$ı$Hopf algebras associated with self-dual Hopf algebras

Abstract

Motivated by the construction of $ı$Hall algebras and $\hat{I}”$-Hall algebras, we introduce $ı$Hopf algebras associated with symmetrically self-dual Hopf algebras. We prove that the $ı$Hopf algebra is an associative algebra with a unit, where the associativity relies on an analogue of Green's formula in the framework of Hopf algebras. As an application, we construct the $ı$Taft algebra of dimension 4, which is proved to be isomorphic to the group algebra of $\mathbb{Z}/4\mathbb{Z}$.