Nichols algebras that are quantum planes

TL;DR

This paper classifies all 2-dimensional Nichols algebras with non-trivial quadratic relations, providing a comprehensive understanding of their structure as quantum planes.

Contribution

It explicitly computes and classifies all such Nichols algebras of dimension 2 with quadratic relations, advancing the understanding of their algebraic structure.

Findings

Complete classification of 2D Nichols algebras with quadratic relations

Identification of conditions for non-trivial quadratic relations

Contribution to the theory of quantum planes

Abstract

We compute all Nichols algebras of rigid vector spaces of dimension 2 that admit a non-trivial quadratic relation.

Tables

Table 1. Table 1. Nichols algebras of rank 2

Name	$R$-matrix	$ℬ\left(V\right)$	Ref.	Conditions
${\Re }_{2,1}$		(3.1)	§3.1	$k^2=-1$ or $pq=1$
${\Re }_{2,2}$		(3.6), (3.8)	§3.2	$k^2=-1$ or $pq=1$
${\Re }_{2,3}$		Table 2	§3.3	$k^2=1$
${\Re }_{1,1}$		Table 3	§3.4	$2p^2=-1$ or $2q^2=1$
${\Re }_{1,2}$		Tables 4, 5, 6, 7	§3.5	$p=-1$ or $q=1$
${\Re }_{1,3}$		Table 8	§3.6	$k^4=1$
${\Re }_{1,4}$		Table 9	§3.7	$k=-1$ or $pq=1$
${\Re }_{0,1}$		(3.39), (3.40)	§3.8	$k^2=1$

Table 2. Table 2. Nichols algebras of type ℜ 2 , 3 subscript ℜ 2 3 \mathfrak{R}_{2,3}

$k$	$p$	$s$	$𝒥\left(V\right)$	Basis	$\mathrm{GK}-\dim$
$-1$	$-q$	$q^2$	$⟨x_1^2,x_2^2-q{x}_1{x}_2,x_1{x}_2+x_2{x}_1⟩$	${\left({\ast }_{2,3}\right)}_1$	$0,dim=4$
		$\ne q^2$	$⟨x_1^2,x_1{x}_2+x_2{x}_1⟩$	${\left({\ast }_{2,3}\right)}_2$	$1$
	$\ne -q$		$⟨x_1^2,x_2{x}_{21}+\left(p-q\right)x_1{x}_{21}-x_{21}{x}_2⟩$	${\left({\ast }_{2,3}\right)}_3$	$2$
$1$			$⟨\frac{q-p}{2}{x}_1^2-x_1{x}_2+x_2{x}_1⟩$	${\left({\ast }_{2,3}\right)}_4$	$2$

Table 3. Table 3. Nichols algebras of type ℜ 1 , 1 subscript ℜ 1 1 \mathfrak{R}_{1,1}

$2p^2$	$2q^2$	$-2pq$	$𝒥\left(V\right)$	Basis	$\mathrm{GK}-\dim$
$-1$	$1$		$⟨r_1,r_2,r_3⟩$	${\left({\ast }_{1,1}\right)}_1$	$0$, $dim=4$
$\ne -1$	$1$	$\in {𝔾}_N^{\prime },N\ge 3$, $N\ne 4$	$⟨r_1,r_2,r_{4,N}⟩$	${\left({\ast }_{1,1}\right)}_2$	$0$, $dim=2N$
		$\notin {𝔾}_{\infty }$	$⟨r_1,r_2⟩$	${\left({\ast }_{1,1}\right)}_3$	$1$
$-1$	$\ne 1$	$\in {𝔾}_N^{\prime },N\ge 3$, $N\ne 4$	$⟨r_1,r_3,r_{4,N}⟩$	${\left({\ast }_{1,1}\right)}_2$	$0$, $dim=2N$
		$\notin {𝔾}_{\infty }$	$⟨r_1,r_3⟩$	${\left({\ast }_{1,1}\right)}_3$	$1$

Table 4. Table 4. Nichols algebras of type ℜ 1 , 2 subscript ℜ 1 2 \mathfrak{R}_{1,2}

$p$	$-q$	$k$	$𝒥\left(V\right)$	Basis	$\mathrm{GK}-\dim$
$-1$	$-1$	$0$	$⟨x_1^2,x_2^2,r_1⟩$	${\left({\ast }_{1,2}\right)}_1$	$0$, $dim=4$
	$\in {𝔾}_N^{\prime },N\ge 3$		$⟨x_1^2,x_2^N,r_1⟩$	${\left({\ast }_{1,2}\right)}_1$	$0$, $dim=2N$
	otherwise		$⟨x_1^2,r_1⟩$	${\left({\ast }_{1,2}\right)}_2$	$1$
$\in {𝔾}_M^{\prime },M\ge 3$	$-1$		$⟨x_1^M,r_1,r_2⟩$	${\left({\ast }_{1,2}\right)}_3$	$0$, $dim=2M$
$\notin {𝔾}_{\infty }^{\prime }$			$⟨r_1,r_2⟩$	${\left({\ast }_{1,2}\right)}_2$	$1$

Table 5. Table 5. Nichols algebras of type ℜ 1 , 2 ​ ( a ) subscript ℜ 1 2 𝑎 \mathfrak{R}_{1,2}\ (a)

$p$	$-q$	$k$	$𝒥\left(V\right)$	Basis	$\mathrm{GK}-\dim$
$-1$	$-1$	$0$	$⟨x_1^2,x_2^2,r_1⟩$	${\left({\ast }_{1,2}\right)}_1$	$0$, $dim=4$
		$\ne 0$	$⟨r_1,r_2⟩$	${\left({\ast }_{1,2}\right)}_4$	$1$
	$\in {𝔾}_N^{\prime },N\ge 3$		$⟨x_2^N,r_1,r_2⟩$	${\left({\ast }_{1,2}\right)}_1$	$0$, $dim=2N$
	$\notin {𝔾}_{\infty }^{\prime }$		$⟨r_1,r_2⟩$	${\left({\ast }_{1,2}\right)}_2$	$1$
$\in {𝔾}_M^{\prime },M\ge 3$	$-1$		$⟨x_1^M,x_2^2,r_1⟩$	${\left({\ast }_{1,2}\right)}_3$	$0$, $dim=2M$
$\notin {𝔾}_{\infty }^{\prime }$			$⟨x_2^2,r_1⟩$	${\left({\ast }_{1,2}\right)}_4$	$1$

Table 6. Table 6. Nichols algebras of type ℜ 1 , 2 ​ ( c ) subscript ℜ 1 2 𝑐 \mathfrak{R}_{1,2}\ (c)

$p$	$-q$	$k$	$𝒥\left(V\right)$	Basis	$\mathrm{GK}-\dim$
$-1$	$-1$	$0$	$⟨x_1^2,x_2^2,r_1⟩$	${\left({\ast }_{1,2}\right)}_1$	$0$, $dim=4$
	$\in {𝔾}_N^{\prime },N\ge 3$		$⟨x_1^2,x_2^N,r_1⟩$	${\left({\ast }_{1,2}\right)}_1$	$0$, $dim=2N$
	otherwise		$⟨x_1^2,r_1⟩$	${\left({\ast }_{1,2}\right)}_2$	$1$
$\in {𝔾}_M^{\prime },M\ge 3$	$-1$		$⟨x_1^M,r_1,r_2⟩$	${\left({\ast }_{1,2}\right)}_3$	$0$, $dim=2M$
$\notin {𝔾}_{\infty }^{\prime }$			$⟨r_1,r_2⟩$	${\left({\ast }_{1,2}\right)}_4$	$1$

Table 7. Table 7. Nichols algebras of type ℜ 1 , 2 ​ ( a ​ c ) subscript ℜ 1 2 𝑎 𝑐 \mathfrak{R}_{1,2}\ (ac)

$p$	$-q$	$k$	$𝒥\left(V\right)$	Basis	$\mathrm{GK}-\dim$
$-1$	$-1$	$0$	$⟨x_1^2,x_2^2,r_1⟩$	${\left({\ast }_{1,2}\right)}_1$	$0$, $dim=4$
		$\ne 0$	$⟨r_1,r_2⟩$	${\left({\ast }_{1,2}\right)}_4$	$1$
	$\in {𝔾}_N^{\prime },N\ge 3$		$⟨x_2^N,r_1,r_2⟩$	${\left({\ast }_{1,2}\right)}_1$	$0$, $dim=2N$
	$\notin {𝔾}_{\infty }^{\prime }$		$⟨r_1,r_2⟩$	${\left({\ast }_{1,2}\right)}_2$	$1$
$\in {𝔾}_M^{\prime },M\ge 3$	$-1$		$⟨x_1^M,x_2^2,r_1⟩$	${\left({\ast }_{1,2}\right)}_3$	$0$, $dim=2M$
$\notin {𝔾}_{\infty }^{\prime }$			$⟨x_2^2,r_1⟩$	${\left({\ast }_{1,2}\right)}_4$	$1$

Table 8. Table 8. Nichols algebras of type ℜ 1 , 3 subscript ℜ 1 3 \mathfrak{R}_{1,3}

$k^2$	$𝒥\left(V\right)$	Basis	$\mathrm{GK}-\dim$
$-1$	$⟨x_1^2,x_2^2-kq{x}_1{x}_2,x_1{x}_2+x_2{x}_1⟩$	${\left({\ast }_{1,3}\right)}_1$	$0$, $dim=4$
$1$	$⟨x_1{x}_2-x_2{x}_1+kp{x}_1^2⟩$	${\left({\ast }_{1,3}\right)}_2$	$2$

Table 9. Table 9. Nichols algebras of type ℜ 1 , 4 subscript ℜ 1 4 \mathfrak{R}_{1,4}

$k$	$pq$	$𝒥\left(V\right)$	Basis	$\mathrm{GK}-\dim$
$-1$	$\in {𝔾}_N^{\prime },N\ge 1$	$⟨r_{1,2},r_{2,2},r_{3,N}⟩$	${\left({\ast }_{1,4}\right)}_1$	$0$, $dim=4N$
	$\notin {𝔾}_{\infty }$	$⟨r_{1,2},r_{2,2}⟩$	${\left({\ast }_{1,4}\right)}_2$	$1$
$\in {𝔾}_N^{\prime },N\ge 3$	$1$	$⟨r_{1,N},r_{2,N},r_{3,1}⟩$	${\left({\ast }_{1,4}\right)}_3$	$0$, $dim=N^2$
$\notin {𝔾}_{\infty }^{\prime }$		$⟨r_{3,1}⟩$	${\left({\ast }_{1,4}\right)}_4$	$2$

Table 10. Table 10. Algebras arising as Nichols algebras of rank 2

Algebra	$𝒥$	Parameters
quantum plane	$⟨y_1{y}_2-\zeta y_2{y}_1⟩$	$\zeta \in {𝕜}^{\times }$
quantum plane	$⟨y_1^M,y_1{y}_2-\zeta y_2{y}_1⟩$	$\zeta \in {𝕜}^{\times }$, $M\in {\mathbb{N}}_{\ge 2}$
quantum plane	$⟨y_1^M,y_1{y}_2-\zeta y_2{y}_1,y_2^N⟩$	$\zeta \in {𝕜}^{\times }$, $M,N\in {\mathbb{N}}_{\ge 2}$
deformation of a quantum plane	$⟨y_2^2-\zeta y_1^2,y_1{y}_2-\eta y_2{y}_1,y_1^N⟩$	$\eta ,\zeta \in {𝕜}^{\times }$, $N\in {\mathbb{N}}_{\ge 3}$
deformation of a quantum plane	$⟨y_2^2-\zeta y_1^2,y_1{y}_2-\eta y_2{y}_1⟩$	$\eta ,\zeta \in {𝕜}^{\times }$
deformation of an exterior algebra	$⟨y_1^2,y_2^2-y_1{y}_2,y_1{y}_2+y_2{y}_1⟩$
	$⟨y_1^2-\zeta y_2^2,y_1{y}_2+ϵy_2{y}_1,y_1{y}_2^N⟩$	$\zeta \in {𝕜}^{\times }$, $ϵ\in {𝔾}_2$, $N\in {\mathbb{N}}_{\ge 2}$
Jordan plane	$⟨y_1^2-y_1{y}_2+y_2{y}_1⟩$
super Jordan plane	$⟨y_1^2,y_2^2{y}_1-y_1{y}_2{y}_1-y_1{y}_2^2⟩$
	$⟨y_1{y}_2,y_2{y}_1⟩$
	$⟨y_1{y}_2,y_2{y}_1,y_1^{2N}-\zeta y_2^{2N}⟩$	$\zeta \in {𝕜}^{\times }$, $N\in \mathbb{N}$
	$⟨y_1^2-\zeta y_2^2⟩$	$\zeta \in {𝕜}^{\times }$
	$⟨y_1^2-\zeta y_2^2,r_{1,N},r_{2,N}⟩$, cf. (3.35), (3.36)	$\zeta \in {𝕜}^{\times }$, $N\in {\mathbb{N}}_{\ge 3}$

Equations

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

$$c(v_i\otimes v_j)$$

$$x_i{x}_j$$

$$x_i^{N_i}$$

$$\{x_1^{a_1}\dots x_n^{a_n}:$$

$$=p^2-q^2,$$

$$GK-dim\mathcal{B}(V)\#H=GK-dim\mathcal{B}(V)+GK-dimH.$$

$$(n{)}_q$$

$$\rho (h\cdot v)$$

$$c_{V,W}(v\cdot w)$$

$${\mathcal{J}}^2(V)$$

$$\begin{array}{rlrl}{\partial }_f(1)& =0,& & {\partial }_f(v)=f(v),\forall v\in V,\\ {\partial }_f(xy)& ={\partial }_f(x)y+\sum _i{x}_i{\partial }_{f_i}(y),& & \text{where }{c}^{-1}(f\otimes x)=\sum _i{x}_i\otimes f_i.\end{array}$$

$${\partial }_f(x)=0\text{ for all }f\in V^{\ast }⟹x\in \mathcal{J}(V).$$

$$R_{12}{R}_{13}{R}_{23}$$

$$R\stackrel{\star }{⟷}c$$

$$R\mapsto R^{\#}$$

$$⟨f\otimes g,v\otimes w⟩$$

$$(c(x_i\otimes x_j){)}_{i,j\in {\mathbb{I}}_2}=\left(\begin{array}{cc}{k}^2{x}_1\otimes x_1& kq{x}_2\otimes x_1+(k^2-pq)x_1\otimes x_2\\ kp{x}_1\otimes x_2& k^2{x}_2\otimes x_2\end{array}\right).$$

$$\begin{array}{rlrl}c(z_0\otimes z_0)& =-z_0\otimes z_0,& c(z_0\otimes z_1)& =-q_{21}(z_1+f_0)\otimes z_0,\\ c(z_0\otimes f_0)& =-q_{21}{f}_0\otimes z_0,& c(z_0\otimes f_1)& =-q_{21}^2{f}_1\otimes z_0,\end{array}$$

$$\begin{array}{rlrl}c(z_1\otimes z_0)& =-q_{12}{z}_0\otimes z_1-(2z_1+f_0)\otimes z_0,& c(z_1\otimes z_1)& =-z_1\otimes z_1,\\ c(z_1\otimes f_0)& =-f_0\otimes z_1+2q_{21}{f}_1\otimes z_0,& c(z_1\otimes f_1)& =q_{21}{f}_1\otimes z_1,\\ c(f_0\otimes z_0)& =-q_{12}{z}_0\otimes f_0-2f_0\otimes z_0,& c(f_0\otimes f_0)& =-f_0\otimes f_0,\\ c(f_0\otimes z_1)& =-z_1\otimes f_0-2q_{21}{f}_1\otimes z_0,& c(f_0\otimes f_1)& =q_{21}{f}_1\otimes f_0,\end{array}$$

$$\begin{array}{rl}c(f_1\otimes z_0)& =-q_{12}^2{z}_0\otimes f_1-2q_{12}{f}_0\otimes z_1+2q_{12}{z}_1\otimes f_0,\\ c(f_1\otimes z_1)& =q_{12}(z_1-f_0)\otimes f_1+f_1\otimes (f_0-2z_1),\end{array}$$

$$\begin{array}{rlrl}c(f_1\otimes f_0)& =q_{12}{f}_0\otimes f_1-2f_1\otimes f_0,& c(f_1\otimes f_1)& =-f_1\otimes f_1.\end{array}$$

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

$$r_i$$

$$GK-dim\mathcal{B}(V)=\{\begin{array}{ll}0& \text{if }N<\infty (\text{and }\dim \mathcal{B}(V)=N^2),\\ 2& \text{otherwise.}\end{array}$$

$$\Delta (u)$$

$$+({\lambda }_3+{\lambda }_{2}kq)x_2\otimes x_1+({\lambda }_{3}kp+{\lambda }_2(1+k^2-pq))x_1\otimes x_2.$$

$$r=\sum _{M_2\le a\le M_1}{c}_a{x}_2^a{x}_1^{n-a},$$

$$\Delta (r)$$

$$=\sum _{M_2\le a\le M_1}\sum _{i=0}^a\sum _{j=0}^{n-a}{c}_a{\left(\genfrac{}{}{0ex}{}{a}{i}\right)}_{k^2}{\left(\genfrac{}{}{0ex}{}{n-a}{j}\right)}_{k^2}(kp{)}^{(a-i)j}{x}_2^i{x}_1^j\otimes x_2^{a-i}{x}_1^{n-a-j}.$$

$${\partial }_1(r)$$

$${\partial }_2(r)$$

$$c_a(n-a{)}_{k^2}$$

$$c_a(a{)}_{k^2}$$

$$(c^{\prime }(x_i\otimes x_j){)}_{i,j\in {\mathbb{I}}_2}=\left(\begin{array}{cc}{k}^2{x}_1\otimes x_1& kp{x}_2\otimes x_1\\ kq{x}_1\otimes x_2+(k^2-pq)x_2\otimes x_1& k^2{x}_2\otimes x_2\end{array}\right).$$

Full text

2000 Mathematics Subject Classification. 16W30.

The work was partially supported by CONICET, Secyt (UNC), the MathAmSud project GR2HOPF, CNPq (Brazil)

1. Introduction

1.1.

Let $V$ be a vector space. We say that $c\in\operatorname{End}(V\otimes V)$ satisfies the braid equation if

[TABLE]

If in addition $c$ is invertible, then we say that $(V,c)$ is a braided vector space and that $c$ is a braiding. Assume that $\dim V<\infty$ and let $(v_{i})$ be a basis of $V$ and $(v^{i})$ its dual basis. Then $c$ (or else $V$) is rigid if the map $c^{\flat}:V^{*}\otimes V\to V\otimes V^{*}$ given by $f\otimes v\longmapsto\sum_{i}(\operatorname{ev}\otimes\operatorname{id}\otimes\operatorname{id})(f\otimes c(v\otimes v_{i})\otimes v^{i})$ is invertible.

Let $(V,c)$ be a rigid braided vector space and let ${\mathcal{B}}(V)$ be its Nichols algebra, see §2.2 for details. An important problem is to determine the presentation and structure of ${\mathcal{B}}(V)$, particularly when it has finite dimension, or Gelfand-Kirillov dimension. For example, let $n\in\mathbb{N}$ and $\mathbb{I}=\mathbb{I}_{n}=\{1,2,\dots,n\}$; let $\mathbf{q}=(q_{ij})_{i,j\in\mathbb{I}}\in(\Bbbk^{\times})^{\mathbb{I}\times\mathbb{I}}$; let $V$ be a vector space with a basis $(v_{i})_{i\in\mathbb{I}}$; and let $c\in\operatorname{End}(V\otimes V)$ be given by

[TABLE]

Then $(V,c)$ is a rigid braided vector space, called of diagonal type. The classification of the finite-dimensional Nichols algebras of diagonal type is known [He]. The following particular case was studied in [AS1]. Assume that $q_{ij}q_{ji}=1$ for all $i\neq j\in\mathbb{I}$ and set $N_{i}=\begin{cases}\operatorname{ord}q_{ii},&\text{if }q_{ii}\neq 1;\\ \infty,&\text{if }q_{ii}=1.\end{cases}$, $i\in\mathbb{I}$. Then ${\mathcal{B}}(V)$ is presented by generators $x_{i}$, $i\in\mathbb{I}$, with relations

[TABLE]

In particular $\dim{\mathcal{B}}(V)=\prod_{i\in\mathbb{I}}N_{i}$ is finite if and only if all $N_{i}$ are finite. Also, $\operatorname{GK-dim}{\mathcal{B}}(V)=|\{i\in\mathbb{I}:N_{i}=\infty\}|$.

An algebra $A$ presented by generators $x_{i}$, $i\in\mathbb{I}$, with relations (1.2), (1.3) is known as a quantum linear space, or a quantum plane when $n=2$. It is well-known that for any quantum linear space $A$, we have

[TABLE]

1.2.

In [GGi, Propositions 4.8, 4.9], there were found braided vector spaces $(V,c)$ of dimension 2 not of diagonal type but such that ${\mathcal{B}}(V)$ are quantum planes with $N_{1}=2$ and $N_{2}=4$. Notice that the coproduct of a Nichols algebra is determined by the braiding $c$, and vice versa the algebra and coalgebra structures determine the braiding [S2]; thus these Nichols algebras are not isomorphic to Nichols algebras of diagonal type as a braided Hopf algebras. Consequently, the following question arises naturally: classify all Nichols algebras (of rigid braided vector spaces) that are isomorphic to quantum linear spaces as algebras. In the present paper we solve this question for quantum planes; we recover the examples in [GGi] in Remarks 3.2 and 3.9.

More generally, we consider braided vector spaces of dimension 2 and compute the quadratic relations of their Nichols algebras, which is straightforward. Roughly speaking, there are four possible scenarios:

1. (A)

There are no quadratic relations. We plan to address (at least some of) these in the future. 2. (B)

There is a quadratic relation close enough to (1.2) that, together with (1.3) for suitable $N_{1}$ and $N_{2}$, is a presentation of ${\mathcal{B}}(V)$. Furthermore there is a PBW-basis (1.4). 3. (C)

There are strange examples with quadratic relations and more relations in higher degree with basis of a very specific type. 4. (D)

There is a unique quadratic relation $x_{1}^{2}=0$ and an extra cubic relation that together present ${\mathcal{B}}(V)$. The super Jordan plane [AAH] fits here.

Here is our main result:

Theorem 1.1**.**

Let $(V,c)$ be a rigid braided vector space of dimension 2, not of diagonal type. If ${\mathcal{B}}(V)$ has quadratic relations, then $(V,c)$ is as in Table 1. Furthermore, the explicit presentation of ${\mathcal{B}}(V)$, a PBW-basis, the dimension and the GK-dimension are given.

In Table 1, the first column follows the conventions of [Hi], cf. §2.3; the third provides the defining relations, while the fourth contains the reference to the proof and the fifth, the conditions needed to have quadratic relations. We use for $\mathfrak{R}_{1,1}$ the notation

[TABLE]

1.3.

We discuss some features of the outcome of Theorem 1.1. First, in all cases where at least one non-trivial quadratic relation exists, the Nichols algebra can be determined. This is neatly different from the diagonal case. Indeed, consider a braided vector space $V$ of diagonal type with generalized Dynkin diagram

\textstyle{\circ^{q}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p}$$\textstyle{\circ^{-1}} (see [He] for details). Then ${\mathcal{B}}(V)$ has the quadratic relation $x_{2}^{2}=0$ but to the best of our knowledge the complete set of defining relations is in general not known.

Second, the list of (isomorphism classes of) algebras underlying the Nichols algebras in our classification is rather short, see the Appendix, and the same algebra underlies various different Nichols algebras, as already present in the motivating examples from [GGi]. This suggests relations between the different Nichols algebras, e.g. twisting.

1.4.

Let $(V,c)$ be a rigid braided vector space and let $R=\tau c$ be the corresponding solution of the QYBE, cf. §2.3. For applications of the determination of ${\mathcal{B}}(V)$, it is necessary to find Hopf algebras $H$ that realize $V$ in ${}^{H}_{H}\mathcal{YD}$, see §2.1. Indeed, ${\mathcal{B}}(V)$ becomes a Hopf algebra in ${}^{H}_{H}\mathcal{YD}$ and the bosonization ${\mathcal{B}}(V)\#H$ is a Hopf algebra that likely inherits properties of ${\mathcal{B}}(V)$ and $H$. For instance, under some mild conditions, cf. [AAH, Lemma 2.2],

[TABLE]

Notice that $\operatorname{GK-dim}{\mathcal{B}}(V)\leq 2$ for all $V$ in Table 1.

Now, if $H$ is a co-quasitriangular Hopf algebra, then the category $\mathcal{M}^{H}$ of right $H$-comodules embed into ${}^{H}_{H}\mathcal{YD}$ [T2, (3.4)]. Thus we may seek for co-quasitriangular Hopf algebras $H$ that realize $(V,c)$ in $\mathcal{M}^{H}$. The celebrated FRT-construction [FRT] provides a universal co-quasitriangular bialgebra $A(R)$ realizing $(V,c)$; see [T2] for details and more references. When $\dim V=2$, $A(R)$ was computed explicitly in [ACDM1, ACDM2, FHR, T1].

Further, the rigidity of $c$ is equivalent to the existence of a Hopf algebra $H(R)$ realizing $(V,c)$ as a $H(R)$-comodule [S1, Ha]. The construction of $H(R)$ is simpler when $A(R)$ has a group-like element $g$ with favourable properties (the so-called quantum determinant) so that $H(R)$ is the localization of $A(R)$ at $g$ [FRT, T1]; but otherwise is involved. Particularly, it is not evident to us whether $\operatorname{GK-dim}H(R)<\infty$ if $\operatorname{GK-dim}A(R)<\infty$, except when localizing at the quantum determinant. When $\dim V=2$, $H(R)$ is a localization of $A(R)$ by the quantum determinant for $\mathfrak{R}_{2,1}$ [T1], but otherwise it is not known, apparently.

Notation

If $j\leq k\in\mathbb{N}_{0}$, then we denote $\mathbb{I}_{j,k}=\{j,j+1,\dots,k\}$ and $\mathbb{I}_{k}=\mathbb{I}_{1,k}$. We denote by $\mathbb{G}_{N}$, $N\geq 1$, the group of $N$-th roots of unity and by $\mathbb{G}_{N}^{\prime}\subset\mathbb{G}_{N}$ the subset of primitive roots. We also set $\mathbb{G}_{\infty}=\bigcup_{N\geq 1}\mathbb{G}_{N}$, $\mathbb{G}^{\prime}_{\infty}=\mathbb{G}_{\infty}-\{1\}$. Given $n\in\mathbb{N}$, $0\leq i\leq n$ and $q\in\Bbbk$, we consider the $q$–numbers

[TABLE]

The multiplication of an algebra $A$ is denoted $\mu$, or $\mu_{A}$ when emphasis is needed.

Acknowledgments

We thank Iván Angiono, Juan Cuadra and Leandro Vendramin for fruitful conversations at different stages of our research.

2. Preliminaries

2.1. Yetter–Drinfeld modules

For more information on the material in this Subsection, see e.g. [AHS, AS2]. Let $H$ be a Hopf algebra. A Yetter-Drinfeld module $V$ over $H$ is simultaneously a left $H$-module and a left $H$-comodule via $\rho$ satisfying the compatibility

[TABLE]

The category of Yetter-Drinfeld modules over $H$, denoted by ${}^{H}_{H}\mathcal{YD}$, is a braided tensor one: the tensor product of $V,W\in{}^{H}_{H}\mathcal{YD}$ has the natural action and coaction, while the braiding is given by

[TABLE]

Thus, every $V\in{}^{H}_{H}\mathcal{YD}$ becomes a rigid braided vector space $(V,c_{V,V})$; but a rigid braided vector space could be realized in ${}^{H}_{H}\mathcal{YD}$ for many possible $H$. As every braided tensor category, ${}^{H}_{H}\mathcal{YD}$ has some special features:

• $\diamondsuit$

Algebras in ${}^{H}_{H}\mathcal{YD}$ are objects in ${}^{H}_{H}\mathcal{YD}$ that are also associative algebras, with the unit and multiplication maps being morphisms in ${}^{H}_{H}\mathcal{YD}$. Same for coalgebras.

• $\diamondsuit$

The tensor product of two algebras $A$ and $B$ in ${}^{H}_{H}\mathcal{YD}$ is the object $A\otimes B\in{}^{H}_{H}\mathcal{YD}$ with the multiplication $\mu_{A\otimes B}=(\mu_{A}\otimes\mu_{B})(\operatorname{id}_{A}\otimes c_{B,A}\otimes\operatorname{id}_{B})$. It is denoted $A\underline{\otimes}B$.

• $\diamondsuit$

A bialgebra in ${}^{H}_{H}\mathcal{YD}$ is an algebra and coalgebra $R$ in ${}^{H}_{H}\mathcal{YD}$ with the comultiplication $\Delta:R\to R\underline{\otimes}R$ and the counit being morphisms of algebras.

• $\diamondsuit$

A Hopf algebra in ${}^{H}_{H}\mathcal{YD}$ is a bialgebra whose identity is convolution-invertible (the inverse is called the antipode).

Here is a distinguished fact of ${}^{H}_{H}\mathcal{YD}$: if $R$ is a Hopf algebra in ${}^{H}_{H}\mathcal{YD}$, then $R\#H=$ the vector space $R\otimes H$ with the semi-direct product and semi-direct coproduct structures is again a Hopf algebra called the (Radford-Majid) bosonization of $R$ by $H$.

2.2. Nichols algebras

Let $V\in{}^{H}_{H}\mathcal{YD}$. The tensor algebra $T(V)$ has a natural structure of (graded) Hopf algebra in ${}^{H}_{H}\mathcal{YD}$, where the comultiplication is the algebra map $\Delta:T(V)\to T(V)\underline{\otimes}T(V)$ given by $\Delta(v)=v\otimes 1+1\otimes v$, $v\in V$. Let $\mathfrak{C}$ be the set of Hopf ideals of $T(V)$ that are generated by homogeneous elements of degree $\geq 2$ and subobjects in ${}^{H}_{H}\mathcal{YD}$.

Definition 2.1**.**

The Nichols algebra ${\mathcal{B}}(V)$ is the quotient of the $T(V)$ by the largest ideal ${\mathcal{J}}(V)$ in $\mathfrak{C}$.

Thus, if $I\in\mathfrak{C}$, then $\mathfrak{B}=T(V)/I$ is a graded Hopf algebra in ${}^{H}_{H}\mathcal{YD}$ and there is a natural epimorphism $\mathfrak{B}\to{\mathcal{B}}(V)$; such $\mathfrak{B}$ are called pre-Nichols algebras over $V$.

We emphasize ${\mathcal{B}}(V,c)$, ${\mathcal{J}}(V,c)$, if needed. As a coalgebra, ${\mathcal{B}}(V)$ depends only on the braided vector space $(V,c)$ and not on the realization in ${}^{H}_{H}\mathcal{YD}$. The ideal ${\mathcal{J}}(V)=\oplus_{n\geq 2}{\mathcal{J}}^{n}(V)$ has alternative descriptions. For instance, ${\mathcal{J}}^{n}(V)$ is the kernel of the quantum symmetrizer associated to $c$; in particular

[TABLE]

Left skew derivations are also quite useful. Let $f\in V^{*}$ and consider $\partial^{L}_{f}=\partial_{f}\in\operatorname{End}T(V)$ given by

[TABLE]

Let $\mathfrak{B}$ be a pre-Nichols algebra over $V$. Then for every $f\in V^{*}$, there is $\partial_{f}\in\operatorname{End}\mathfrak{B}$ satisfying (2.2), compatible with the projection $T(V)\to\mathfrak{B}$. This was proved in [AAH] albeit previous partial formulations appeared much before.

We now outline the method to determine Nichols algebras, already used in many papers.

First, one looks for relations in ${\mathcal{J}}(V)$: the quadratic ones are given by (2.1); if $c(v\otimes v)=qv\otimes v$ for some $q\in\mathbb{G}^{\prime}_{n}$, $N\geq 2$, then $v^{N}\in{\mathcal{J}}(V)$; for more relations the following fact is convenient: Let $x\in T(V)$. Then

[TABLE]

So, one disposes of a collection of homogeneous relations $(r_{\ell})_{\ell\in\Lambda}$. Let $I=\langle r_{\ell}:\ell\in\Lambda\rangle$. We may assume that $I$ is a Hopf ideal, since usually the $r_{\ell}$’s are either primitive, or primitive modulo other relations.

Second, one considers the pre-Nichols algebra $\mathfrak{B}=T(V)/I$ and the natural projection $\pi:\mathfrak{B}\twoheadrightarrow{\mathcal{B}}(V)$. Assume that there is $0\neq r\in\ker\pi$ that we choose homogeneous of the lowest degree. Then $\partial_{f}(r)\in\ker\pi$, hence $\partial_{f}(r)=0$ by minimality of the degree. If this leads to a contradiction, then $\pi$ is actually an isomorphism.

2.3. The braid equation in rank 2

We say that $R\in GL(V\otimes V)$ satisfies the quantum Yang-Baxter equation (QYBE)

[TABLE]

Let $\tau:V\otimes V\to V\otimes V$ be the usual flip, $\tau(x\otimes y)=y\otimes x$.

The QYBE (2.4) is equivalent to the braid equation (1.1) in two ways:

[TABLE]

Therefore, we have involutions of the set of solutions of the QYBE (2.4), respectively the braid equation (1.1), given by

[TABLE]

Obviously, $c=\tau R$ iff $c=R^{\#}\tau$, etc.

2.3.1. Classification

Assume that $\dim V=2$. The classification of the solutions of (2.4) was performed by J. Hietarinta in [Hi], up to the equivalence generated by the following relations, listed exactly as in loc. cit.:

1. (a)

$R\leftrightarrow R^{t}$ (the transpose). 2. (b)

The change of basis $R\mapsto(F\otimes F)R(F\otimes F)^{-1}$, $F\in GL(V)$ given by $x_{1}\mapsto x_{2}$, $x_{2}\mapsto x_{1}$; a particular case of (e) below. 3. (c)

A change of basis given by $x_{1}\otimes x_{1}\mapsto x_{1}\otimes x_{1}$, $x_{2}\otimes x_{1}\mapsto x_{1}\otimes x_{2}$, $x_{1}\otimes x_{2}\mapsto x_{2}\otimes x_{1}$, $x_{2}\otimes x_{2}\mapsto x_{2}\otimes x_{2}$. This is just the assignment $R\mapsto R^{\#}$ defined above. 4. (d)

Homotheties $R\mapsto\kappa R$, $\kappa\in\Bbbk^{\times}$. 5. (e)

The change of basis $R\mapsto(\varphi\otimes\varphi)R(\varphi\otimes\varphi)^{-1}$, $\varphi\in GL(V)$.

Thus, we proceed case-by-case following the classification in [Hi] up to $\star$. It turns out that there are 23 families of solutions, numbered $(\mathfrak{R}_{0,j})_{j\in\mathbb{I}_{6}}$, $(\mathfrak{R}_{1,j})_{j\in\mathbb{I}_{12}}$, $(\mathfrak{R}_{2,j})_{j\in\mathbb{I}_{4}}$ and $\mathfrak{R}_{3,1}$. We are interested in those invertible and rigid; we also exclude the diagonal braidings. The remaining solutions are listed in Table 1. Notice that we homogeneize $\mathfrak{R}_{0,1}$ from [Hi] for the discussion in §2.3.2 below.

Remark 2.2*.*

The $R$-matrices $(\mathfrak{R}_{0,j})_{j=4,5,6}$, $(\mathfrak{R}_{1,j})_{j=5,\dots,12}$ and $\mathfrak{R}_{2,4}$ are not invertible; $(\mathfrak{R}_{0,j})_{j=2,3}$ are invertible but not rigid; $\mathfrak{R}_{3,1}$ is of diagonal type.

We shall denote by $\mathbf{V}_{i,j}$ the braided vector space corresponding to $R$-matrix $\mathfrak{R}_{i,j}$. We have fixed a basis $x_{1},x_{2}$ of $\mathbf{V}_{i,j}$ and the $R$-matrix is described in the ordered basis $x_{1}\otimes x_{1}$, $x_{2}\otimes x_{1}$, $x_{1}\otimes x_{2}$, $x_{2}\otimes x_{2}$ of $\mathbf{V}_{i,j}\otimes\mathbf{V}_{i,j}$. In Section 3 we determine the quadratic relations of these 8 families and describe the Nichols algebras of them, when there are quadratic relations.

2.3.2. Behaviour of Nichols algebras up to equivalence

It remains to discuss the relations between the Nichols algebras under the equivalence in [Hi]. Let $R$, $R^{\prime}$ be solutions of (2.4) on $V$ and $V^{\prime}$ respectively (where $\dim V=\dim V^{\prime}<\infty$), and $c=\tau R$, $c^{\prime}=\tau R^{\prime}$. Assume that $c$, $c^{\prime}$ are rigid. Notice that there are two ways of identifying $(V\otimes V)^{*}$ with $V^{*}\otimes V^{*}$, namely

[TABLE]

$v,w\in V$, $f,g\in V^{*}$. The map $(V\otimes V)^{*}\to(V\otimes V)^{*}$ induced by these two identifications turns out to be the change of basis in (c). In short, we have:

• (a)

Let $R^{\prime}=R^{t}$. Then $c^{t}=R^{t}\tau$, $(c^{t})^{\#}=\tau R^{t}=c^{\prime}$.

• (c)

Let $R^{\prime}=R^{\#}$. Then $c^{\prime}=R\tau$ and $(c^{\prime})^{\#}=c$.

• (d)

if $R^{\prime}=\kappa R$ for $\kappa\in\Bbbk$, then $c^{\prime}=\kappa c$. Observe that if $R$ belongs to the family $\mathfrak{R}_{i,j}$, then so does $\kappa R$; this is why we homogeneize $\mathfrak{R}_{1,1}$. (Actually ${\mathcal{J}}^{2}(V,c^{\prime})=\ker(\operatorname{id}+c^{\prime})=\ker(\operatorname{id}+\kappa c)=\ker(\kappa^{-1}+c)$. Therefore, ${\mathcal{J}}^{2}(V,c^{\prime})\neq 0\iff-\kappa^{-1}$ is an eigenvalue of $c$).

• (e)

Let $\varphi\in GL(V)$. If $R^{\prime}=(\varphi\otimes\varphi)R(\varphi\otimes\varphi)^{-1}$, then $c^{\prime}=(\varphi\otimes\varphi)c(\varphi\otimes\varphi)^{-1}$. Hence the map $T(\varphi):T(V)\to T(V)$ is an algebra isomorphism that induces a Hopf algebra isomorphism ${\mathcal{B}}(\varphi):{\mathcal{B}}(V,c)\to{\mathcal{B}}(V,c^{\prime})$.

In conclusion, the new braided vector spaces to be considered from a solution $\mathfrak{R}_{i,j}$ arise by the three transformations (a), (c) and (a) composed with (c). Indeed the transformations (a) and (c) are commuting involutions. Sometimes we will use (b), when it coincides with any of those.

3. Nichols algebras of rank two with quadratic relations

This Section contains the case-by-case analysis of the Nichols algebras with quadratic relations of the braided vector spaces $(V,c)$ in the list in [Hi]. We fix a basis $(x_{i})_{i\in\mathbb{I}_{2}}$ of $V$; let $(f_{i})_{i\in\mathbb{I}_{2}}$ be its dual basis. We denote $\partial_{i}=\partial_{f_{i}}$.

3.1. Case $\mathfrak{R}_{2,1}$

We assume that $k,p,q\neq 0$ and $k^{2}\neq pq$. The associated braiding is

[TABLE]

Remark 3.1*.*

The braided vector space $K^{1}=\Bbbk\{z_{0},z_{1},f_{0},f_{1}\}$ associated to the Nichols algebra $K$ of [AAH, Lemma 4.37] has the following braiding

[TABLE]

Observe that $V=\Bbbk\{f_{0},f_{1}\}$ is a braided vector subspace and $K^{1}/V$ is a quotient braided space that fit in this case.

Remark 3.2*.*

The braidings of the braided vector spaces $V_{i,0}$ and $V_{i,2}$ considered in [GGi, Prop. 3.4] belong to this case.

Here is our main result for this family. Let $N=\begin{cases}\operatorname{ord}k^{2},&\text{if }1\neq\operatorname{ord}k^{2};\\ \infty,&\text{otherwise.}\end{cases}$

Proposition 3.3**.**

If $k^{2}\neq-1$ and $pq\neq 1$, then there are no quadratic relations. Otherwise, the Nichols algebras are

[TABLE]

Also $B=\{x_{2}^{a_{2}}x_{1}^{a_{1}}:\,0\leq a_{i}<N\}$ is a PBW-basis of ${\mathcal{B}}(V)$ and

[TABLE]

Proof.

Set $u=\lambda_{1}x_{1}^{2}+\lambda_{2}x_{1}x_{2}+\lambda_{3}x_{2}x_{1}+\lambda_{4}x_{2}^{2}$. Then

[TABLE]

From (3.3), all assertions made about quadratic relations hold. The relations (3.2) are also clear. Let $\mathfrak{B}$ be the pre-Nichols algebra in the right-hand side of (3.1) and let $\pi:\mathfrak{B}\to{\mathcal{B}}(V)$ be the natural projection. A standard argument shows that $B$ is a system of linear generators of $\mathfrak{B}$. We claim that the image of $B$ (that we call $B$ again) in ${\mathcal{B}}(V)$ is linearly independent; the claim implies that $\pi$ is an isomorphism and that $B$ is a basis as claimed. Assume that $B$ is not linearly independent and pick $r$ a homogeneous relation of minimal degree $n>2$:

[TABLE]

where $M_{1}=\min\{n,N-1\}$, $M_{2}=\max\{0,n+1-N\}$. Then

[TABLE]

Thus

[TABLE]

By minimality of $n$, $\partial_{1}(r)=\partial_{2}(r)=0$, hence

[TABLE]

If any $c_{a}\neq 0$, then either $(a)_{k^{2}}$ or $(n-a)_{k^{2}}=0$, but this contradicts the definition of $N$. Therefore the coefficients $c_{a}$ are trivial and $\pi$ is bijective. ∎

We close this Subsection by a discussion of the Nichols algebras arising from the equivalence in [Hi]. First, (b) and (c) give rise to the same braiding, that is

[TABLE]

Since (b) is a change of basis, the Nichols algebras are isomorphic. Second, (a) gives rise to the braiding

[TABLE]

But (3.5) is (3.4) up to $p\leftrightarrow q$, so no new Nichols algebra arises.

Third, (a) composed with (c) gives the initial $\mathfrak{R}_{2,1}$ up to $p\leftrightarrow q$, so no new Nichols algebra arises.

3.2. Case $\mathfrak{R}_{2,2}$

We assume that $k,p,q\neq 0$ and $k^{2}\neq pq$. The associated braiding is

[TABLE]

Let $N_{1}=\begin{cases}\operatorname{ord}k^{2},&\text{if }1\neq\operatorname{ord}k^{2};\\ \infty,&\text{otherwise.}\end{cases}$ and $N_{2}=\begin{cases}\operatorname{ord}(-pq),&\text{if }1\neq\operatorname{ord}(-pq);\\ \infty,&\text{otherwise.}\end{cases}$

Proposition 3.4**.**

If $k^{2}\neq-1$ and $pq\neq 1$, then there are no quadratic relations. Otherwise, the Nichols algebras are

[TABLE]

where $r_{i}=x_{i}^{N_{i}},\,i\in\mathbb{I}_{2}$, only if $N_{i}<\infty$. Also $\{x_{2}^{a_{2}}x_{1}^{a_{1}}:a_{i}\in\mathbb{I}_{0,N_{i}-1}\}$ is a PBW-basis and $\operatorname{GK-dim}{\mathcal{B}}(V)=|\{i\in\mathbb{I}_{2}:N_{i}=\infty\}|$; if 0, then $\dim{\mathcal{B}}(V)=N_{1}N_{2}$.

Proof.

Similar to the proof of Proposition 3.3. ∎

We next discuss the Nichols algebras arising from the equivalence in [Hi]. First, (a) gives rise to the braiding

[TABLE]

Let $N_{1}$ and $N_{2}$ be as above.

Proposition 3.5**.**

If $k^{2}\neq-1$ and $pq\neq 1$, then there are no quadratic relations. Otherwise, the Nichols algebras are

[TABLE]

where $r_{i}=x_{i}^{N_{i}},\,i\in\mathbb{I}_{2}$, only if $N_{i}<\infty$. Also $\{x_{2}^{a_{2}}x_{1}^{a_{1}}:a_{i}\in\mathbb{I}_{0,N_{i}-1}\}$ is a PBW-basis and $\operatorname{GK-dim}{\mathcal{B}}(V)=|\{i\in\mathbb{I}_{2}:N_{i}=\infty\}|$; if 0, then $\dim{\mathcal{B}}(V)=N_{1}N_{2}$.

Proof.

Similar to the proof of Proposition 3.3. ∎

Second, (c) gives rise to the braiding

[TABLE]

But (3.9) is (3.7) up to $p\leftrightarrow q$, so no new Nichols algebra arises here.

Third, (a) composed with (c) gives the initial $\mathfrak{R}_{2,1}$ up to $p\leftrightarrow q$, so no new Nichols algebra arises.

3.3. Case $\mathfrak{R}_{2,3}$

We assume that $k\neq 0$, and either $p\neq 0$, or $q\neq 0$, or $s\neq 0$. The associated braiding is $(c(x_{i}\otimes x_{j}))_{i,j\in\mathbb{I}_{2}}=$

[TABLE]

Remark 3.6*.*

The braided vector spaces $\mathcal{V}(\epsilon,2)$ considered in [AAH, §1.2] fit in this case taking $k=\epsilon$, $q=1$ and $p=s=0$. In particular, the Jordan plane $\mathcal{V}(1,2)$ and the super Jordan plane $\mathcal{V}(-1,2)$ belong to this case.

To state the next result, we need the notation

[TABLE]

Proposition 3.7**.**

If $k\neq\pm 1$, then there are no quadratic relations. Otherwise, the Nichols algebras are as in Table 2, where

[TABLE]

Proof.

Set $u=\lambda_{1}x_{1}^{2}+\lambda_{2}x_{1}x_{2}+\lambda_{3}x_{2}x_{1}+\lambda_{4}x_{2}^{2}$. Then

[TABLE]

Then all the assertions on quadratic relations hold. The claim in the first row of Table 2 follows then easily. To simplify the discussion of the rest, we consider three cases:

1. (i)

$k=-1$, $p=-q$ and $s\neq q^{2}$; 2. (ii)

$k=-1$, $p\neq-q$; 3. (iii)

$k=1$.

Case (i): We first prove by induction that for $n\geq 2$

[TABLE]

From this, we prove again by induction that, for $n\geq 2$

[TABLE]

Let $\mathfrak{B}=T(V)/\langle x_{1}^{2},x_{1}x_{2}+x_{2}x_{1}\rangle$ and let $\pi:\mathfrak{B}\to{\mathcal{B}}(V)$ be the natural projection. A standard argument shows that $(*_{2,3})_{2}$ generates linearly $\mathfrak{B}$. Assume that the image $B=\pi(B)$ is not linearly independent and pick a linear homogeneous relation of minimal degree $n>2$

[TABLE]

Assume that $n$ is odd. Then

[TABLE]

Assume that $n$ is even. Then

[TABLE]

Hence

[TABLE]

The system above has non trivial solution iff $q^{2}=s$. The claim in row 2 of Table 2 is established.

Case (ii): By analogy with the super Jordan plane $\mathcal{V}(1,2)$, we look for cubic relations and obtain the following one by (2.3):

[TABLE]

Let $\mathfrak{B}=T(V)/\langle x_{1}^{2},x_{2}x_{21}+(p-q)x_{1}x_{21}-x_{21}x_{2}\rangle$. Observe that $x_{1}x_{21}=x_{21}x_{1}$ in $\mathfrak{B}$. Arguing as in [AAH], that is using the commutation relations, we see that $(*_{2,3})_{3}$ is a system of linear generators of $\mathfrak{B}$. We need the formulae of the derivations on the elements of this basis. First,

[TABLE]

For $i\in\mathbb{I}_{2}$, $c\geq 0$, set

[TABLE]

Straightforward calculations show that, for $b\geq 1$, $c\geq 0$,

[TABLE]

Assume that the image of $B$ under the projection $\pi:\mathfrak{B}\to{\mathcal{B}}(V)$ is not linearly independent. Pick $r$ a non-trivial linear combination homogeneous of minimal degree $N\geq 4$.

Suppose first that $N$ is odd. Then there are scalars $\lambda_{b}$, $\mu_{t}$ such that

[TABLE]

Applying $\partial_{2}$ to $r$, we obtain

[TABLE]

From this, we see that $\lambda_{b}=0$, $b=0,1,\cdots,\frac{N-1}{2}$. Therefore

[TABLE]

hence $r=0$.

Assume next that $N$ is even. Then there are scalars $\lambda_{b}$, $\mu_{t}$ such that

[TABLE]

Thus

[TABLE]

Applying $\partial_{1}$ to $r$, we obtain for some $z\in{\mathcal{B}}^{N-2}(V)$

[TABLE]

In particular,

[TABLE]

Also observe that if $n\geq 2$ is even, then we obtain that for some $w_{i}\in\Bbbk\{x_{21}^{b}x_{2}^{d}:b,d\geq 0,\,b+d=n+i-5\}$, $i\in\mathbb{I}_{2}$,

[TABLE]

Looking at the terms $x_{1}x_{2}^{N-2}$ in (3.15) and $x_{2}^{N-1}$ in (3.16), we get

[TABLE]

whose determinant is $\frac{N}{2}(p+q)\neq 0$. Thus $\lambda_{0}=\mu_{0}=0$. Similarly, we prove that $\lambda_{i}=\mu_{i}=0$, $i=0,1,\cdots,\frac{N-2}{2}$. It remains $\lambda_{\frac{N}{2}}$, but

[TABLE]

hence $r=0$.

Case (iii): We start by the following claim, whose proof is straightforward:

[TABLE]

Let $\mathfrak{B}=T(V)/\langle\frac{q-p}{2}x_{1}^{2}-x_{1}x_{2}+x_{2}x_{1}\rangle$. By a standard argument, $(*_{2,3})_{4}$ generates linearly $\mathfrak{B}$. We need the formulae of the derivations on $(*_{2,3})_{4}$. First, we set

[TABLE]

and claim that

[TABLE]

Indeed, the case $n=1$ is clear. Assume that (3.18) holds for $n$. Projecting $\Delta(x_{2}^{n+1})$ to $V\otimes{\mathcal{B}}^{n}(V)$, we get

[TABLE]

Hence, by the inductive hypothesis,

[TABLE]

and the claim is proved.

Observe that, by (3.17), for $n\geq 3$

[TABLE]

Assume that the image of $B$ under the projection $\pi:\mathfrak{B}\to{\mathcal{B}}(V)$ is not linearly independent. Pick $r=\sum_{i=0}^{n}c_{i}x_{1}^{i}x_{2}^{n-i}$ a non-trivial homogeneous relation of minimal degree $n>2$. By (3.19)

[TABLE]

Observe that the term $x_{2}^{n-1}$ appears only one time in (3.20). Furthermore, by (3.18), we can rewrite (3.20) as

[TABLE]

for some $m_{j}\in\Bbbk$. By minimality of $n$, we obtain $c_{0}=0$. Similarly, we can replace (3.21) by

[TABLE]

what gives us $c_{1}=0$. Inductively, we get $c_{i}=0,\,0\leq i<n$. Thus, $r=c_{n}x_{1}^{n}$. But $\partial_{1}(x_{1}^{n})=nx_{1}^{n-1}$ which implies $r=0$. ∎

We next discuss the Nichols algebras arising from the equivalence in [Hi]. First, (a) and (b) give rise to the same braiding, that is $(c^{\prime}(x_{i}\otimes x_{j}))_{i,j\in\mathbb{I}_{2}}$

[TABLE]

Since (b) is a change of basis, this Nichols algebra is isomorphic to the original one. Second, (c) gives rise to the braiding $(c^{\prime\prime}(x_{i}\otimes x_{j}))_{i,j\in\mathbb{I}_{2}}$

[TABLE]

But this is the initial $\mathfrak{R}_{2,3}$ up to $p\leftrightarrow q$, so no new Nichols algebra arises. Thus, (a) composed with (c) gives the braiding $c^{\prime}$ above up to $p\leftrightarrow q$, so no new Nichols algebra arises.

3.4. Case $\mathfrak{R}_{1,1}$

We assume that $p,q\neq 0$ and $p^{2}\neq q^{2}$ with a and b as in (1.5). The associated braiding is $(c(x_{i}\otimes x_{j}))_{i,j\in\mathbb{I}_{2}}=$

[TABLE]

Proposition 3.8**.**

If $2p^{2}\neq-1$ and $2q^{2}\neq 1$, then there are no quadratic relations. Otherwise, the Nichols algebras are as in Table 3, where

[TABLE]

[TABLE]

Proof.

To start with, observe that $N\neq 1,2$ in rows $2$ and $4$ of Table 3 because $\texttt{a}\neq 0$ by hypothesis. Moreover, $N\neq 4$ since this case is already covered by row 1.

As previously, we set $u=\lambda_{1}x_{1}^{2}+\lambda_{2}x_{1}x_{2}+\lambda_{3}x_{2}x_{1}+\lambda_{4}x_{2}^{2}$. Then

[TABLE]

Then all the assertions on quadratic relations hold. In particular, row $1$ is established. To simplify the exposition, we consider two cases:

1. (i)

$2p^{2}\neq-1$ and $2q^{2}=1$; 2. (ii)

$2p^{2}=-1$ and $2q^{2}\neq 1$.

Case (i): We consider the braided Hopf algebra $\widetilde{{\mathcal{B}}}=T(V)/\langle r_{1},r_{2}\rangle$; clearly

[TABLE]

generates linearly $\widetilde{{\mathcal{B}}}$. By induction, it follows that for $n\geq 2$

[TABLE]

here the functions $f_{j}$ and $g_{j}$, found explicitly with WolframAlpha, are

[TABLE]

Let $0\neq r=c_{1}x_{2}^{n}+c_{2}x_{1}x_{2}^{n-1}\in\ker(\widetilde{{\mathcal{B}}}\to{\mathcal{B}}(V))$ with $n\geq 3$ minimal. Then

[TABLE]

In other words, $\ker(\widetilde{{\mathcal{B}}}\to{\mathcal{B}}(V))\neq 0$ iff there exists $n\in\mathbb{N}$ such that either $f_{1}(n)=g_{1}(n)=0$ or $f_{2}(n)=g_{2}(n)=0$. But

[TABLE]

Hence, if $-2pq\notin\mathbb{G}_{\infty}$, then $\widetilde{{\mathcal{B}}}\simeq{\mathcal{B}}(V)$ and arguing as in all preceding cases, $(*_{1,1})_{3}$ is a basis of ${\mathcal{B}}(V)$. This establishes the third row of Table 3.

Next, we assume that $-2pq\in\mathbb{G}_{\infty}$ and set $N=\operatorname{ord}(-2pq)$. As explained above, we suppose that $N\neq 1,2,4$. Then:

[TABLE]

Therefore, the relation $r_{4,N}$, i.e. (3.26), holds in ${\mathcal{B}}(V)$. Let us consider the braided Hopf algebra ${\mathcal{B}}=T(V)/\langle r_{1},r_{2},r_{4,N}\rangle\simeq\widetilde{{\mathcal{B}}}/\langle r_{4,N}\rangle$; clearly $(*_{1,1})_{2}$ generates linearly ${\mathcal{B}}$. Finally, $(*_{1,1})_{2}$ is linearly independent in ${\mathcal{B}}(V)$, as any linear relation would appear in a degree higher than $\deg r_{4,N}$ by (3.28).

Case (ii): By induction, we obtain equations (3.27) but this time with

[TABLE]

The rest of the proof is similar to the previous case. ∎

Finally, there are no new Nichols algebras arising from the equivalence in [Hi], because all changes (a), (c) and (a) composed with (c) give rise to the same initial braiding $\mathfrak{R}_{1,1}$.

3.5. Case $\mathfrak{R}_{1,2}$

We assume that $p,q\neq 0$, and either $p\neq q$, or $k\neq 0$. The associated braiding is

[TABLE]

Remark 3.9*.*

The braidings of the braided vector spaces $V_{i,1}$ and $V_{i,3}$ in [GGi, Prop. 3.4] belong to this case. If $i=0$ or $i=2$, then the braidings fit directly. But if $i=1$ or $i=3$, then the braidings have the shape

[TABLE]

which is equivalent to $\mathfrak{R}_{1,2}$ using the equivalences (a) and (b) in §2.3, namely

[TABLE]

Proposition 3.10**.**

If $p\neq-1$ and $q\neq 1$, then there are no quadratic relations. Otherwise, the Nichols algebras are as in Table 4, where

[TABLE]

[TABLE]

Proof.

Set $u=\lambda_{1}x_{1}^{2}+\lambda_{2}x_{1}x_{2}+\lambda_{3}x_{2}x_{1}+\lambda_{4}x_{2}^{2}$. Then

[TABLE]

Then all the assertions on quadratic relations hold. In particular, row $1$ is established. To simplify the exposition, we consider two cases:

1. (i)

$p=-1$; 2. (ii)

$q=1$.

Case (i): We start by the braided Hopf algebra $\widetilde{{\mathcal{B}}}=T(V)/\langle x_{1}^{2},r_{1}\rangle$, that covers ${\mathcal{B}}(V)$ and is linearly generated by

[TABLE]

We compute the derivations of the elements in $(*_{1,2})_{2}$; we have for $n\geq 2$

[TABLE]

here the coefficients $\texttt{a}_{n}$, found explicitly with WolframAlpha, are

[TABLE]

On the other hand, we have for $n\geq 2$

[TABLE]

We claim that $(*_{1,2})_{2}$ is linearly independent in ${\mathcal{B}}(V)$. Otherwise, we pick a non-trivial relation $r=c_{1}x_{2}^{n}+c_{2}x_{2}^{n-1}x_{1}$ with $n\geq 3$ minimal. Then

[TABLE]

what implies $c_{2}=0$. Therefore, we get a new relation $r=x_{2}^{n}$ iff

[TABLE]

Hence, row $3$ of Table 4 holds. Next, suppose that $-q\in\mathbb{G}^{\prime}_{\infty}$. Observe that

• •

If $q=1$, then, by (3.29), $\texttt{a}_{n}=0$ only if $k=0$, but this is row $1$.

• •

If $N=\operatorname{ord}(-q)\geq 3$, then it follows that $\texttt{a}_{N}=0$ by (3.29).

By (3.30) and arguing as in previous cases, we conclude that $(*_{1,2})_{1}$ is a basis of ${\mathcal{B}}(V)$. Hence, row $2$ of Table 4 holds.

Case (ii): Again we start by the braided Hopf algebra $\widetilde{{\mathcal{B}}}=T(V)/\langle r_{1},r_{2}\rangle$, that covers ${\mathcal{B}}(V)$ and is linearly generated by

[TABLE]

It is easy to see that for $n\geq 2$

[TABLE]

Let $r=c_{1}x_{1}^{n}+c_{2}x_{2}x_{1}^{n-1}$ be a relation in ${\mathcal{B}}(V)$ with $n\geq 3$ minimal. Then

[TABLE]

what gives us $c_{2}=0$. Hence,

[TABLE]

and we have the relation $x_{1}^{n}$ iff $p\in\mathbb{G}_{n}^{\prime},$ $n\geq 2$; but $p=-1$ is excluded since it turns out to be case (i). The rest of the proof goes as in the case (i). ∎

We next discuss the Nichols algebras arising from the equivalence in [Hi]. Here the equivalences (a), (c) and (a) $\circ$ (c) give different braidings.

First, (a) gives rise to the braiding

[TABLE]

Proposition 3.11**.**

If $p\neq-1$ and $q\neq 1$, then there are no quadratic relations. Otherwise, the Nichols algebras are as in Table 5, where

[TABLE]

[TABLE]

Proof.

Similar to the proof of Proposition 3.10. ∎

Second, (c) gives rise to the braiding

[TABLE]

Proposition 3.12**.**

If $p\neq-1$ and $q\neq 1$, then there are no quadratic relations. Otherwise, the Nichols algebras are as in Table 6, where

[TABLE]

[TABLE]

Proof.

Similar to the proof of Proposition 3.10. ∎

Finally, (a) composed with (c) gives rise to the braiding

[TABLE]

Proposition 3.13**.**

If $p\neq-1$ and $q\neq 1$, then there are no quadratic relations. Otherwise, the Nichols algebras are as in Table 7, where

[TABLE]

[TABLE]

Proof.

Similar to the proof of Proposition 3.10. ∎

3.6. Case $\mathfrak{R}_{1,3}$

We assume that $k\neq 0$, and either $p\neq 0$, or $q\neq 0$. The associated braiding is $(c(x_{i}\otimes x_{j}))_{i,j\in\mathbb{I}_{2}}=$

[TABLE]

Proposition 3.14**.**

If $k^{4}\neq 1$, then there are no quadratic relations. Otherwise, the Nichols algebras are as in Table 8, where

[TABLE]

Proof.

Set $u=\lambda_{1}x_{1}^{2}+\lambda_{2}x_{1}x_{2}+\lambda_{3}x_{2}x_{1}+\lambda_{4}x_{2}^{2}$. Then

[TABLE]

Thus the quadratic relations above are clear; hence, row $1$ is established. By a routine argument, $(*_{1,3})_{2}$ generates linearly $\mathfrak{B}:=T(V)/\langle x_{1}x_{2}-x_{2}x_{1}+kpx_{1}^{2}\rangle$. As in the (proof of) Proposition 3.7 case (iii), we set

[TABLE]

In a similar way as for (3.18), we prove that

[TABLE]

Also observe that, by induction, we have for $n\geq 1$

[TABLE]

Then, by (3.32), for $n\geq 3$ and $0<i\leq n$,

[TABLE]

Assume that the image of $B$ under the projection $\pi:\mathfrak{B}\to{\mathcal{B}}(V)$ is not linearly independent. Pick $r=\sum_{i=0}^{n}c_{i}x_{1}^{i}x_{2}^{n-i}$ a non-trivial homogeneous relation of minimal degree $n>2$. By (3.33),

[TABLE]

Looking at the monomials $x_{2}^{n-i}$in (3.34) as in case (iii) of the proof of Proposition 3.7, we get $c_{i}=0,\,0\leq i<n$. Also $\partial_{1}(x_{1}^{n})=nx_{1}^{n-1}$, then $r=0$. ∎

We next discuss the Nichols algebras arising from the equivalence in [Hi]. First, (c) gives rise to the braiding $(c^{\prime}(x_{i}\otimes x_{j}))_{i,j\in\mathbb{I}_{2}}$

[TABLE]

But this is $\mathfrak{R}_{1,3}$ up to $p\mapsto-p$, $q\mapsto-q$, so no new Nichols algebra arises.

Second, (a) gives rise to the braiding $(c^{\prime\prime}(x_{i}\otimes x_{j}))_{i,j\in\mathbb{I}_{2}}$

[TABLE]

But this is (b) up to $p\mapsto-q$, $q\mapsto-p$, so no new Nichols algebra arises. Therefore, (a) composed with (c) is (b) up to $p\mapsto q$, $q\mapsto p$, so again no new Nichols algebra arises.

3.7. Case $\mathfrak{R}_{1,4}$

We assume that $k,p,q\neq 0$. The associated braiding is

[TABLE]

Proposition 3.15**.**

If $k\neq-1$ and $pq\neq 1$, then there are no quadratic relations. Otherwise, the Nichols algebras are as Table 9, where

[TABLE]

The bases are

[TABLE]

Proof.

Set $u=\lambda_{1}x_{1}^{2}+\lambda_{2}x_{1}x_{2}+\lambda_{3}x_{2}x_{1}+\lambda_{4}x_{2}^{2}$. Then

[TABLE]

Then all assertions made about quadratic relations hold. To simplify, we consider two cases:

1. (i)

$k=-1$; 2. (ii)

$k\neq-1$ and $pq=1$.

Case (i): We consider the braided Hopf algebra

[TABLE]

clearly it is linearly generated by

[TABLE]

Note that, for $n\geq 1$

[TABLE]

Then, for $n\geq 2$, we prove that

[TABLE]

Let $0\neq r=c_{1}x_{1}^{n}+c_{2}x_{2}^{n}\in\ker(\widetilde{{\mathcal{B}}}\to{\mathcal{B}}(V))$ with $n\geq 3$ minimal. If $n$ is odd, then it is clear that $r=0$ (just apply $\partial_{1}$ and $\partial_{2}$). So, suppose that $n$ is even. Then

[TABLE]

This implies that

[TABLE]

The system above has a non trivial solution iff $(pq)^{\frac{n}{2}}=1$. Hence, if $pq\notin\mathbb{G}_{\infty}$, then $\widetilde{{\mathcal{B}}}\simeq{\mathcal{B}}(V)$ and arguing as before, we conclude that $(*_{1,4})_{2}$ is a basis of ${\mathcal{B}}(V)$. This establishes the second row of Table 9.

Next, we assume that $pq\in\mathbb{G}_{\infty}$ and set $N=\operatorname{ord}(pq)$. Then, the relation $r_{3,N}$ (3.37) holds in ${\mathcal{B}}(V)$, and we may consider the braided Hopf algebra

[TABLE]

clearly $(*_{1,4})_{1}$ generates linearly ${\mathcal{B}}$. But $(*_{1,4})_{1}$ is linearly independent in ${\mathcal{B}}(V)$, as any linear relation would appear in a degree higher than $\deg r_{3,N}$ by the definition of $N$ and the discussion above on (3.38). This establishes the first row of Table 9.

Case (ii): We consider the braided Hopf algebra

[TABLE]

We claim that $(*_{1,4})_{4}$ linearly generates $\widetilde{{\mathcal{B}}}$.

*Proof of the claim. * It is enough to show that $I=$ linear span of $(*_{1,4})_{4}$ is left ideal, since clearly $1\in I$. Obviously, $x_{1}I\subset I$, so we only need to show that $x_{2}I\subset I$. Observe first that

[TABLE]

Thus we multiply $x_{2}x_{1}^{a}(x_{2}x_{1})^{b}x_{2}^{c}$ and get

[TABLE]

The claim is proved. Then we check, by an inductive argument, that

[TABLE]

$a\in\mathbb{N}$. Also, we have that, for $b\geq 1$

[TABLE]

Therefore, for $a,b\geq 1$

[TABLE]

and then

[TABLE]

We then proceed as in the previous case. Namely, let $r\in\ker(\widetilde{{\mathcal{B}}}\to{\mathcal{B}}(V))$ be an homogeneous relation of degree $n$ with $n\geq 3$ minimal. We consider separately the cases $n$ odd and $n$ even. Using the derivations, we see that the relations $r_{1,N}$ and $r_{2,N}$ hold. Also, any other relation arises in higher degree. In this way, rows $3$ and $4$ of Table 9 are established. ∎

We finally discuss the Nichols algebras arising from the equivalence in [Hi]. First, (a) and (a) composed with (c) give rise to the same braiding

[TABLE]

which is $\mathfrak{R}_{1,4}$ up to $p\leftrightarrow q$, so no new Nichols algebra arises.

Second, (c) gives the initial $\mathfrak{R}_{1,4}$, so no new Nichols algebra arises.

3.8. Case $\mathfrak{R}_{0,1}$

The associated braiding is

[TABLE]

Proposition 3.16**.**

If $k^{2}\neq 1$, then there are no quadratic relations.

If $k=1$, then

[TABLE]

$B_{1}=\{x_{1}^{a_{1}}x_{2}^{a_{2}}:a_{i}\in\mathbb{N}_{0}\}$* is a PBW-basis of ${\mathcal{B}}(V)$ and $\operatorname{GK-dim}{\mathcal{B}}(V)=2$.*

If $k=-1$, then

[TABLE]

$B_{2}=\{x_{1}^{a_{1}}x_{2}^{a_{2}}:a_{1}\in\mathbb{I}_{0,2},\,a_{2}\in\mathbb{N}_{0}\}$* is a PBW-basis of ${\mathcal{B}}(V)$. Hence

$\operatorname{GK-dim}{\mathcal{B}}(V)=1$.*

Proof.

Set $u=\lambda_{1}x_{1}^{2}+\lambda_{2}x_{1}x_{2}+\lambda_{3}x_{2}x_{1}+\lambda_{4}x_{2}^{2}$. Then

[TABLE]

From here, all assertions about quadratic relations hold.

Case $k=1$: Let $\mathfrak{B}=T(V)/\langle x_{1}x_{2}+x_{2}x_{1}\rangle$. By a standard argument, $B_{1}$ is a basis of $\mathfrak{B}$. Note that, for $n\geq 2$

[TABLE]

Let $n\geq 3$ and $i\in\mathbb{I}_{2,n-2}$. Then

[TABLE]

Observe that the formulae (3.41) also hold for $i\in\{0,1,n-1,n\}$ taking by [math] the terms that are not well defined.

Assume that the image of $B_{1}$ under the projection $\pi:\mathfrak{B}\to{\mathcal{B}}(V)$ is not linearly independent. Pick $0\neq r=\sum_{i=0}^{N}c_{i}x_{1}^{i}x_{2}^{N-i}$ a homogeneous relation of minimal degree $N>2$. Thus

[TABLE]

Then, $c_{i}=0$ for $i\in\mathbb{I}_{0,N-1}$. But $\partial_{1}(x_{1}^{N})=Nx_{1}^{N-1}$, hence $r=0$.

Case $k=-1$: Let $\mathfrak{B}=T(V)/\langle x_{1}^{2},x_{1}x_{2}-x_{2}x_{1}\rangle$. By a standard argument, $B_{2}$ is a basis of $\mathfrak{B}$. Note that, for $n\geq 2$

[TABLE]

Then, for $n\geq 2$,

[TABLE]

Suppose that the image of $B$ under the projection $\pi:\mathfrak{B}\to{\mathcal{B}}(V)$ is not linearly independent. Pick $r=c_{1}x_{2}^{N}+c_{2}x_{1}x_{2}^{N-1}$ a homogeneous non-trivial relation of minimal degree $N>2$. Applying $\partial_{1}$ to $r$, we obtain $r=0$. ∎

We next discuss the Nichols algebras arising from the equivalence in [Hi]. First, (a), (b) and (a) composed with (c) give rise to the same braiding

[TABLE]

Since (b) is a change of basis, the Nichols algebras are isomorphic.

Finally, (c) gives the initial braiding $\mathfrak{R}_{0,1}$, so no new Nichols algebra arises.

4. Appendix

Here we collect all isomorphism classes of algebras arising as Nichols algebras in Theorem 1.1, see the information in Table 1. In many cases a change of variables is needed, and we leave to the reader its explicit calculation. All the algebras are of the form $T(W)/{\mathcal{J}}$, where $W$ has a basis $y_{1},y_{2}$.