Some Results On Leibniz Algebra

This paper proves that Derivaion algebras of perfect Leibniz algebra with O right annihilator is complete and discusses the decomposition of complete Leibniz algebra by providing some elementary definitions and properties of Leibniz algebra. Further, this kind of decomposition is uniqueness under the condition of disregarding the decomposition order. On the other hand, some elementary definitions of Leibniz superalgebra and quadratic Leibniz algebra are provided and the decomposition of quadratic Leibniz algebra are proved.The main results in this paper are the following:Theoreml: Let L be a perfect Leib algebra with zero right annihilator(i.e. [L,L]=L, Z(L)=0. ) Then we have the derivation algebra DerL is complete. Theorem2: Let L be a complete Leib algebra, then we havewhere every L_i is simple complete Leib algebra and is an ideal of L, and if L_t only have the decomposition L_i = L_i (?) {0}, we can prove this kind of decomposition is uniqueness under the condition of disregarding the decomposition order. Theorem3: Let (L,B) be a quadratic Leib algebra. Thensuch that for all 1 ≤ i ≤ r(1) L_i is a non-degenerate ideal,(2) L_i contains no nontrivial non-degenerate ideal of L,(3) for all i ≠ j,L_i and L_j are orthogonal.