Almost involutive Hopf algebras

Abstract

We define the concept of companion automorphism of a Hopf algebra H as an automorphism σ : H Ñ H: σ 2 " S 2 -where S denotes the antipode-. A Hopf algebra is said to be almost involutive (AI) if it admits a companion automorphism that can be viewed as a special additional symmetry. We present examples and study some of the basic properties and constructions of AI-Hopf algebras centering the attention in the finite dimensional case. In particular we show that within the family of Hopf algebras of dimension smaller or equal than 15, only in dimension eight and twelve, there are non almost involutive Hopf algebras.

Almost involutive Hopf-Ore extensions of low dimension

“…An element of the Ore extension is uniquely expressed as a finite sum i h i t i , where h i ∈ H (see [21]), and h i t i h j t j is computed using the relation (1). This definition was extended to Hopf algebras by A. N. Panov as follows.…”

“…Examples of involutive Hopf algebras are commutative or cocommutative or semisimple Hopf algebras. This concept was generalized in [1] As we have seen, all non-semisimple Hopf algebras of dimension up to 23 can be obtained by the constructions of Sect. 2, so it will be useful to know when a Hopf-Ore extension is almost involutive.…”

“…To answer this question, in [1] was introduced the concept of almost involutive Hopf algebra, that is a Hopf algebra H that has an automorphism σ such that σ 2 = S 2 , being S the antipode of H . In [1] there are several examples of almost involutive Hopf algebras, and there is also the classification of all the almost involutive Hopf algebras with dimension up to 15. The first dimension for which the Hopf algebras are not completely known is 24, so in this paper we want to go further and classify all the almost involutive Hopf algebras up to dimension 23.…”