On The Classification Of Lie Triple Systems

As an algebraic system, Lie triple systems arise upon consideration of certain sub-spaces of Riemannian manifolds, the totally geodesic submanifolds. Lie triple systems with a ternary composition are intimately linked with Lie algebras. On one hand, Lie triple systems can be treated as a natural ternary generalization of Lie algebras, that is, a Lie algebra g with [xyz] = [[x, y],z] is a Lie triple system. On the other hand, a Lie triple system can be imbedded in a Lie algebra. In this paper, we mainly investigate the classification of simple Lie triple systems over the complex and real fields, and give some results about the complexification of a real Lie triple system and the real forms of a complex Lie triple system.We first discuss the classification of complex simple Lie triple systems using the classification of finite order automorphisms for each complex simple Lie algebra given by Kac.In another part of this paper, we mainly study the classification of real simple Lie triple systems and obtain that a real simple Lie triple system is either isomorphic to a real form of a complex simple Lie triple system or to the rcalification of a complex simple Lie triple system.