The Hom-Long dimodule category and nonlinear equations

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 $^{B}_{H} \Bbb L$ is an autonomous category. Second, we prove that the category $^{B}_{H} \Bbb L$ 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 $^{B}_{H} \Bbb L$ can be viewed as a subcategory of the Hom-Yetter-Drinfeld category $^{HøB}_{HøB} \Bbb {HYD}$. Finally, we obtain a solution of the Hom-Long equation from the Hom-Long dimodules.

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