_{1}}$ as in §3.2. Let $\boldsymbol{\lambda}\in\mathcal{R}_{{\mathtt{HV}_{1}}}$, see (5.2). Then

1. (a)

$\mathcal{L}(\boldsymbol{\lambda})\simeq L(\mathcal{A}(\boldsymbol{\lambda}),\mathcal{B}({\mathtt{HV}_{1}})\#\Bbbk\Gamma)$. 2. (b)

$\mathcal{L}(\boldsymbol{\lambda})$* is a lifting of $\mathcal{B}({\mathtt{HV}_{1}})$ over $\Bbbk\Gamma$.* 3. (c)

$\mathcal{L}(\boldsymbol{\lambda})$* is a cocycle deformation of $\mathcal{B}({\mathtt{HV}_{1}})\#\Bbbk\Gamma$.*

Conversely, if $L$ is lifting of $\mathcal{B}({\mathtt{HV}_{1}})$ over $\Bbbk\Gamma$, then there exist $\boldsymbol{\lambda}\in\mathcal{R}_{{\mathtt{HV}_{1}}}$ such that $L\simeq\mathcal{L}(\boldsymbol{\lambda})$.

Proof.

First, (a) follows again by [A+, Corollary 5.12] since $z_{4}$ and $z_{124134}$ are skew primitive in $\mathcal{H}_{4}$, and this implies that (c) holds. Now (b) is a consequence of Lemma 5.4 and [A+, Proposition 4.14. (c)].

Conversely, assume that $L$ is a lifting of $\mathcal{B}({\mathtt{HV}_{1}})$ over $\Bbbk\Gamma$. Let $\phi:\mathcal{T}=T({\mathtt{HV}_{1}})\#\Bbbk\Gamma\twoheadrightarrow L$ be a lifting map as in [AV, Proposition 2.4]. As in [AAG, Theorem 3.5], we shall attach to $\phi$ a family of scalars $\boldsymbol{\lambda}\in\mathcal{R}_{{\mathtt{HV}_{1}}}$ as follows. Since ${\mathtt{HV}_{1}}\#\Gamma\subset\sum_{i\in\mathbb{I}_{4},\,g\in\Gamma}\Bbbk g_{i}g\wedge\Bbbk g$, it follows that the first term of the coradical filtration of $L$ satisfies

[TABLE]

Let $r\in{\mathcal{G}}_{0}\cup{\mathcal{G}}_{1}\cup{\mathcal{G}}_{2}$. Since $r\in\mathcal{T}$ is $(g_{r},1)$-primitive, (5.5) implies that $\phi(r)\in\Bbbk(1-g_{r})+\Bbbk g_{r}\wedge\Bbbk\subset L_{1}$. By Lemma 5.1, $g_{r}\neq g_{i}$ for all $i\in\mathbb{I}_{4}$. Thus $\phi(r)=\lambda_{r}(1-g_{r})$ for some $\lambda_{r}\in\Bbbk$.

Let $r=a_{4}a_{h4}-q_{2}a_{h4}a_{4}\in{\mathcal{G}}_{0}$, for $h\in\mathbb{I}_{3}$. Suppose that $\lambda_{r}\neq 0$. Then

[TABLE]

which implies that $q_{1}q_{2}\in\mathbb{G}^{\prime}_{12}$, a contradiction. Hence $\lambda_{r}=0$.

Next we consider $a_{h}^{2}\in{\mathcal{G}}_{1}$: we denote $\phi(a_{h}^{2})=\lambda_{1h}(1-g_{h}^{2})$. For $h\in\mathbb{I}_{2,3}$,

[TABLE]

where we use that $g_{1}^{2}=g_{h}^{2}$. Hence $\lambda_{11}=\lambda_{1h}$, and we denote the common scalar simply by $\lambda_{1}$. A similar computation shows that $\lambda_{1}$ satisfies (5.2).

A similar argument shows that $\phi(a_{1}a_{2}+a_{3}a_{1}+a_{2}a_{3})=\lambda_{2}(1-g_{1}g_{2})$ for some $\lambda_{2}\in\Bbbk$ satisfying (5.2). Thus $\phi$ descends to a Hopf algebra map $\phi:\mathcal{L}_{3}(\boldsymbol{\lambda})\twoheadrightarrow L$ for any choice of $\lambda_{3}$ and $\lambda_{4}$.

Next, $a_{214}-\omega a_{134}-\lambda_{2}a_{4}\in\mathcal{L}_{3}(\boldsymbol{\lambda})$ is $(g_{1}g_{3}g_{4},1)$-primitive. By Lemma 5.1 there exists $\mu\in\Bbbk$ such that $\phi(a_{214}-\omega a_{134}-\lambda_{2}a_{4})=\mu(1-g_{1}g_{3}g_{4})$. Suppose that $\mu\neq 0$. Conjugation by $g_{1}$ and $g_{4}$ (as we applied for the previous relations) imply that $q_{1}=1$ and $-q_{2}^{2}\omega^{2}=1$ respectively, but this is a contradiction since $q_{1}q_{2}=-\omega$. Thus $\mu=0$ and $\phi$ descends to a Hopf algebra map $\phi:\mathcal{L}_{4}(\boldsymbol{\lambda})\twoheadrightarrow L$ for any choice of $\lambda_{3}$ and $\lambda_{4}$.

Finally, $a_{4}^{6}$ is $(g_{4}^{6},1)$-primitive and $\mathtt{a}_{124134}$ is $(g_{1}^{12}g_{4}^{6},1)$-primitive. An argument analogous to the previous relations shows that

[TABLE]

for some $\lambda_{3},\lambda_{4}\in\Bbbk$ satisfying (5.2). Thus $\phi$ descends to a Hopf algebra map $\phi:\mathcal{L}(\boldsymbol{\lambda})\twoheadrightarrow L$; as the restriction of $\phi$ to the first term $\mathcal{L}(\boldsymbol{\lambda})_{1}$ of the coradical filtration is injective, $\phi$ is an isomorphism by [Mo, Theorem 5.3.1]. ∎