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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 \(\mathfrak {g}\) be the finite dimensional simple Lie algebra associated to an indecomposable and symmetrizable generalized Cartan matrix \(C=(a_{ij})_{n\times n}\) of finite type and let \(\mathfrak {d}\) be a finite dimensional Lie algebra related to a quantum group \(D_{q,p^{-1}}(\mathfrak {g})\) obtained by Hodges et al. (Adv Math 126:52–92, 1997) by deforming the quantum group \(U_q(\mathfrak {g})\). Here we see that \(\mathfrak {d}\) is a generalization of \(\mathfrak {g}\) and give a \(\mathfrak {d}\)-invariant symmetric bilinear form on \(\mathfrak {d}\).

【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.】

Let R be a commutative noetherian algebra and let – be a nonzero derivation. Here we determine the prime ideals ofthe skew polynomial algebra R [ z ; – ] and the Poisson prime ideals of R [ z ; – ] p 1. IntroductionIt is very di–cult to flnd prime ideals of R [ z ; fi;– ] and Poisson primeideals of R [ z ; fi;– ] p generally. Here we determine the prime ideals of R [ z ; – ] and the Poisson prime ideals of R [ z ; – ] p in the case that R is acommutative noetherian algebra and – is a nonzero derivation of R . If – = 0 then R [ z ; – ] = R [ z ] = R [ z ; – ] p and thus the prime ideals of R [ z ]are determined by commutative algebra theory. So we concentrate onthe case – 6= 0.Assume throughout the paper that the base fleld is the complex num-ber fleld Cand that all algebras considered have unities.Definition 1.1. (1) Given an automorphism fi on a C-algebra R ,a C-linear map – is said to be a (left) fi-derivation on R if – ( ab ) = – ( a ) b + fi ( a ) – ( b ) for all a;b 2 R . For such pair (

Fix elements of the quotient field B of the polynomial algebra . It is proved that B is a Poisson algebra with Poisson bracket defined by for any , where det(Jac) is the determinant of a Jacobian matrix.

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

We give a result on the density of certain additive arithmetic functions which count the number of restricted irreducible polynomials on the polynomial ring.

Let A be a co-Poisson Hopf algebra with Poisson co-bracket . Here it is shown that the Hopf dual A ◦ is a Poisson Hopf algebra withPoisson bracket ff;gg ( x ) = ⟨ ( x ) ;f g⟩ for any f;g 2 A ◦ and x 2 A if A is an almost normalizing extension over the ground eld. Moreover weget, as a corollary, the fact that the Hopf dual of the universal envelopingalgebra U (g) for a nite dimensional Lie bialgebra g is a Poisson Hopfalgebra. Let G be a Lie group with Lie algebra g. Then its coordinate ring O ( G ) isa Hopf algebra and can be replaced by the Hopf dual U (g) ◦ of the universalenveloping algebra U (g). In fact, it is well-known that U (g) ◦ is equal to O ( G )if G is connected and simply connected. Moreover it is convenient to work on U (g) ◦ instead of O ( G ) since U (g) ◦ has a natural grading. For instance, see [3,Chapter 2] and [2].Recall that a Lie group G is said to be a Poisson Lie group if its coordinatering O ( G ) is a Poisson Hopf algebra. If G is a Poisson Lie group, then its Liealgebra g becomes a nite dimensional Lie bialgebra with a co-bracket

It is shown that the prime and primitive spectra of the multiparameter quantized algebra of odd-dimensional euclidean spaces are homeomorphic to the Poisson prime and Poisson primitive spectra of the multiparameter Poisson algebra of odd-dimensional euclidean spaces in the case when the multiplicative subgroup of a base field generated by the parameters is torsion free. As a corollary, it is shown that the prime and primitive spectra of the multiparameter quantized algebra of odd-dimensional euclidean spaces are topological quotients of the prime and maximal spectra of the corresponding commutative polynomial ring.

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).

A Poisson ℂ-algebra R appears in classical mechanical system and its quantized algebra appearing in quantum mechanical system is a ℂ[[ħ]]-algebra Q = R[[ħ]] with star product * such that for any a,b Є R ⊆ Q, a*b = ab + B1(a,b)ħ + B2(a,b)ħ2 + … subject to {a,b}= ħ-1(a * b ‒ b * a)|ħ=0, … (**) where Bi : R ⨯ R → R are bilinear products. The given Poisson algebra R is recovered from its quantized algebra Q by R = Q/ħQ with Poisson bracket (**), which is called its semiclassical limit. But it seems that the star product in Q is complicate and that Q is difficult to understand at an algebraic point of view since it is too big. For instance, if λ is a nonzero element of ℂ then ħ - λ is a unit in Q and thus a so-called deformation of R, Q/(ħ - λ)Q, is trivial. Hence it seems that we need an appropriate 픽-subalgebra A of Q such that A contains all generators of Q, ħ є A and A is understandable at an algebraic point of view, where 픽 is a subring of C[[ħ]]. Here we discuss how to find nontrivial deformations from quantized algebras and the natural map in [6] from a class of infinite deformations onto its semiclassical limit. The results are illustrated by examples.

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 be a Lie bialgebra. Here we give an explicit formula for the Poisson bracket on a subalgebra of induced by the given cobracket .

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.

It is shown that every almost ∗–homomorphism h : \({\fancyscript A}\) → ℬ of a unital JC*–algebra \({\fancyscript A}\) to a unital JC*–algebra ℬ is a ∗–homomorphism when h(rx) = rh(x) (r > 1) for all x ∈ \({\fancyscript A}\) , and that every almost linear mapping h : \({\fancyscript A}\) → ℬ is a ∗–homomorphism when h(2nu ∘y) = h(2nu) ∘ h(y), h(3nu ∘ y) = h(3nu) ∘ h(y) or h(qnu ∘ y) = h(qnu) ∘ h(y) for all unitaries u ∈ \({\fancyscript A}\) , all y ∈ \({\fancyscript A}\) , and n = 0, 1, . . . . Here the numbers 2, 3, q depend on the functional equations given in the almost linear mappings.

The universal mapping property and the Gelfand- Kirillov dimension of a skew enveloping algebra are studied and it is proved that every Poisson enveloping algebra is a homomorphic image of a skew enveloping algebra.

The group of Poisson automorphisms of the coordinate ring of Poisson symplectic 2n-space is isomorphic to the algebraic torus (κ✶)n+1 and it confirms that the algebra constructed by K.L. Horton (2003) is a quantisation of the coordinate ring of Poisson symplectic 2n-space.

The non-commutative torus C*(ℝn,ω) is realized as the C*-algebra of sections of a locally trivial C*-algebra bundle over Sω with fibres isomorphic to C*ℝn/Sω, ω1) for a totally skew multiplier ω1 on ℝn/Sω. D. Poguntke [9] proved that Aω is stably isomorphic to C(Sω) ⊗ C(*(∝ Zn/Sω, ω1) ≅ C(Sω) ⊗ Aφ ⊗ Mkl(∝ C) for a simple non-commutative torus Aφ and an integer kl. It is well-known that a stable isomorphism of two separable C*-algebras is equivalent to the existence of equivalence bimodule between them. We construct an Aω-C(Sω) ⊗ Aφ-equivalence bimodule.

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...

Assume throughout the paper that k denotes an algebraically closed field of characteristic zero and that all algebras have unity. A Poisson algebra R is always a commutative k-algebra with k-biline...

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

(1999). Symplectic ideals of poisson algebras and the poisson structure associated to quantum matrices. Communications in Algebra: Vol. 27, No. 5, pp. 2163-2180.

A twisted group algebra kσP on a free Abelian group P with finite rank and a Poisson structure on kP are studied. As an application, the primitive spectrum of , the coordinate ring of quantum Euclidean space, is described and a Poisson algebra A is constructed so that there is a bijection between the primitive spectrum of and the symplectic spectrum of R.

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

The simple modules of , the coordinate ring of quantum affine space, are classified in the case when q is a root of unity.

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.

We prove the generalized Hyers-Ulam-Rassias stability of general- ized Jensen’s equations in Banach modules over a unital C∗-algebra associ- ated with its unitary group. It is applied to show the stability of generalized Jensen’s equations in a Hilbert module over a unital C∗-algebra associated with its unitary group.

We prove the generalized Hyers-Ulam-Rassias stability of the invertible mapping in a Banach module over a unital Banach algebra in the spirit of G\u avruta, and prove the generalized Hyers-Ulam-Rassias stability of linear functional equations in a Hilbert module over a unital $C^*$-algebra in the spirit of G\u avruta.