Group gradings on simple lie algebras of cartan and melikyan type
Thesis (Ph.D.)--Memorial University of Newfoundland, 2010. Mathematics and Statistics Includes bibliographical references (leaves 88-91) In this thesis we explore the gradings by groups on the simple Cartan type Lie algebras and Melikyan algebras over algebraically closed fields of positive characte...
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ftmemorialunivdc:oai:collections.mun.ca:theses4/59216 2023-05-15T17:23:33+02:00 Group gradings on simple lie algebras of cartan and melikyan type McGraw, Jason Melvin Memorial University of Newfoundland. Dept. of Mathematics and Statistics 2010 vi, 91 leaves Image/jpeg; Application/pdf http://collections.mun.ca/cdm/ref/collection/theses4/id/59216 Eng eng Electronic Theses and Dissertations (9.04 MB) -- http://collections.mun.ca/PDFs/theses/McGraw_JasonMelvin.pdf a3496973 http://collections.mun.ca/cdm/ref/collection/theses4/id/59216 The author retains copyright ownership and moral rights in this thesis. Neither the thesis nor substantial extracts from it may be printed or otherwise reproduced without the author's permission. Paper copy kept in the Centre for Newfoundland Studies, Memorial University Libraries Abelian groups Automorphisms Finite groups Lie algebras Text Electronic thesis or dissertation 2010 ftmemorialunivdc 2015-08-06T19:22:05Z Thesis (Ph.D.)--Memorial University of Newfoundland, 2010. Mathematics and Statistics Includes bibliographical references (leaves 88-91) In this thesis we explore the gradings by groups on the simple Cartan type Lie algebras and Melikyan algebras over algebraically closed fields of positive characteristic p > 2 (p = 5 for the Melikyan algebras). -- We approach the gradings by abelian groups without p-torsion on a simple Lie algebra L by looking at the dual group action. This action defines an abelian semisimple algebraic subgroup (quasi-torus) of the automorphism group of L. A result of Platonov says that any quasi-torus of an algebraic group is contained in the normalizer of a maximal torus. We show that if L is a simple graded Cartan or Melikyan type Lie algebra, then any quasi-torus of the automorphism group of L is contained in a maximal torus. Thus all gradings by groups without p-torsion are, up to isomorphism, coarsenings of the eigenspace decomposition of a maximal torus in the automorphism group. We also give examples of gradings by the cyclic group of order p which do not follow the pattern of the general description of gradings by groups without p-torsion as well as describe gradings by arbitrary groups on the restricted Witt algebra W(1; 1). Thesis Newfoundland studies University of Newfoundland Memorial University of Newfoundland: Digital Archives Initiative (DAI) |
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Memorial University of Newfoundland: Digital Archives Initiative (DAI) |
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Abelian groups Automorphisms Finite groups Lie algebras |
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Abelian groups Automorphisms Finite groups Lie algebras McGraw, Jason Melvin Group gradings on simple lie algebras of cartan and melikyan type |
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Abelian groups Automorphisms Finite groups Lie algebras |
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Thesis (Ph.D.)--Memorial University of Newfoundland, 2010. Mathematics and Statistics Includes bibliographical references (leaves 88-91) In this thesis we explore the gradings by groups on the simple Cartan type Lie algebras and Melikyan algebras over algebraically closed fields of positive characteristic p > 2 (p = 5 for the Melikyan algebras). -- We approach the gradings by abelian groups without p-torsion on a simple Lie algebra L by looking at the dual group action. This action defines an abelian semisimple algebraic subgroup (quasi-torus) of the automorphism group of L. A result of Platonov says that any quasi-torus of an algebraic group is contained in the normalizer of a maximal torus. We show that if L is a simple graded Cartan or Melikyan type Lie algebra, then any quasi-torus of the automorphism group of L is contained in a maximal torus. Thus all gradings by groups without p-torsion are, up to isomorphism, coarsenings of the eigenspace decomposition of a maximal torus in the automorphism group. We also give examples of gradings by the cyclic group of order p which do not follow the pattern of the general description of gradings by groups without p-torsion as well as describe gradings by arbitrary groups on the restricted Witt algebra W(1; 1). |
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Memorial University of Newfoundland. Dept. of Mathematics and Statistics |
format |
Thesis |
author |
McGraw, Jason Melvin |
author_facet |
McGraw, Jason Melvin |
author_sort |
McGraw, Jason Melvin |
title |
Group gradings on simple lie algebras of cartan and melikyan type |
title_short |
Group gradings on simple lie algebras of cartan and melikyan type |
title_full |
Group gradings on simple lie algebras of cartan and melikyan type |
title_fullStr |
Group gradings on simple lie algebras of cartan and melikyan type |
title_full_unstemmed |
Group gradings on simple lie algebras of cartan and melikyan type |
title_sort |
group gradings on simple lie algebras of cartan and melikyan type |
publishDate |
2010 |
url |
http://collections.mun.ca/cdm/ref/collection/theses4/id/59216 |
genre |
Newfoundland studies University of Newfoundland |
genre_facet |
Newfoundland studies University of Newfoundland |
op_source |
Paper copy kept in the Centre for Newfoundland Studies, Memorial University Libraries |
op_relation |
Electronic Theses and Dissertations (9.04 MB) -- http://collections.mun.ca/PDFs/theses/McGraw_JasonMelvin.pdf a3496973 http://collections.mun.ca/cdm/ref/collection/theses4/id/59216 |
op_rights |
The author retains copyright ownership and moral rights in this thesis. Neither the thesis nor substantial extracts from it may be printed or otherwise reproduced without the author's permission. |
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