Sei-Qwon Oh

A POINCARÉ-BIRKHOFF-WITT THEOREM FOR POISSON ENVELOPING ALGEBRAS

Poisson algebras have recently become extremely important in many areas of mathematics and have been studied by many people. In particular, Joseph, Vancliff, Hodges and Levasseur proved that symplectic leaves of certain Poisson varieties correspond bijectively to primitive ideals of the corresponding quantum algebras (see [1, 2, 3, 9]). Hence one can ask if there is a Poisson algebraic concept which corresponds to the primitive ideal. In [7], the first author gave a definition of a symplectic ideal in a Poisson algebra as follows: Let R be a Poisson algebra. A Poisson ideal Q of R is said to be symplectic if there is a maximal ideal M of the commutative algebra R such that Q is the largest Poisson ideal contained in M . Moreover, he proved in [7] that there is a one to one correspondence between the primitive ideals of quantum 2×2 matrices algebra and the symplectic ideals of a Poisson algebra constructed appropriately. Note that a primitive ideal of an algebra is the annihilator of a simple module. As a Poisson version for primitive ideals, this paper concerns the question: Is a symplectic ideal of a Poisson algebra R the annihilator of a simple Poisson Rmodule? Let us describe our approach in more details. Let R be a Poisson algebra over a field k and let U(R) be the Poisson enveloping algebra of R. Since a k-vector space M is a Poisson R-module if and only if M is a left U(R)-module by [8, 6],

Poisson Polynomial Rings

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 brackets and Poisson spectra in polynomial algebras

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.

A natural map from a quantized space onto its semiclassical limit and a multi-parameter Poisson Weyl algebra

ABSTRACT 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 multiparameter quantized Weyl algebra onto the Poisson spectrum of its semiclassical limit.

Poisson Hopf algebra related to a twisted quantum group

ABSTRACT 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. This establishes that all Poisson prime and primitive ideals of ℂ[G] are characterized. Further it is shown that ℂ[G] satisfies the Poisson Dixmier-Moeglin equivalence and that Zariski topology on the space of Poisson primitive ideals of ℂ[G] agrees with the quotient topology induced by the natural surjection from the maximal ideal space of ℂ[G] onto the Poisson primitive ideal space.

Relationship Between Quantum and Poisson Structures of Odd Dimensional Euclidean Spaces

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.

Poisson Prime Ideals of Poisson Polynomial Rings

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].

Semiclassical limits of Ore extensions and a Poisson generalized Weyl algebra

We observe [Launois and Lecoutre, Trans. Am. Math. Soc. 368:755–785, 2016, Proposition 4.1] that Poisson polynomial extensions appear as semiclassical limits of a class of Ore extensions. As an application, a Poisson generalized Weyl algebra A1, considered as a Poisson version of the quantum generalized Weyl algebra, is constructed and its Poisson structures are studied. In particular, a necessary and sufficient condition is obtained, such that A1 is Poisson simple and established that the Poisson endomorphisms of A1 are Poisson analogues of the endomorphisms of the quantum generalized Weyl algebra.

SKEW ENVELOPING ALGEBRAS AND POISSON ENVELOPING ALGEBRAS

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.

Simple modules over the coordinate ring of quantum affine space

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