Brauer configuration algebras: A generalization of Brauer graph algebras

TL;DR

This paper introduces Brauer configuration algebras, a broad generalization of Brauer graph algebras, establishing their structural properties and relationships with other algebraic and graph-theoretic objects.

Contribution

It defines Brauer configuration algebras, proves they are finite dimensional symmetric algebras, and analyzes their structure and connections to graphs and matrices.

Findings

Brauer configuration algebras are finite dimensional symmetric algebras.

They are multiserial, with radicals decomposable into uniserial modules.

Established links between radical cubed zero algebras, graphs, and matrices.

Abstract

In this paper we introduce a generalization of a Brauer graph algebra which we call a Brauer configuration algebra. As with Brauer graphs and Brauer graph algebras, to each Brauer configuration, there is an associated Brauer configuration algebra. We show that Brauer configuration algebras are finite dimensional symmetric algebras. After studying and analysing structural properties of Brauer configurations and Brauer configuration algebras, we show that a Brauer configuration algebra is multiserial; that is, its Jacobson radical is a sum of uniserial modules whose pairwise intersection is either zero or a simple module. The paper ends with a detailed study of the relationship between radical cubed zero Brauer configuration algebras, symmetric matrices with non-negative integer entries, finite graphs and associated symmetric radical cubed zero algebras.