The associated graded algebras of Brauer graph algebras I: finite representation type

Abstract

It is well known that Brauer graph algebras coincide with symmetric special biserial algebras and there has been a lot of work on Brauer graph algebras and their representation theory. Given a Brauer graph algebra A associated with a Brauer graph G, we denote by gr(A) the graded algebra associated with the radical filtration of A. We give a criterion for gr(A) to be representation-finite in terms of the graded degrees of vertices in G. Moreover, when gr(A) is representation-finite, we give the precise relationship between the Auslander–Reiten quiver of A and the Auslander–Reiten quiver of gr(A).

Keywords:

• Auslander–Reiten quiver
• Brauer graph algebra
• finite representation type
• graded algebra associated with the radical filtration
• graded degrees of vertices in a Brauer graph

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 16G10
• 16G20
• 16G60
• 16G70

Acknowledgments

We would like to thank Yu Ye and Steffen Koenig for their comments and suggestions on a previous version of this article. We would like to thank an anonymous referee for careful reading and many constructive suggestions, which have led to substantial changes and significant improvement on the presentation of this article.

Additional information

Funding

The authors are supported by NCET Program from MOE of China and by NSFC (No.12031014, No.11571329).