  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.
