Almost gentle algebras and their trivial extensions

TL;DR

This paper introduces almost gentle algebras, a generalization of gentle algebras, and explores their trivial extensions, showing their relation to symmetric special multiserial and Brauer configuration algebras with combinatorial criteria.

Contribution

It defines almost gentle algebras, establishes their trivial extensions as Brauer configuration algebras, and links them via hypergraphs to combinatorial criteria for isomorphism.

Findings

Trivial extension of an almost gentle algebra is a symmetric special multiserial algebra.

Every almost gentle algebra is an admissible cut of a unique Brauer configuration algebra.

A hypergraph associated to an almost gentle algebra determines its trivial extension's Brauer configuration.

Abstract

In this paper we define almost gentle algebras. They are monomial special multiserial algebras generalizing gentle algebras. We show that the trivial extension of an almost gentle algebra by its minimal injective co-generator is a symmetric special multiserial algebra and hence a Brauer configuration algebra. Conversely, we show that any almost gentle algebra is an admissible cut of a unique Brauer configuration algebra and as a consequence, we obtain that every Brauer configuration algebra with multiplicity function identically one, is the trivial extension of an almost gentle algebra. We show that to every almost gentle algebra A is associated a hypergraph, and that this hypergraph induces the Brauer configuration of the trivial extension of A. Amongst other things, this gives a combinatorial criterion to decide when two almost gentle algebras have isomorphic trivial extensions.