Fractional Brauer configuration algebras I: definitions and examples

TL;DR

This paper introduces fractional Brauer configuration algebras, extending previous algebraic structures, and characterizes their properties and classifications, especially in relation to Frobenius and representation-finite algebras.

Contribution

It generalizes Brauer configuration algebras to fractional versions and explores their algebraic properties and classification in relation to Frobenius and representation-finite algebras.

Findings

Fractional Brauer configuration algebras are locally bounded but not necessarily finite-dimensional.

Type S fractional Brauer configuration algebras are locally bounded Frobenius algebras.

Finite-dimensional indecomposable representation-finite fractional Brauer configuration algebras of type S match certain self-injective algebras.

Abstract

In 2017, Green and Schroll introduced a generalization of Brauer graph algebras which they call Brauer configuration algebras. In the present paper, we further generalize Brauer configuration algebras to fractional Brauer configuration algebras by generalizing Brauer configurations to fractional Brauer configurations. The fractional Brauer configuration algebras are locally bounded but neither finite-dimensional nor symmetric in general. We show that if the fractional Brauer configuration is of type S (resp. of type MS), then the corresponding fractional Brauer configuration algebra is a locally bounded Frobenius algebra (resp. a locally bounded special multiserial Frobenius algebra). Moreover, we show that over an algebraically closed field, the class of finite-dimensional indecomposable representation-finite fractional Brauer configuration algebras in type S coincides with the class of basic indecomposable finite-dimensional standard representation-finite self-injective algebras.