On Lie Triple Systems

In this report, we shall give a structure theory of Lie triple systems in the first part as a collection of the work have been done including the relation between Lie triple systems and Lie algebras(Jordan algebras), some results on the solvable, the nilpotent and the semi-simple. Invariant bilinear forms will be also concerned. The second part we shall mainly connect Lie triple systems with some other algebras and then try to find more new results on Lie triple systems. W.G. Lister proved that a semi-simple Lie triple system can decomposed into direct sum of simple Lie triple systems with the tool of standard imbedding Lie algebra and gave the uniqueness of the decomposition. In the first section of this part we shall give the uniqueness of decomposition from Lie triple system's structure directly, and cover the result of the uniqueness of semi-simple Lie triple systems. In section part we shall discuss some properties of the tensor product of a communicative associative algebra and a Lie triple system, and then construct some finite dimensional Lie triple systems from Laurent-polynomial algebra and Novikov algebra. L. J. Santharouabane gave the definition of nilpotent Lie algebras of maximal rank and have shown some relationships between Kac-Moody algebras and such special kind of nilpotent Lie algebras as follows. For any nilpotent Lie algebra g of maximal rank and of type l ,with a minimal system of generators X = {e₁,e₂,…e_l}, there exists an l × l generalized Cartan matrix A = (a_ij) whose equivalence class is an invariant of g such that(ade_i)^-a_ije_j≠ 0, (ade_i)^-a_ij+1e_j = 0.Therefore, from the Kac-Moody algebra g(A) associated to A, a special kind of nilpotent Lie algebras of maximal rank with some universal property were constructed . In the third section, we shall generalize these notions and method to the research on nilpotent Lie triple systems.