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.