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

TL;DR

This paper constructs the Drinfel'd double for finite-dimensional Hom-Hopf algebras using two methods, explores its relation to the Heisenberg double, and provides novel examples not derived from traditional Hopf algebras.

Contribution

It introduces two approaches to construct the Drinfel'd double in Hom-Hopf algebras and investigates its relationship with the Heisenberg double, expanding the theory beyond classical Hopf algebras.

Findings

Constructed the Drinfel'd double via bicrossproduct and dual pairs methods.

Established the relation between Drinfel'd and Heisenberg doubles in Hom-Hopf algebras.

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