Shephard-Todd-Chevalley Theorem for skew polynomial rings

TL;DR

This paper generalizes the Shephard-Todd-Chevalley Theorem to skew polynomial rings, characterizing when fixed subrings have finite global dimension based on the generating set of quasi-reflections.

Contribution

It extends the classical theorem to noncommutative skew polynomial rings and provides conditions for the structure of fixed subrings under group actions.

Findings

Fixed subring has finite global dimension iff generated by quasi-reflections.

Fixed subring is isomorphic to a skew polynomial ring.

Theorem applies to abelian groups acting on quantum polynomial rings.

Abstract

We prove the following generalization of the classical Shephard-Todd-Chevalley Theorem. Let $G$ be a finite group of graded algebra automorphisms of a skew polynomial ring $A:=k_{p_{ij}}[x_1,...,x_n]$. Then the fixed subring $A^G$ has finite global dimension if and only if $G$ is generated by quasi-reflections. In this case the fixed subring $A^G$ is isomorphic a skew polynomial ring with possibly different $p_{ij}$'s. A version of the theorem is proved also for abelian groups acting on general quantum polynomial rings.