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Title:  Dimension of the Mesh Algebra of a Finite Auslander–Reiten Quiver 
Authors:  Buchweitz, RagnarOlaf Liu, Shiping 
Keywords:  Translation quiver Mesh algebra AuslanderReiten quiver 
Issue Date:  Jan2003 
Publisher:  Taylor & Francis 
Citation:  Communications in Algebra, 31, (5), 2207  2217 
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 representationfinite 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. 
URI:  http://hdl.handle.net/1807/9839 
Appears in Collections:  Mathematics

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