Group Gradings on Simple Lie Algebras of Cartan and Melikyan Type

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 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).;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.