The Hom-Long dimodule category and nonlinear equations

Shengxiang Wang; Xiaohui Zhang; Shuangjian Guo; Shengxiang Wang; Xiaohui Zhang; Shuangjian Guo

doi:10.3934/era.2022019

Electronic Research Archive

2022, Volume 30, Issue 1: 362-381. doi: 10.3934/era.2022019

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Research article

The Hom-Long dimodule category and nonlinear equations

1.
School of Mathematics and Finance, Chuzhou University, Chuzhou, 239000, China
2.
School of Mathematical Sciences, Qufu Normal University, Qufu, 273165, China
3.
School of Mathematics and Statistics, Guizhou University of Finance and Economics, Guiyang, 550025, China

Received: 08 June 2021 Revised: 22 November 2021 Accepted: 22 November 2021 Published: 13 January 2022

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{\otimes} B}_{H{\otimes} B} \Bbb {HYD} $. Finally, we obtain a solution of the Hom-Long equation from the Hom-Long dimodules.
- Hom-Long dimodule,
- Hom-Yetter-Drinfeld category,
- Yang-Baxter equation,
- Hom-Long equation
Citation: Shengxiang Wang, Xiaohui Zhang, Shuangjian Guo. The Hom-Long dimodule category and nonlinear equations[J]. Electronic Research Archive, 2022, 30(1): 362-381. doi: 10.3934/era.2022019

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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{\otimes} B}_{H{\otimes} B} \Bbb {HYD} $. Finally, we obtain a solution of the Hom-Long equation from the Hom-Long dimodules.

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