Radford (m, n)-biproduct and (m+n)-Yetter-Drinfeld category

Abstract

In this article, firstly, we introduce the notions of m-smash product monoidal Hom-algebra $B{\#}^{m}H$ and n-smash coproduct monoidal Hom-coalgebra $B{\times }_{n}H.$ Furthermore, the necessary and sufficient conditions for $B{\#}^{m}H$ and $B{\times }_{n}H$ on $B\otimes H$ to be a monoidal Hom-bialgebra structure are derived, where the conditions are equivalent to that $(B,\beta )$ is a bialgebra in the $(m+n)$-Yetter-Drinfeld category. Secondly, we prove that the category is a braided monoidal category. At last, we give two mapping descriptions of Radford (0, 0)-biproduct.

Communicated by Jason P. Bell

Keywords:

• Mapping description
• Radford biproduct
• Yetter-Drinfeld category

MATHEMATICS SUBJECT CLASSIFICATION:

• 16T05
• 16T99

Acknowledgments

The authors are deeply indebted to the referee for his/her very useful suggestions and some improvements to the original manuscript.

Additional information

Funding

This work was partially supported by Natural Science Foundation of Henan Province (No. 20A110019) and National Natural Science Foundation of China (No. 11771069).