Shephard-Todd-Chevalley Theorem for skew 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.