The Hom-Long dimodule category and nonlinear equations

Abstract

In this paper, we construct a kind of new braided monoidal category over two Hom-Hopf algerbas (H, α) and (B, β) and associate it with two nonlinear equations. We first introduce the notion of an (H, B)-Hom-Long dimodule and show that the Hom-Long dimodule category B H L is an autonomous category. Second, we prove that the category B H L is a braided monoidal category if (H, α) is quasitriangular and (B, β) is coquasitriangular and get a solution of the quantum Yang-Baxter equation. Also, we show that the category B H L can be viewed as a subcategory of the Hom-Yetter-Drinfeld category H⊗B H⊗B HYD. Finally, we obtain a solution of the Hom-Long equation from the Hom-Long dimodules.