The Killing Forms Of Lie Triple Systems

A Lie triple system, which is a ternary composition, can be treated as a generalization of the Lie algebra. Lie triple systems arose in Cartan's studies of Riemannian geometry, i.e., the totally geodesic submanifolds. A Lie algebra, together with [xyz] = [[x, y], z], is a Lie triple system. Moreover, we can get the Lie algebras by the standard imbedding Lie algebras.In this paper, we mainly study the qualities of the Killing forms of Lie triple systems, and give some results about the complexification of a real Lie triple system and the real forms of a complex Lie triple system.Accoding to the definitions of the Killing form that define by Merberg([1]), we mainly discuss some relations of the Killing form of T and its standard imbedding Lie algebra L(T). And using the same methods in Lie algebras, we prove that T is solvable if and only if the Killing form is 0 and T is semi-simple if and only if the Killing form is non-degenerate.In another part of this paper, as in Lie algebras, we give definitions of the complexification of a real Lie triple systems and real forms of a complex Lie triple system and also make a definition of compatible complex structure. Then, by using the Killing form and non-degeneracy for the Killing form of Lie triple system, we investigate the semisimplicity of T₀, T, T^R.