Almost involutive Hopf algebras

We define the concept of companion automorphism of a Hopf algebra H as an automorphism σ :H→ H such that σ ^2=𝒮^2 , where 𝒮 denotes the antipode. This automorphism can be viewed as a special additional symmetry. A Hopf algebra is said to be almost involutive (AI) if it admits a companion automorphism. We present examples and study some of the basic properties and constructions of AI-Hopf algebras centering the attention in the finite dimensional case. In particular we show that within the family of Hopf algebras of dimension smaller or equal than 15, only in dimension eight and twelve, there are non-almost involutive Hopf algebras.