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Almost Gentle Algebras and their Trivial Extensions

Part of: Algebraic combinatorics Representation theory of rings and algebras

Published online by Cambridge University Press:  29 November 2018

Edward L. Green  and
Sibylle Schroll
Edward L. Green*
Affiliation:
Department of Mathematics, Virginia Tech, Blacksburg, VA 24061, USA ([email protected])
Sibylle Schroll
Affiliation:
Department of Mathematics, University of Leicester, University Road, Leicester LE1 7RH, UK ([email protected])
*
*Corresponding author.
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Abstract

In 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.

Keywords

gentle algebraspecial biserial algebrasymmetric special multiserial algebraBrauer configuration algebra

MSC classification

Primary: 16G20: Representations of quivers and partially ordered sets
Secondary: 16G10: Representations of Artinian rings 05E10: Combinatorial aspects of representation theory
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 62 , Issue 2 , May 2019 , pp. 489 - 504
Copyright
Copyright © Edinburgh Mathematical Society 2018 

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