Poisson Algebraic Structure Analysis And Research

The deformation theory of algebras was introduced by Gerstenhaber.It has subse-quently been extended by Gerstenhaber and Schack to contravariant functors from a smallcategory to algebras,and to bialgebras and Hopf algebras. Most recently, it has beenapplied by Flato et al. to algebras which carry both an associative and a Lie structurewith the Lie bracket acting as derivations of the associative structure, e.g., Poisson alge-bras (where the associative multiplication is commutative). They consider a pair (A,L)consisting of an associative algebra A, a Lie algebra L, together with a Lie morphism pfrom L into the Lie algebra of derivations of A; note that L need not be identical as avector space with A. They call this a "Leibniz pair" and consider the problem of si-multaneously deforming the triple. It can happen that L is identical with A. When thisis so, one has non-commutative Poisson algebras.For finite-dimensional ones, the mostimportant examples of finite-dimensional non-commutative Poisson algebras are the n×n full matrix algebra M_n(k),it has a standard structure: {-,-}=[-,-]. Using two differentmeanings,I make a concrete proof, moreover I show that how p comes from. Next we discusswhen its subalgebras have a standard structure. Here we show that if A is semisimple asassociative algebra, then it has a standard structure, on the other hand, if it is semisimpleas Lie algebras then its associative products are trivial. We also give the descriptions of thestructures of finite-dimensional non-commutative Poisson algebras whose Lie structuresare reductive. At last, we get a structure theorem of a finite-dimensional non-commutativePoisson algebra.Poisson algebra play an important role in many mathematical branches related toquantized algebras. Many quantum groups have been constructed from Poisson algebraswhich are polynomial rings with certain Poisson brackets. For a Poisson algebra with Pois-son bracket {-,-}, Letα,δbe linear maps from A into itself. Here we find a necessary andsufficient condition such that the polynomial ring A[x] has Poisson bracket.Next we givea Poisson structure of skew polynomial algebra and prove that it has no other structure.The universal mapping property of a skew enveloping algebra are studied and it isproved that every Poisson enveloping algebra is a homomorphic image of a skew envelop-ing algebra.The paper also give a natural Hopf structure of the Poisson enveloping algebra U(A) for a Poisson Hopf algebra A.We introduce and characterize the inner Poisson algebras on a given associative al-gebra A by a classes of linear transformations of A. This gives a way of constructing non-commutative Poisson structures. Applying these to the finite-dimensional path algebrask(?),together with the decomposition into indecomposable Lie ideals of the standard Pois-son structure on k(?),we classify all the inner Poisson structures on k(?),which turn out tobe the piecewise scale Poisson algebras.In this paper I give some examples to learn thatimportant theorems.