Valued Graphs and the Representation Theory of Lie Algebras

dc.contributor.authorLemay, Joel
dc.description.abstractQuivers (directed graphs) and species (a generalization of quivers) as well as 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 thesis, 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 K-species (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 that do not appear in the literature. 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.
dc.publisherUniversité d'Ottawa / University of Ottawa
dc.subjectLie algebra
dc.subjectRepresentation theory
dc.subjectRoot system
dc.subjectValued graph
dc.subjectValued quiver
dc.subjectModulated quiver
dc.subjectTensor algebra
dc.subjectPath algebra
dc.subjectRingel-Hall algebra
dc.titleValued Graphs and the Representation Theory of Lie Algebras
dc.faculty.departmentMathématiques et statistique / Mathematics and Statistics
dc.contributor.supervisorSavage, Alistair
dc.embargo.termsimmediate / Science / Science
uottawa.departmentMathématiques et statistique / Mathematics and Statistics
CollectionThèses, 2011 - // Theses, 2011 -

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