The Isomorphism Theorems Of Lie Triple Systems

The concept of Lie triple systems is a natural generalization of that of Lie algebras and has intimate relations with Lie algebras. The paper presents results concerning the relationships and isomorphisms among Lie triple systems, standard imbedding Lie algebras and involutions of Lie algebras.Section 1,begins with the basic concepts and examples of Lie triple system,such as definitions of subsystems, ideals, simple Lie triple systems, solvability, semi-simplicity, standard imbedding Lie algebras,and homomorphisms.In section 2, we firstly give the relationship between the involutionary automorphism of Lie algebra and standard imbedding Lie algebra of Lie triple system. Then we prove that the isomorphism of Lie triple systems implies the isomorphism of their standard imbedding Lie algebras. At last, we get the classification of conjugate classes of the involutions of simple Lie algebra is equivalent to the classification of the simple Lie triple systems determined by involutions of Lie algebra.