Semisimple Hopf algebras of dimension $2q^3$

TL;DR

This paper classifies semisimple Hopf algebras of dimension 2q^3 over an algebraically closed field of characteristic zero, showing they are constructed from group algebras or Radford's biproducts.

Contribution

It provides a complete classification of semisimple Hopf algebras of dimension 2q^3, detailing their construction methods from known algebraic structures.

Findings

H can be built from group algebras and their duals via extensions.

H can be realized as a Radford biproduct involving a group algebra of order 2.

Explicit classification of semisimple Hopf algebras of dimension 2q^3.

Abstract

Let $q$ be a prime number, $k$ an algebraically closed field of characteristic 0, and $H$ a non-trivial semisimple Hopf algebra of dimension $2q^3$. This paper proves that $H$ can be constructed either from group algebras and their duals by means of extensions, or from Radford's biproduct $H\cong R#kG$, where $kG$ is the group algebra of $G$ of order 2, $R$ is a semisimple Yetter-Drinfeld Hopf algebra in ${}^{kG}_{kG}\mathcal{YD}$ of dimension $q^3$.