Representations of Modular Skew Group Algebras

TL;DR

This paper investigates the representation theory of skew group algebras formed from finite-dimensional algebras and finite groups, providing classifications of their global dimension, representation type, and conditions for being generalized Koszul.

Contribution

It characterizes skew group algebras with finite global dimension or finite representation type and classifies representation types of transporter categories, also establishing conditions for generalized Koszul property preservation.

Findings

Characterization of skew group algebras with finite global dimension.

Classification of representation types of transporter categories.

Equivalence of generalized Koszul property between $a0$ and $a0$.

Abstract

In this paper we study representations of skew group algebras $\Lambda G$, where $\Lambda$ is a connected, basic, finite-dimensional algebra (or a locally finite graded algebra) over an algebraically closed field $k$ with characteristic $p \geqslant 0$, and $G$ is an arbitrary finite group each element of which acts as an algebra automorphism on $\Lambda$. We characterize skew group algebras with finite global dimension or finite representation type, and classify the representation types of transporter categories for $p \neq 2,3$. When $\Lambda$ is a locally finite graded algebra and the action of $G$ on $\Lambda$ preserves grading, we show that $\Lambda G$ is a generalized Koszul algebra if and only if so is $\Lambda$.