Generalised Taft algebras and pairs in involution

TL;DR

This paper studies a class of finite-dimensional Hopf algebras generalizing Taft algebras, providing conditions for the absence of pairs in involution, and demonstrating the existence of infinitely many such examples with implications for related algebraic theories.

Contribution

It establishes necessary and sufficient conditions for these Hopf algebras to lack pairs in involution, and proves the existence of infinitely many such algebras.

Findings

Identified conditions for Hopf algebras to omit pairs in involution.

Proved the existence of infinitely many finite-dimensional Hopf algebras without pairs in involution.

Implications for anti-Yetter-Drinfeld modules and biduality of representations.

Abstract

A class of finite-dimensional Hopf algebras which generalise the notion of Taft algebras is studied. We give necessary and sufficient conditions for these Hopf algebras to omit a pair in involution, that is, to not have a group-like and a character implementing the square of the antipode. As a consequence we prove the existence of an infinite set of examples of finite-dimensional Hopf algebras without such pairs. This has implications for the theory of anti-Yetter-Drinfeld modules as well as biduality of representations of Hopf algebras. This article has been accepted for publication in Communications in Algebra, published by Taylor & Francis.