Almost Gentle Algebras and their Trivial Extensions

Edward L Green,Sibylle Schroll

doi:10.1017/s001309151800055x

Abstract

AbstractIn this paper we define almost gentle algebras, which 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 a hypergraph is associated with every almost gentle algebra A, and that this hypergraph induces the Brauer configuration of the trivial extension of A. Among other things, this gives a combinatorial criterion to decide when two almost gentle algebras have isomorphic trivial extensions.

Highlights

In this paper we introduce a new class of multiserial algebras called almost gentle algebras
In the other direction we show that every symmetric special multiserial algebra with no powers in the relations or equivalently that every Brauer configuration algebra with multiplicity function equal to one, is the trivial extension of an almost gentle algebra
KQ/(IΛ ∩ KQ) is a special multiserial algebra and we have shown that KQ/(IΛ ∩ KQ) is an almost gentle algebra

Summary

Introduction

In this paper we introduce a new class of multiserial algebras called almost gentle algebras. Examples of trivial extensions of almost gentle algebras appear, for example, in the derived equivalence classification of symmetric algebras of finite and tame representation type, see [29] and the references within. In the other direction we show that every symmetric special multiserial algebra with no powers in the relations or equivalently that every Brauer configuration algebra with multiplicity function equal to one, is the trivial extension of an almost gentle algebra (see [16] for the definition of Brauer configuration algebras). We note that this almost gentle algebra is not unique. It follows that two almost gentle algebras have the same trivial extensions if and only if they have the same associated hypergraph

Almost gentle algebras

Trivial extension of an almost gentle algebra

Admissible cuts

The oriented hypergraph of an almost gentle algebra