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

Maximal Abelian subgroups of diagonalizable automorphisms of Lie algebra (s o-called MAD-groups) play a crucial role in the construction of fine gradin gs 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 subgroup s 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 prob lem 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 e quivalence classes. The classification of these equivalence classes is pres ented here. (C) 2000 Published by Elsevier Science Inc. All rights reserved .