Lie On The System

In this paper, we first show that every Lie supertriple system can be obtainded by some a Lie superalgebra. And then, we obtain some basic solutions of Lie supertriple system. Finally, we show that an invariant supersymmetric bilinear form on a Lie supertriple system can be uniquely extended to its standard imbedding Lie superalgebra.The main results in this paper are the following:Theorem 1 If V is a Lie supertriple system such that L = V [V, V], and the Lie superalgebra L whose Killing form is non-degenerate, then erery derivation of V is inner.Theorem 2 Let Φ be a supersymmetric right invariant bilinear form on a Lie supertriple system. There is a unique supersymmetric invariant bilinear form Φ on the standard imbedding Lie superalgebra L(V'). satisfying1) Φ |v= 02)Φ (H,V) =0Moreover, 0 is nondegenerate if and only if Φ is nondegenerate.