The Drinfel'd codouble constuction for monoidal Hom-Hopf algebra

Abstract

Let $(H,\beta )$ be a monoidal Hom-Hopf algebra with the bijective antipode $S$, In this paper, we mainly construct the Drinfel'd codouble $T(H)=(H^{op}\otimes H^{\ast },\beta \otimes {\beta }^{\ast -1})$ and $\widehat{T(H)}=(H^{\ast }\otimes H^{op},{\beta }^{\ast -1}\otimes \beta )$ in the setting of monoidal Hom-Hopf algebras. Then we prove both $T(H)$ and $\widehat{T(H)}$ are coquasitriangular. Finally, we discuss the relation between Drinfel'd codouble and Heisenberg double in the setting of monoidal Hom-Hopf algebras, which is a generalization of the part results in \cite{L94}.