Skew group algebras, invariants and Weyl Algebras

TL;DR

This paper explores the structure and relationships of skew group algebras, invariants, and Weyl algebras, extending known results from polynomial rings to homogenized Weyl algebras and analyzing module categories.

Contribution

It generalizes results on invariants and skew group algebras from polynomial rings to homogenized Weyl algebras, especially for the case n=1.

Findings

Established relations between skew group algebras and invariant rings for homogenized Weyl algebras.

Extended known results from polynomial rings to homogenized Weyl algebras.

Analyzed module categories over invariant rings and skew group algebras.

Abstract

The aim of this paper is two fold: First to study finite groups $G$ of automorphisms of the homogenized Weyl algebra $B_{n}$, the skew group algebra $B_{n}\ast G$, the ring of invariants $B_{n}^{G}$, and the relations of these algebras with the Weyl algebra $A_{n}$, with the skew group algebra $A_{n}\ast G$, and with the ring of invariants $A_{n}^{G}$. Of particular interest is the case $n=1$. In the on the other hand, we consider the invariant ring $\QTR{sl}{C}[X]^{G}$ of the polynomial ring $K[X]$ in $n$ generators, where $G$ is a finite subgroup of $Gl(n,\QTR{sl}{C}$) 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 $\QTR{sl}{C}[X]^{G}$ and the corresponding category over the skew group algebra $\QTR{sl}{C}% [X]\ast G$. 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 $B_{n}$.