On Lie gradings III. Gradings of the real forms of classical Lie algebras

Abstract

Maximal Abelian subgroups of diagonalizable automorphisms of Lie algebra (so-called MAD-groups) play a crucial role in the construction of fine gradings of Lie algebra. Our aim is to give a description of MAD-groups for real forms of classical Lie algebras. We introduce four types of matrix subgroups of Gl(n, C) called Out-groups, Ad-groups, Out * -groups and Ad * -groups. For each type of these subgroups, we define a relation of equivalence. The problem of classifying of all non-conjugate MAD-groups on real forms of sl(n, C) , o(n, C) or sp(n, C) is transformed to the problem of classifying these equivalence classes. The classification of these equivalence classes is presented here.