Semiclassical limits of generic multiparameter quantized coordinate rings A = Oq(k ) of affine spaces are constructed and related to A, for k an algebraically closed field of characteristic zero and q a multiplicatively antisymmetric matrix whose entries generate a torsionfree subgroup of k. A semiclassical limit of A is a Poisson algebra structure on the corresponding classical coordinate ring...

By Langlands duality one usually understands a correspondence between automorphic representations of a reductive group G over the ring of adels of a field F , and homomorphisms from the Galois group Gal(F/F ) to the Langlands dual group GL. It was originally introduced in the case when F is a number field or the field of rational functions on a curve over a finite field [1]. Recently A. Beilins...

Semiclassical limits of generic multiparameter quantized coordinate rings A = Oq(k) of affine spaces are constructed and related to A, for k an algebraically closed field of characteristic zero and q a multiplicatively antisymmetric matrix whose entries generate a torsionfree subgroup of k×. A semiclassical limit of A is a Poisson algebra structure on the corresponding classical coordinate ring...

In this note we consider an arbitrary submanifold C of a Poisson manifold P and ask whether it can be embedded coisotropically in some bigger submanifold of P . We define the classes of submanifolds relevant to the question (coisotropic, Poisson-Dirac, pre-Poisson ones), present an answer to the above question and consider the corresponding picture at the level of Lie groupoids, making concrete...

We find a one parameter family of quadratic Poisson structures on R 4 × SL(2, C) which satisfies the property a) that it is preserved under the Lie-Poisson action of the Lorentz group, as well as b) that it reduces to the standard Poincaré algebra for a particular limiting value of the parameter. (The Lie-Poisson transformations reduce to canonical ones in that limit, which we therefore refer t...

This paper studies certain relations among vertex algebras, vertex Lie algebras and vertex poisson algebras. In this paper, the notions of vertex Lie algebra (conformal algebra) and vertex poisson algebra are revisited and certain general construction theorems of vertex poisson algebras are given. A notion of filtered vertex algebra is formulated in terms of a notion of good filtration and it i...

The category of Banach Lie-Poisson spaces is introduced and studied. It is shown that the category of W ∗-algebras can be considered as one of its subcategories. Examples and applications of Banach Lie-Poisson spaces to quantization and integration of Hamiltonian systems are given. The relationship between classical and quantum reduction is discussed.

We prove new upper and lower bounds for Banach space-valued stochastic integrals with respect to a compensated Poisson random measure. Our estimates apply to Banach spaces with non-trivial martingale (co)type and extend various results in the literature. We also develop a Malliavin framework to interpret Poisson stochastic integrals as vector-valued Skorohod integrals, and prove a Clark-Ocone r...

The category of Banach Lie-Poisson spaces is introduced and studied. It is shown that the category of W ∗-algebras can be considered as one of its subcategories. Examples and applications of Banach Lie-Poisson spaces to quantization and integration of Hamiltonian systems are also given.

The structure of Poisson polynomial algebras of the type obtained as semiclassical limits of quantized coordinate rings is investigated. Sufficient conditions for a rational Poisson action of a torus on such an algebra to leave only finitely many Poisson prime ideals invariant are obtained. Combined with previous work of the first-named author, this establishes the Poisson Dixmier-Moeglin equiv...