Generalised Taft algebras and pairs in involution

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.