Valued Graphs and the Representation Theory of Lie Algebras
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,...
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| Format: | Article |
| Language: | English |
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MDPI AG
2012-07-01
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| Series: | Axioms |
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| Online Access: | http://www.mdpi.com/2075-1680/1/2/111 |
| _version_ | 1818269012245610496 |
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| author | Joel Lemay |
| author_facet | Joel Lemay |
| author_sort | Joel Lemay |
| collection | DOAJ |
| description | 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 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. 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. |
| first_indexed | 2024-12-12T20:47:37Z |
| format | Article |
| id | doaj.art-374cd7cf3769441fb8dfa40206203da8 |
| institution | Directory Open Access Journal |
| issn | 2075-1680 |
| language | English |
| last_indexed | 2024-12-12T20:47:37Z |
| publishDate | 2012-07-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Axioms |
| spelling | doaj.art-374cd7cf3769441fb8dfa40206203da82022-12-22T00:12:31ZengMDPI AGAxioms2075-16802012-07-011211114810.3390/axioms1020111Valued Graphs and the Representation Theory of Lie AlgebrasJoel LemayQuivers (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 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. 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.http://www.mdpi.com/2075-1680/1/2/111quiverspecieslie algebrarepresentation theoryroot systemvalued graphmodulated quivertensor algebrapath algebraRingel–Hall algebra |
| spellingShingle | Joel Lemay Valued Graphs and the Representation Theory of Lie Algebras Axioms quiver species lie algebra representation theory root system valued graph modulated quiver tensor algebra path algebra Ringel–Hall algebra |
| title | Valued Graphs and the Representation Theory of Lie Algebras |
| title_full | Valued Graphs and the Representation Theory of Lie Algebras |
| title_fullStr | Valued Graphs and the Representation Theory of Lie Algebras |
| title_full_unstemmed | Valued Graphs and the Representation Theory of Lie Algebras |
| title_short | Valued Graphs and the Representation Theory of Lie Algebras |
| title_sort | valued graphs and the representation theory of lie algebras |
| topic | quiver species lie algebra representation theory root system valued graph modulated quiver tensor algebra path algebra Ringel–Hall algebra |
| url | http://www.mdpi.com/2075-1680/1/2/111 |
| work_keys_str_mv | AT joellemay valuedgraphsandtherepresentationtheoryofliealgebras |