The Ext algebra of a Brauer graph algebra

Abstract

In this paper we study finite generation of the Ext algebra of a Brauer graph algebra by determining the degrees of the generators. As a consequence we characterize the Brauer graph algebras that are Koszul and those that are K2.

Dessins d'enfants and Brauer configuration algebras

“…The relations ρ D are not necessarily a minimal set of relations for I D . The relations of type one and three are always minimal, however relations of type two are often redundant; this is a generalisation of a similar result for Brauer graph algebras [10].…”

“…Evidently this observation extends to I Dn . Ideals generated by elements of length 2 are called quadratic and the corresponding Brauer configuration algebra is a Brauer graph algebra and by [10] it is Koszul. 6.…”

The Ext-algebra of the Brauer tree algebra associated to a line

“…Here, unlike in [4] we assume that there are no exceptional vertex. The edges are labelled by the simple A-modules S 1 , .…”

“…Let us emphasise that Ext-algebras for more general biserial algebras were explicitly computed by Green-Schroll-Snashall-Taillefer in [4], but under some assumption on the multiplicity of the the vertices, assumption which is not satisfied for the simple example treated in this note. Other general results relying on Auslander-Reiten theory were obtained by Antipov-Generalov [1] and Brown [3].…”

Special multiserial algebras, Brauer configuration algebras and more: A survey

“…We show that the generalizations satisfy properties that are analogous to the properties of the algebras they generalize. Biserial, special biserial, and Brauer graph algebras have been extensively studied, see, for example, [1,2,5,7,10,11,13,14,19,20,23,25,27,28,32,31,34] and also the survey articles [29,30] and the references therein.…”