Group gradings on simple lie algebras of cartan and melikyan type

• Main Author: McGraw, Jason Melvin
• Other Authors: Memorial University of Newfoundland. Dept. of Mathematics and Statistics
• Format: Thesis
• Language: English
• Published: 2010
• Subjects: Abelian groups; Automorphisms; Finite groups; Lie algebras; Newfoundland studies; University of Newfoundland
• Online Access: http://collections.mun.ca/cdm/ref/collection/theses4/id/59216

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).