The Unique Decomposition And The Universal Central Extension Of Lie Poisson Algebra

Lie Poisson algebras have been developed from Lie algebras and Poisson algebras,which have two structures.The aim of this paper is to study the decomposition and universal central extensions of Lie Pois-son algebras, In section 1,we first present all necessary definitions,such as subalgebras,ideals,homomorphism in Lie Poisson algebras. Then by introducing T-endomorphisms,we discuss the uniqueness of the decom-position of Lie Poisson algebras with trival center. In section 2, we give some properties of the central extensions and universal central extensions of Lie Poisson algebras,by constructing universal central ex-tension,we obtain that the existence of the universal covering if and only if the Lie Poisson algebras are perfect. Last we study the lifting of automorphisms and derivations of Lie Poisson algebras.We have the following theorems:Theorem 1:Let T be a Lie Poisson algebra,and CT=0, T has the ideals decompositions T=K1⊕and can not be divded. Then r=s and exchange the order,we have Ki=Li,i=1,2,..., r.Theorem 2:A Lie Poisson algebra has the universal covering if and only if the Lie Poisson algebras is perfect.Theorem 3:Let T be a perfect Lie Poisson algebra,then Aut(T)→{h∈Aut(u(T))|h(Keru)=Keru}: g→u(g) is a group isomorphism. In particular, if CT=0,Then Aut(T)=Aut(u(T)).