Representations of equipped graphs: Auslander-Reiten theory
William Crawley-Boevey
TL;DR
This paper explores the representation theory of equipped graphs, extending classical concepts like roots and Auslander-Reiten theory from quivers to equipped graphs without fixed orientations.
Contribution
It surveys previous results on dimension vectors of indecomposables and develops Auslander-Reiten theory for equipped graphs, including examples of Auslander-Reiten quivers.
Findings
Dimension vectors of indecomposables are positive roots.
Development of Auslander-Reiten theory for equipped graphs.
Examples of Auslander-Reiten quivers provided.
Abstract
Representations of equipped graphs were introduced by Gelfand and Ponomarev; they are similar to representation of quivers, but one does not need to choose an orientation of the graph. In a previous article we have shown that, as in Kac's Theorem for quivers, the dimension vectors of indecomposable representations are exactly the positive roots for the graph. In this article we begin by surveying that work, and then we go on to discuss Auslander-Reiten theory for equipped graphs, and give examples of Auslander-Reiten quivers.
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Taxonomy
TopicsAlgebraic structures and combinatorial models · Advanced Topics in Algebra · Homotopy and Cohomology in Algebraic Topology