The Hom-Long dimodule category and nonlinear equations

TL;DR

This paper constructs a new braided monoidal category over Hom-Hopf algebras, introduces Hom-Long dimodules, and derives solutions to nonlinear equations including the quantum Yang-Baxter and Hom-Long equations.

Contribution

It introduces the notion of Hom-Long dimodules, establishes their category as braided monoidal, and connects these structures to solutions of important nonlinear equations.

Findings

The Hom-Long dimodule category is autonomous.

The category becomes braided monoidal under certain conditions.

Explicit solutions to the quantum Yang-Baxter and Hom-Long equations are obtained.

Abstract

In this paper, we construct a kind of new braided monoidal category over two Hom-Hopf algerbas $(H,\alpha)$ and $(B,\beta)$ 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 $(H,\alpha)$ is quasitriangular and $(B,\beta)$ 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 $^{H\o B}_{H\o B} \Bbb {HYD}$. Finally, we obtain a solution of the Hom-Long equation from the Hom-Long dimodules.