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

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 \(H(A)=A\# A^{*}\), generalizing the main result in \cite{Lu}. Especially, the examples given in the paper are not obtained from the usual Hopf algebras.