Generalized Taft algebras and pairs in involution

Abstract

A class of finite-dimensional Hopf algebras which generalize 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. Implications for the theory of anti-Yetter–Drinfeld modules as well as biduality of representations of Hopf algebras are discussed.

Keywords:

• Anti-Drinfeld double
• Hopf algebra
• pair in involution
• pivotal category

2020 Mathematics Subject Classification:

• Primary: 16T05
• Secondary: 57T05

Acknowledgements

The author would like to thank P. Hajac for his kind invitation to IMPAN. He would also like to thank I. Heckenberger and U. Krähmer for many stimulating discussions.

Notes

1 The definition of the Drinfeld and anti-Drinfeld double given here varies from [14, Chapter IX.4] and [9, Proposition 4.1] to accomodate our choice of (anti-)Yetter-Drinfeld modules. In terms of the literature listed above our definition would read as $D{(H^{\text{cop}})}^{\text{cop}}$ and $A{(H^{\text{cop}})}^{\text{cop}}.$