Grothendieck Rings of 2$n^2$ dimensional Hopf Algebras $H_{2n^2}$

TL;DR

This paper constructs and describes the Grothendieck rings of a family of semisimple Hopf algebras of dimension 2n^2, generalizing known low-dimensional cases, and classifies their irreducible modules.

Contribution

It explicitly constructs the Grothendieck ring for the class of 2n^2-dimensional Hopf algebras and classifies all their irreducible modules, extending previous specific cases.

Findings

Explicit description of the Grothendieck ring $r(H_{2n^2})$

Classification of all irreducible modules of $H_{2n^2}$

Generalization of the 8-dimensional Kac-Paljutkin algebra

Abstract

In this paper, we construct the Grothendieck ring of a class of 2$n^2$-dimension semisimple Hopf Algebras $H_{2n^2}$, which can be viewed as a generalization of the 8-dimension Kac-Paljutkin Hopf algebra $K_8$. All irreducible $H_{2n^2}$-modules are classified. Furthermore, we describe the Grothendieck ring $r(H_{2n^2})$ by generators and relations explicitly.