Hopf quasicomodules and Yetter-Drinfel’d quasicomodules

Abstract

Let H be a Hopf coquasigroup over a field k possessing an adjoint quasicoaction. We first show that if M is any right H-module and N is any right H-quasicomodule such that ${\tau }_{M,N}°{\tau }_{N,M}=i{d}_{N\otimes M}$, where ${\tau }_{N,M}:N\otimes M\to M\otimes N$ is a favorable map, then we have H = k. As an application of this result, we get that symmetric category ${}_H^H\mathcal{Y}\mathcal{D}\mathcal{Q}$ of Yetter-Drinfeld quasicomodules over H is trivial, as a generalization of Pareigis’ Theorem. Furthermore, let (H, R) be a quasitriangular Hopf coquasigroup and $(B,\sigma )$ coquasitriangular Hopf coquasigroup. Then, we show that the category of generalized Long quasicomodules ${}_H^B\mathcal{L}\mathcal{Q}$ is a braided monoidal subcategory of Yetter-Drinfeld category ${}_{H\otimes B}^{H\otimes B}\mathcal{Y}\mathcal{D}\mathcal{Q}$. Finally, we give a new approach to a braided monoidal category by generalizing one of Schauenburg’s main results in the setting of Hopf coquasigroups introduced by Klim and Majid. This yields new sources of braidings that provide solutions to the Yang-Baxter equation playing an important role in various areas of mathematics.

Keywords:

• Braided monoidal category
• Hopf coquasigroup
• Hopf quasicomodule
• long quasicomodule
• Yang-Baxter equation
• Yetter-Drinfel’d quasicomodule

MATHEMATICS SUBJECT CLASSIFICATIONS (2010):

• 16W30
• 16T05
• 16T25
• 16T05
• 17A32
• 18D10

Acknowledgments

The authors are very grateful to the anonymous referee for his/her thorough review of this work and his/her comments. The authors thank Prof. S. Majid for a discussion about this topic and his very helpful comments. The authors also thank Tao Zhang for her helpful discussion.

Additional information

Funding

The work was partially supported by the NSF of China (no. 11371088, 11571173, and 11871144), and the NSF of Jiangsu Province (BK20171348).