Dimension of the Mesh Algebra of a Finite Auslander–Reiten Quiver

Abstract

Translation quivers appear naturally in the representation theory of finite dimensional algebras; see, for example, Bongartz and Gabriel (Bongartz, K., Gabriel, P., (1982). Covering spaces in representation theory. Invent. Math. 65:331–378.). A translation quiver defines a mesh algebra over any field. A natural question arises as to whether or not the dimension of the mesh algebra depends on the field. The purpose of this note is to show that the dimension of the mesh algebra of a finite Auslander–Reiten quiver over a field is a purely combinatorial invariant of this quiver. Indeed, our proof yields a combinatorial algorithm for computing this dimension. As a further application, one may use then semicontinuity of Hochschild cohomology of algebras as in Buchweitz and Liu (Buchweitz, R.-O., Lui, S. Hochschild cohomology and representation-finite algebras. Preprint.) to conclude that a finite Auslander–Reiten quiver contains no oriented cycle if its mesh algebra over some field admits no outer derivation.

Key Words:

• Translation quiver
• Mesh algebra
• Auslander-Reiten quiver

Acknowledgment

Both authors gratefully acknowledge partial support from the Natural Sciences and Engineering Research Council of Canada.