Bi-isotropic decompositions of semisimple Lie algebras and homogeneous bi-Lagrangian manifolds

摘要

Let g be a real semisimple Lie algebra with Killing form B and l a B-nondegenerate subalgebra of g of maximal rank. We give a description of all -invariant decompositions g=l+m++m- such that B|m±=0, B(l,m++m-)=0 and l+m± are subalgebras. It is reduced to a description of parabolic subalgebras of g with given reductive part l. This is obtained in terms of crossed Satake diagrams. As an application, we get a classification of invariant bi-Lagrangian (or equivalently para-Kahler) structures on homogeneous manifolds G/K of a semisimple group G.