Brauer configuration algebras: A generalization of Brauer graph algebras

Sibylle Schroll,Edward L Green

doi:10.1016/j.bulsci.2017.06.001

Sibylle Schroll, Edward L Green

Open Access

https://doi.org/10.1016/j.bulsci.2017.06.001

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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.

Highlights

The classification of algebras into finite, tame and wild representation type has led to many structural insights in the representation theory of finite dimensional algebras
In this paper we introduce a new class of mostly wild algebras, called Brauer configuration algebras
As a first step in this direction, we show that the Brauer configuration yields the Loewy structure of the indecomposable projective modules of a Brauer configuration algebra and that the dimension of the algebra can be directly read off the Brauer configuration (Proposition 3.13)

Summary

Introduction

The classification of algebras into finite, tame and wild representation type has led to many structural insights in the representation theory of finite dimensional algebras. If an algebra has tame representation type its representation theory usually still exhibits a certain regularity in its structure making calculations and establishing and proving conjectures often possible One example of this is the class of Brauer graph algebras, which are, depending on their presentation, known as symmetric special biserial algebras [R, S]. Brauer configuration algebras contain another class of well-studied algebras that have a combinatorial presentation in the form of a finite graph, namely that of symmetric algebras with radical cube zero [GS]. Almost all representatives of derived equivalence classes of symmetric algebras of finite representation type in the classification by Skowronski et al, see [Sk] and the references within, are Brauer configuration algebras.

Brauer Configurations

Brauer configuration algebras

Properties of Brauer configuration algebras

Radical cubed zero Brauer configuration algebras