McGraw, Jason Melvin (2010) Group gradings on simple lie algebras of cartan and melikyan type. Doctoral (PhD) thesis, Memorial University of Newfoundland.
- Accepted Version
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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).
|Item Type:||Thesis (Doctoral (PhD))|
|Additional Information:||Includes bibliographical references (leaves 88-91).|
|Department(s):||Science, Faculty of > Mathematics and Statistics|
|Library of Congress Subject Heading:||Abelian groups; Automorphisms; Finite groups; Lie algebras|
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