The Drinfel'd Double versus the Heisenberg Double for Hom-Hopf Algebras

Abstract

Let $(A,\alpha )$ be a finite-dimensional Hom-Hopf algebra. In this paper we mainly construct the Drinfel'd double $D(A)=(A^{op}\bowtie A^{\ast },\alpha \otimes ({\alpha }^{-1}{)}^{\ast })$ in the setting of Hom-Hopf algebras by two ways, one of which generalizes Majid's bicrossproduct for Hopf algebras (see \cite{M2}) and another one is to introduce the notion of dual pairs of of Hom-Hopf algebras. Then we study the relation between the Drinfel'd double $D(A)$ and Heisenberg double $H(A)=A\#A^{\ast }$, generalizing the main result in \cite{Lu}. Especially, the examples given in the paper are not obtained from the usual Hopf algebras.

References in corpus (1)

• Module Hom-algebras