Drinfel'd double for monoidal Hom-Hopf algebras

In this paper we mainly construct bicrossproduct for finite-dimensional monoidal Hom-Hopf algebra $(H,\alpha)$, generalizing the Majid's bicrossproduct. Naturally the Hom-type bicrossproduct leads to Drinfel'd double $(H^{op}\bowtie H^{\ast}, \alpha \otimes (\alpha^{-1})^\ast)$ with a quasitriangular structure $R$ satisfying the quantum Hom-Yang-Baxter equations.