Skew group algebras, invariants and Weyl algebras

Abstract

The aim of this paper is two fold. We study first finite groups G of automorphisms of the homogenized Weyl algebra Bn, the skew group algebra Bn ∗ G, the ring of invariants BG n , and the relations of these algebras with the Weyl algebra An, with the skew group algebra An ∗G, and with the ring of invariants An . Of particular interest is the case n=1. On the other hand, we consider the invariant ring C[X]G of the polynomial ring C[X] in n generators, where G is a finite subgroup of Gl(n,C) such that any element in G different from the identity does not have one as an eigenvalue. We study the relations between the category of finitely generated modules over C[X]G and the corresponding category over the skew group algebra C[X]∗G. We obtain a generalization of known results for n=2 and G a finite subgroup of Sl(2,C ). In the last part of the paper we extend the results for the polynomial algebra C[X] to the homogenized Weyl algebra Bn Mathematics Subject Classification: Primary 16S30, 16P40; Secondary 16G70, 18E30