Actions of Drinfeld doubles of finite groups on Artin-Schelter regular algebras

TL;DR

This paper studies how Drinfeld doubles of finite groups act on Artin-Schelter regular algebras, providing explicit calculations for quaternion groups of order 8 and 16, revealing their invariant structures.

Contribution

It introduces the action of Drinfeld doubles on Artin-Schelter regular algebras and computes invariants explicitly for specific quaternion groups.

Findings

Explicit invariants for quaternion groups of order 8 and 16

Inner faithful actions of Drinfeld doubles on regular algebras

Structural insights into invariant subalgebras

Abstract

For a finite group $G$ and $\Bbbk$ an algebraically closed field of characteristic zero we consider Artin-Schelter regular algebras $A$ on which the Drinfeld double $D(G)$ acts inner faithfully, and its associated algebras of invariants $A^{D(G)}$. Explicit computations for the cases when $G$ is the (generalized) quaternion group of order 8 and 16 are given.