Almost involutive Hopf-Ore extensions of low dimension

Abstract

The purpose of this paper is to complete the classification of almost involutive Hopf algebras up to dimension 23. For this, we first present a brief survey of these algebras, showing that most of them can be obtained by iterated Hopf-Ore extensions. Then the classification follows using a technique we developed for determining when a Hopf-Ore extension is almost involutive.

Some advances about the existence of compact involutions in semisimple Hopf algebras

“…Conversely, if we assume that H is a C * -Hopf algebra, then we have a decomposition of the * -algebra H as in (35), which by duality implies a decomposition of the •-coalgebra H ∨ as in (34). Hence Proposition 3.9 implies that H ∨ is a compact quantum group and therefore so it is H by Remark 3.3 (4). products C G # σ,τ CF, where the cocycles verify |σ | = |τ | = 1.…”

“…For the convenience of the reader, all of the above description is summarized in Theorem 2.8. Note that this survey together with the one in [4] provide an explicit description of all the non-trivial Hopf algebras of dimension less than 24. In Sect.…”