Three infinite families of reflection Hopf algebras

Abstract

Let H be a semisimple Hopf algebra acting on an Artin-Schelter regular algebra A, homogeneously, inner-faithfully, preserving the grading on A, and so that A is an H-module algebra. When the fixed subring A H is also AS regular, thus providing a generalization of the Chevalley-Shephard-Todd Theorem, we say that H is a reflection Hopf algebra for A. We show that each of the semisimple Hopf algebras H 2n 2 of Pansera, and A 4m and B 4m of Masuoka is a reflection Hopf algebra for an AS regular algebra of dimension 2 or 3.2010 Mathematics Subject Classification. Primary 16T05, 16E65, 16G10.The Hopf algebras A 4m and B 4m can be viewed as deformations of kQ 4m (see [2]), and are extensions of the form:The examples of reflection Hopf algebras that we have computed indicate that there are an abundance of examples. The properties that characterize such a pair (A, H) are not clear, and invite further investigation. One obvious question is:Question 0.1. When is a bicrossed product H = K# τ σ H a reflection Hopf algebra for some AS regular algebra A?The method that is used in this paper is as follows. First, we compute the Grothendieck ring of finite-dimensional H-modules for each Hopf algebra H. The results are summarized in the following table.

Semisimple Reflection Hopf Algebras of Dimension Sixteen

“…In [17] actions of duals of group algebras H = kG • (or equivalently, group coactions) were considered, and some dual reflection groups were constructed. In [10] three infinite families of Hopf algebras were shown to be reflection Hopf algebras. The goal of this paper is to provide further data toward a better understanding of reflection Hopf algebras.…”

“…For primes p and q it is known that semisimple Hopf algebras of dimension p [27], 2p [18], p 2 [19], and pq [9,11] are trivial. The families considered in [10] include the two nontrivial semisimple Hopf algebras of dimension twelve. Hence the next dimension to consider is sixteen.…”

Examples of non-semisimple Hopf algebra actions on Artin-Schelter regular algebras

“…In the commutative case see, for example, [OT,Theorem 6.19(2,3)]. In the noncommutative case, this was verified for many examples, see [FKMW1,FKMW2]. However, in the non-semisimple case, T is never a free H-module as H T 0 ∼ = H k cannot be projective.…”

“…Artin-Schelter regular algebras [AS], viewed as a natural noncommutative generalization of the commutative polynomial rings, play an important role in noncommutative algebraic geometry, representation theory, and the study of noncommutative algebras [ATV1,ATV2,CV]. Hopf actions (including group actions) on Artin-Schelter regular algebras have been studied extensively by many authors in recent years, see [CKWZ1,CKWZ2,CKWZ3,CG,FKMW1,FKMW2,FKMP,KKZ1,KKZ2,KWZ,KZ] and so on. A very nice survey was given by Kirkman [Ki] a few years ago.…”

A note on the discriminant of reflection Hopf algebras

“…Now assume that (i 2 − j 2 , n) = 1, n is odd and i − j is even. Then the fixed subring A H is the polynomial ring k[u n + v n , u n v n ] ( [10,Theorem 3.10]). This implies that H 2n 2 acts on A as a reflection Hopf algebra.…”