Almost gentle algebras and their trivial extensions

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.