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Axioms 2012, 1(2), 111148; doi:10.3390/axioms1020111
Article
Valued Graphs and the Representation Theory of Lie Algebras
Department of Mathematics and Statistics, University of Ottawa, Ottawa, K1N 6N5, Canada
Received: 13 February 2012 / Revised: 20 June 2012 / Accepted: 20 June 2012 / Published: 4 July 2012
(This article belongs to the Special Issue Hopf Algebras, Quantum Groups and YangBaxter Equations)
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Abstract
Quivers (directed graphs), species (a generalization of quivers) and their representations play a key role in many areas of mathematics including combinatorics, geometry, and algebra. Their importance is especially apparent in their applications to the representation theory of associative algebras, Lie algebras, and quantum groups. In this paper, we discuss the most important results in the representation theory of species, such as Dlab and Ringel’s extension of Gabriel’s theorem, which classifies all species of finite and tame representation type. We also explain the link between species and Kspecies (where K is a field). Namely, we show that the category of K species can be viewed as a subcategory of the category of species. Furthermore, we prove two results about the structure of the tensor ring of a species containing no oriented cycles. Specifically, we prove that two such species have isomorphic tensor rings if and only if they are isomorphic as “crushed” species, and we show that if K is a perfect field, then the tensor algebra of a K species tensored with the algebraic closure of K is isomorphic to, or Morita equivalent to, the path algebra of a quiver.Keywords:
quiver; species; lie algebra; representation theory; root system; valued graph; modulated quiver; tensor algebra; path algebra; Ringel–Hall algebra
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