Quivers and Lattices

Kevin MeGerty (University of Chicago)
March 10, 2005

Abstract: A quiver is simply a directed graph. By a representation of a quiver we mean a collection of vector spaces indexed by the nodes of the graph, together with linear maps corresponding to the arrows of the graph. We will introduce the notions of simple and indecomposable representations of a quiver (which may be considered part of the Abelian category structure of quiver representations). Gabriel asked which quivers have "finite representation type", that is, which quivers have finitely many indecomposables, and discovered a beautiful connection to Lie theory: The quivers with this property are precisely those whose underlying undirected graph is the Dynkin diagram of a Lie algebra. Moreover the indecomposable objects themselves are indexed by the roots of the associated Lie algebra. We will give an account of this theorem.