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Keywords = BiHom-Hopf algebra

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15 pages, 316 KB  
Article
On the BiHom-Type Nonlinear Equations
by Hui Wu and Xiaohui Zhang
Mathematics 2022, 10(22), 4360; https://doi.org/10.3390/math10224360 - 19 Nov 2022
Viewed by 1562
Abstract
In this paper, the Heisenberg doubles and Long dimodules of a BiHom-Hopf algebra are introduced. Then, we discussed the relation between BiHom-Hopf equation and BiHom-pentagon equation, and we obtain the solutions of BiHom-Hopf equation from Heisenberg doubles. We also showed that the parametric [...] Read more.
In this paper, the Heisenberg doubles and Long dimodules of a BiHom-Hopf algebra are introduced. Then, we discussed the relation between BiHom-Hopf equation and BiHom-pentagon equation, and we obtain the solutions of BiHom-Hopf equation from Heisenberg doubles. We also showed that the parametric generalized Long dimodules can provide the solutions of BiHom-Yang-Baxter equation and generalized D-equation. Full article
(This article belongs to the Special Issue Hopf-Type Algebras, Lie Algebras, Quantum Groups and Related Topics)
17 pages, 295 KB  
Article
The Braiding Structure and Duality of the Category of Left–Left BiHom–Yetter–Drinfeld Modules
by Ling Liu and Bingliang Shen
Mathematics 2022, 10(4), 621; https://doi.org/10.3390/math10040621 - 17 Feb 2022
Viewed by 1575
Abstract
Let (H,μH,ΔH,αH,βH,ψH,ωH,SH) be a BiHom–Hopf algebra. First, we provide a non-trivial example of a left–left BiHom–Yetter–Drinfeld module and show that [...] Read more.
Let (H,μH,ΔH,αH,βH,ψH,ωH,SH) be a BiHom–Hopf algebra. First, we provide a non-trivial example of a left–left BiHom–Yetter–Drinfeld module and show that the category HHBHYD is a braided monoidal category. We also study the connection between the category HHBHYD and the category HM of the left co-modules over a coquasitriangular BiHom–bialgebra (H,σ). Secondly, we prove that the category of finitely generated projective left–left BiHom–Yetter–Drinfeld modules is closed for left and right duality. Full article
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