Solvable Lie Triple Supersystem And The Tensor Product

We give some elementary conception and properties of Lie triple supersys-tem, and discuss the tensor product of a commutative associative superal-gebra and a Lie triple supersystem, then we discuss some properties of the solvable and nilpotent Lie triple supersystem.The main results:Theorem 1 Let a Lie triple supersystem T be decomposed into a direct sum of two ideals, i.e. T = T₁ (?) T₂. We have(1) C(T) have the decomposition(2) If in addition C(T) = 0, thenTheorem 2 If A is a commutative associative superalgebra with a unit 1, and T is a Lie triple supersystem, then the derivation algebra of A(?)T is generated by DerA(?) id_T and A(?) DerT.Theorem 3 Any enveloping Lie superalgebra of a solvable Lie triple supersystem is solvable, and if a Lie triple supersystem has some solvable enveloping Lie superalgebra, it is solvable.Theorem 4 (1)Suppose (?) is an ideal of a Lie triple supersystem T. Then (?) is T-nilpotent if and only if that I((?)) is nilpotent.(2)N(L(T)) ∩T is unique maximal T-nilpotent ideal of T.(3)If (?)1 and (?)2 are T-nilpotent ideals of T, then (?)1 + (?)2 is a T-nilpotent ideal of T.(4) N(L(T)) = J(N(T)).