Multiserial and special multiserial algebras and their representations

TL;DR

This paper explores multiserial and special multiserial algebras, generalizing biserial types to wild representation algebras, and establishes their properties and connections to Brauer configuration algebras.

Contribution

It introduces the concept of multiserial modules, proves all modules over special multiserial algebras are multiserial, and links symmetric special multiserial algebras to Brauer configuration algebras.

Findings

All finitely generated modules over special multiserial algebras are multiserial.

Symmetric special multiserial algebras coincide with Brauer configuration algebras.

Any symmetric algebra with radical cube zero is a Brauer configuration algebra.

Abstract

In this paper we study multiserial and special multiserial algebras. These algebras are a natural generalization of biserial and special biserial algebras to algebras of wild representation type. We define a module to be multiserial if its radical is the sum of uniserial modules whose pairwise intersection is either 0 or a simple module. We show that all finitely generated modules over a special multiserial algebra are multiserial. In particular, this implies that, in analogy to special biserial algebras being biserial, special multiserial algebras are multiserial. We then show that the class of symmetric special multiserial algebras coincides with the class of Brauer configuration algebras, where the latter are a generalization of Brauer graph algebras. We end by showing that any symmetric algebra with radical cube zero is special multiserial and so, in particular, it is a Brauer configuration algebra.