Uniqueness Of Decomposition Of Lie Supertriple Systems And Quadratic Lie Supertriple Systems

This paper first shows the dedecomposition of the center, the derivation superalgebras and inner derivation superalgebras respectively according to the decomposition of Lie Sup- ertriple Systems, it mainly consider the decomposition of Lie Supertriple Systems and qu- adratic Lie Supertriple Systems with trivial center and uniqueness , and the extension of the automorphism of a Lie Supertriple Systems.The main results in this paper are the following:Theorem 1 Let a Lie Supertriple Systems V be decomposed into a direct sum of two ideals,i.e. V = V1⊕V2,we have (1) ( ) ( ) ( )C V = C V1⊕C V2 , (2) If in addition C (V ) = 0,then ( ) ( ) ( )D V = D V1⊕D V2, ( ) ( ) ( )D0 V = D0 V1⊕D0 V2 ..Theorem 2 If Lie Supertriple Systems V has trivial center ,then (1)V has decompositions of indecomposable ideals, (2)If Lie Supertriple Systems V has decompositions of ideals where M i ,N jare indecomposable, i = 1,2, , m; j = 1,2, , n,then m = nand M i = N i, i = 1,2, ,nafter changing the orders.Theorem 3 An automorphism of a Lie Supertriple SystemsV can be extended to be an automorphism of ( ) ( )L0 V = D0 V⊕V.Theorem 4 Let (V ,φ) be a quadratic Lie Supertriple Systems with trivial center C (V ).Then V =⊕i =n1Vi such that for all 1≤i≤n, (1) Vi is a nondegenerate ideal of V ;(2) For all i≠j,Vi and V j are orthogonal relative toφ;(3) The above decomposition is unique up to permutations.