One Kind Of Classifications Of N-dimensional Lie Triple Systems

The concept of Lie triple system is a natural generalization of that of Lie algebra.In this paper, We classify a class of n-dimensional Lie triple systems over the complex number field. Let T be a n-dimensional vector-space with the multiplication such that [x, y,z]= f(y, z)x - f(x, z)y. Where f(x,y) is a symmetric bilinear functions . In theprogress of the paper. We exploit the standard form of the symmetric bilinear functions, and draw the conclusions that this kind of n-dimensional Lie triple system can be classified to n+1 types . then we discuss the simple, Solvable and Abel. Finally, We write out the matrix form of their derivation algebra.Section 1, begins with some basic definitions and basic properties of n-dimensional Lie triple systems, including subalgebras ideals, solvable ideals, simple Lie triple systems, semi-simple Lie triple systems, radicals and abelian Lie triple systems and so on ,and prove that R(T) is the maximal solvable ideal and T/R(T) is semi-simple .In section 2, we investigate the symmetric bilinear functions and their standard form over the complex (real) field .Section 3 takes up the classification of a n-dimensional Lie triple system over complex field and give a necessary and sufficient condition for which two n-dimensional, and discuss their simplicity solvability and their abelian properties .In the last section ,we discuss the derivation algebras and the properties of this class of Lie triple systems and get the matrix forms of the derivation algebras for the simple Lie triple systems in this class.