# Auslander's Theorem for permutation actions on noncommutative algebras

## TL;DR

This paper generalizes Auslander's Theorem to various noncommutative algebras with permutation group actions, establishing isomorphisms of skew group algebras and fixed rings as graded algebras and singularities.

## Contribution

It extends Auslander's Theorem to noncommutative settings like skew polynomial rings, quantum Weyl algebras, and Sklyanin algebras, revealing new algebraic isomorphisms and singularity properties.

## Key findings

- Isomorphism of skew group algebra and endomorphism algebra for new classes
- Identification of fixed rings as graded isolated singularities
- Extension of Auslander's Theorem to noncommutative algebras

## Abstract

When $A = \mathbb{k}[x_1, \ldots, x_n]$ and $G$ is a small subgroup of $\operatorname{GL}_n(\mathbb{k})$, Auslander's Theorem says that the skew group algebra $A \# G$ is isomorphic to $\operatorname{End}_{A^G}(A)$ as graded algebras. We prove a generalization of Auslander's Theorem for permutation actions on $(-1)$-skew polynomial rings, $(-1)$-quantum Weyl algebras, three-dimensional Sklyanin algebras, and a certain graded down-up algebra. We also show that certain fixed rings $A^G$ are graded isolated singularities in the sense of Ueyama.