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Let A be a Poisson algebra over a field $$\mathbf{k}$$ with characteristic zero, let $$\gamma $$ , $$\alpha $$ be Poisson derivations on A such that $$\gamma \alpha =\alpha \gamma $$ and $$0\ne \rho \in \mathbf{k}$$ . Here the notion of a $$\gamma $$ -Poisson normal element is introduced, it is proved that the polynomial algebra A[y, x] has a Poisson structure defined by $$\{y,a\}=\alpha (a)y, \{x,a\}=\beta (a)x, \{x,y\}=\beta (y)x+\delta (y)$$ for $$a\in A$$ , where $$\beta $$ is a Poisson derivation on A[y] defined by $$\beta |_A=\gamma -\alpha $$ , $$\beta (y)=\rho y$$ and $$\delta $$ is a derivation on A[y] such that $$\delta |_A=0$$ , and its Poisson simplicity criterion is established and endorsed by examples.

Let R be an affine Poisson algebra over a field of characteristic zero and let A be a Poisson order over R. It is proved that every Poisson primitive ideal of A is annihilator of a simple Poisson A-module.

It is proved that the Poisson enveloping algebra of a double Poisson-Ore extension is an iterated double Ore extension. As an application, properties that are preserved under iterated double Ore extensions are invariants of the Poisson enveloping algebra of a double Poisson-Ore extension., 15 pages

Let $\widehat{\Gamma}$ be the natural map given in \cite[\S1]{Oh12}. Here, we construct a deformation $B_q$ of a Poisson algebra $B_1$ and a prime ideal $P$ of $B_q$ such that $\widehat{\Gamma}(P)$ is not a Poisson prime ideal of $B_1$., Comment: 4 pages

A natural map from a quantized space onto its semiclassical limit is obtained. As an application, we see that an induced map by the natural map is a homeomorphism from the spectrum of the multi-parameter quantized Weyl algebra onto the Poisson spectrum of its semiclassical limit., 17 pages

A Poisson algebra ℂ[G] considered as a Poisson version of the twisted quantized coordinate ring ℂq,p[G], constructed by Hodges et al. [11], is obtained and its Poisson structure is investigated. Th...

【Let H be a color Poisson bialgebra. Here we find a canonical nondegenerate bilinear form $\mathfrak{g}(H){\times}\mathfrak{m/m^2}$ , where $\mathfrak{g}(H)$ and $\mathfrak{m/m^2}$ are certain color Lie algebras induced by H.】

We observe \cite[Proposition 4.1]{LaLe} that Poisson polynomial extensions appear as semiclassical limits of a class of Ore extensions. As an application, a Poisson generalized Weyl algebra $A_1$ considered as a Poisson version of the quantum generalized Weyl algebra is constructed and its Poisson structures are studied. In particular, it is obtained a necessary and sufficient condition such that $A_1$ is Poisson simple and established that the Poisson endomorphisms of $A_1$ are Poisson analogues of the endomorphisms of the quantum generalized Weyl algebra., Comment: 10 pages

It gives a method to obtain a natural Lie bialgebra from a Poisson bialgebra by an algebraic point of view. Let g be a coboundary Lie bialgebra associated to a Poission Lie group G. As an application, we obtain a Lie bialgebra from a sub-Poisson bialgebra of the restricted dual of the universal enveloping algebra U(g).

Let A have a locally finite and multiparameter indexed filtration ℱ, and let B be a homomorphic image of A. Thus B has the locally finite and multiparameter indexed filtration induced from ℱ. Here we study a relation between the associated graded algebra of A and that of B and use this result to calculate the Gelfand–Kirillov dimension of several algebras related to quantized algebras and Poisson enveloping algebras.

A class of Poisson algebras A n , Γ P , Q considered as a Poisson version of the multiparameter quantized coordinate rings K n , Γ P , Q of symplectic and Euclidean 2n-spaces is constructed and Poisson structures of A n , Γ P , Q are described. In particular, it is proved that the prime Poisson and Poisson primitive spectra of A n , Γ P , Q are homeomorphic to the prime and primitive spectra of K n , Γ P , Q in the case when the multiplicative subgroup of k × generated by all parameters in K n , Γ P , Q is torsion free and, as a corollary, that the prime and primitive spectra of K n , Γ P , Q are topological quotients of the prime and maximal spectra of the corresponding commutative polynomial ring.

Let A be a finitely generated Poisson algebra over a field of characteristic zero. Here we prove that every Poisson prime ideal of A is prime and give a method to find all Poisson prime ideals in an arbitrary Poisson polynomial ring A[x; α, δ].

Let A be a Poisson algebra with Poisson bracket {·, ·} A and let α,δ be linear maps from A into itself. Here we find a necessary and sufficient condition for the pair (α,δ) such that the polynomial ring A[x] has the Poisson bracket for all a, b ∊ A and construct a class of Poisson algebras including the coordinate rings of Poisson 2 × 2-matrices and Poisson symplectic 4-space.

Poisson algebras have recently become extremely important in many of areas f mathematics and have been studied by many people. In particular, Joseph, Vancliff, Hodges and Levasseur proved that symp...

(1999). Poisson enveloping algebras. Communications in Algebra: Vol. 27, No. 5, pp. 2181-2186.

Poisson brackets on the polynomial algebra C[x,y,z] are studied. A description of all such brackets is given and, for a significant class of Poisson brackets, the Poisson prime ideals and Poisson primitive ideals are determined. The results are illustrated by numerous examples., includes minor corrections to published version

A significant class of Poisson brackets on the polynomial algebra C [ x 1 , … , x n ] is studied and, for this class of Poisson brackets, the Poisson prime ideals, Poisson primitive ideals and symplectic cores are determined. Moreover it is established that these Poisson algebras satisfy the Poisson Dixmier–Moeglin equivalence.