A class of semisimple Hopf algebras acting on quantum polynomial algebras

Abstract

We construct a class of non-commutative, non-cocommutative, semisimple Hopf algebras of dimension 2n 2 and present conditions to define an inner faithful action of these Hopf algebras on quantum polynomial algebras, providing, in this way, more examples of semisimple Hopf actions which do not factor through group actions. Also, under certain condition, we classify the inner faithful Hopf actions of the Kac-Paljutkin Hopf algebra of dimension 8, H8, on the quantum plane.1991 Mathematics Subject Classification. 12E15; 13A35; 16T05; 16W70. Key words and phrases. semisimple Hopf algebras, inner faithful action, quantum polynomial rings. I would like to thank Christian Lomp very much for many clarifications and helpful insights. The author was partially supported by CMUP (UID/MAT/00144/2013), which is funded by FCT (Portugal) with national (MEC) and European structural funds through the programs FEDER, under the partnership agreement PT2020, and also supported by CAPES, Coordination of Superior Level Staff Improvement -Brazil.Definition 2.1. Let R be a bialgebra and J be an invertible element in R ⊗ R. J is called a right twist (or a Drinfel'd twist) for R if J satisfies:If J is a left twist for R, then J −1 is a right twist for R.Definition 2.2 ([8, 2.1]). Let R be a bialgebra. Let J be a right twist for R and σ ∈ End(R). We say that the pair (σ, J) is a twisted homomorphism for R if σ satisfies: (i) J(σ ⊗ σ)∆(h) = ∆(σ(h))J for all h ∈ R;

Three infinite families of reflection Hopf algebras

“…We denote by R the group algebra k[Z n × Z n ], and think of it as a Hopf subalgebra of H 2n 2 generated by x and y. Then I ∩ R is a Hopf ideal of R. By [20,Lemma 1.4] there is N a normal subgroup of Z n × Z n such that…”

“…In this paper we consider three infinite families of Hopf algebras: the Hopf algebras H 2n 2 of dimension 2n 2 defined by Pansera [20], and the two families A 4m and B 4m of Hopf algebras of dimension 4m defined by Masuoka [18]. We begin in section 2 by considering the Kac-Palyutkin algebra, which occurs as H 8 (n = 2) in the Pansera construction, as well as B 8 (m = 2) in one of the Masuoka constructions.…”

Semisimple Reflection Hopf Algebras of Dimension Sixteen

“…Before specializing to the two cases, we prove a lemma which helps us to identify inner faithful representations for algebras of this form. It is inspired by a lemma of Pansera [23,Lemma 2.12].…”

On the quasitriangular structures of abelian extensions of $Z_2$

“…(i) w 1 be given by w 1 (a 2i , a 2j ) = w 1 (a 2i b, a 2j ) = (λ 2j,1 ) 2i (αβ) 2ij [σ(a 2j )] i w 1 (a 2i , a 2j b) = w 1 (a 2i b, a 2j b) = (λ 2j,1 ) 2i (αβ) 2ij [σ(a 2j )] i , (ii) w 2 be given by w 2 (a 2i , a 2j+1 ) = w 2 (a 2i , a 2j+1 b) = λ 2i,2j+1 [S j,0 S j,1 ] i w 2 (a 2i b, a 2j+1 ) = w 2 (a 2i b, a 2j+1 b) = τ (b,a)β τ (b,a 2i )α λ 2i,2j+1 [S j,0 S j,1 ] i , (iii) w 3 be given by w 3 (a 2i+1 , a 2j ) = w 3 (a 2i+1 b, a 2j ) = λ 2j,2i+1 [S i,0 S i,1 ] j w 3 (a 2i+1 , a 2j b) = w 3 (a 2i+1 b, a 2j b) = τ (b,a)β τ (b,a 2j )α λ 2j,2i+1 [S i,0 S i,1 ] j , (iv) w 4 be given by…”

“…To state the applications of such observations, we give three classes of Hopf algebras which are denoted by H 2n 2 , A 2n 2 ,t and K(8n, σ, τ ) respectively. We need point out that the first two classes of Hopf algebras were studied by some authors [3,8] before. As the main conclusion of this paper, all universal R-matrices on Hopf algebras H 2n 2 , A 2n 2 ,t and K(8n, σ, τ ) are given explicitly.…”