The Central Extension Of Leibnize _n-Lie Algebre

In this paper, the aim of us is to study the central extension of Leibnize n-Lie algebra and the decomposition and uniqueness of finite dimensional Leibnize n-Lie algebras with trivial center, and according to the decomposition, we obtain the decomposition of inner algebras and derivation algebras respectively.Theorem2.1.For a Leibnize n-Lie algebra L the following are equivalent.(1) L is simply connected. i.e. every central extension f :L′→L splits uniquely.(2) L is central closed, i.e. Id: L→L is a universal central extension, If u : L→M is a central extension, then (1) and (2) are also equivalent to.(3) u : L→Mis a universal central extension of M , In this case.(a)both L and M are perfect.(b) Z ( L) = U?1 (Z ( M)), U( Z(M )) =Z(M ).Theorem3.1. A is a finite-dimensional Leibnize n-Lie algebra with a decomposition A = A1⊕A2, where each Ai is a nonzero ideal of A . Then the following statements hold:(1) ( ) ( ) ( )Z A= ZA1⊕ZA2. (3.1)(2)If Z ( A) =0, then DerA = DerA1⊕DerA2. (3.2)(3) ( ) ( ) ( )L A= LA1⊕LA2. (3.3)Theorem3.2. Let A be a finite dimensional Leibnize n-Lie algebra with trivial center. Then we have(1) A can be decomposed into the direct sum of its indecomposable ideals.(2)The decomposition of A is uniqueness, that is if A = A1⊕A2⊕⊕Am, (3.10) and A = B1⊕B2⊕⊕Bs, (3.11) where A1 ,,Am and B 1 ,,BS are indecomposable. Then m= s and by an ordering of summands Ai = Bi(i =1,2,,m).