The Grothendieck ring of Yetter-Drinfeld modules over a class of 2n²-dimensional Kac-Paljutkin Hopf algebras

Abstract

As is well known, a class of $2n^2$-dimensional Kac-Paljutkin Hopf algebras $H_{2n^2}$ was introduced by Pansera. It is the generalization of the 8-dimensional Kac-Paljutkin Hopf algebra H₈. In this paper, the classification of Yetter-Drinfeld modules over $H_{2n^2}$ is given by Radford’s method of constructing Yetter-Drinfeld modules for a Hopf algebra. Furthermore, the tensor product of Yetter-Drinfeld modules over $H_{2n^2}$ is established, and the Grothendieck ring $r{(}_{H_{2n^2}}^{H_{2n^2}}\mathcal{Y}\mathcal{D})$ is described explicitly by generators and relations.

Communicated by Julia Plavnik

Keywords:

• Grothendieck ring
• Kac-Paljutkin algebra
• Yetter-Drinfeld module
• tensor product

2020 Mathematics Subject Classification:

• 16D70
• 16T05
• 16T99

Additional information

Funding

Supported by National Natural Science Foundation of China (gant no. 11671024).