It is well-known that the question of existence of a star product on a Poisson manifold is open and only some partial results are known [see the author, J. Geom. Phys. 9, No. 1, 45-73 (1992; Zbl 0761.16012)].In the paper under review, the author proves the existence of the star products for the Poisson structures of the following type with , for some .
It is shown how the universal enveloping algebra of a Lie algebra can be obtained as a formal deformation of the Kirillov-Souriau Poisson algebra of smooth functions on the dual of . This deformation process may be viewed as a “quantization” in the sense of and [Ann. Phys. 111, 61-110 (1978; Zbl 0377.53024) and ibid., 111-151 (1978; Zbl 0377.53025)]. The result presented is a somewhat more elaborate version of earlier findings by [Lett. Math. Phys. 7, 249-258 (1983; Zbl 0522.58019)] and...
The notion of Poisson Lie group (sometimes called Poisson Drinfel'd group) was first introduced by Drinfel'd [1] and studied by Semenov-Tian-Shansky [7] to understand the Hamiltonian structure of the group of dressing transformations of a completely integrable system. The Poisson Lie groups play an important role in the mathematical theories of quantization and in nonlinear integrable equations. The aim of our lecture is to point out the naturality of this notion and to present basic facts about...
We present a general theorem describing the isomorphisms of the local Lie algebra structures on the spaces of smooth (real-analytic or holomorphic) functions on smooth (resp. real-analytic, Stein) manifolds, as, for example, those given by Poisson or contact structures. We admit degenerate structures as well, which seems to be new in the literature.
The main result is a Pursell-Shanks type theorem describing isomorphism of the Lie algebras of vector fields preserving generalized foliations. The result includes as well smooth as real-analytic and holomorphic cases.
We discuss a concept of loopoid as a non-associative generalization of Brandt groupoid. We introduce and study also an interesting class of more general objects which we call semiloopoids. A differential version of loopoids is intended as a framework for Lagrangian discrete mechanics.
Results on characterization of manifolds in terms of certain Lie algebras growing on them, especially Lie algebras of differential operators, are reviewed and extended. In particular, we prove that a smooth (real-analytic, Stein) manifold is characterized by the corresponding Lie algebra of linear differential operators, i.e. isomorphisms of such Lie algebras are induced by the appropriate class of diffeomorphisms of the underlying manifolds.
We characterize Poisson and Jacobi structures by means of complete lifts of the corresponding tensors: the lifts have to be related to canonical structures by morphisms of corresponding vector bundles. Similar results hold for generalized Poisson and Jacobi structures (canonical structures) associated with Lie algebroids and Jacobi algebroids.
Binary operations on algebras of observables are studied in the quantum as well as in the classical case. It is shown that certain natural compatibility conditions with the associative product imply properties which are usually additionally required.
Results on derivations and automorphisms of some quantum and classical Poisson algebras, as well as characterizations of manifolds by the Lie structure of such algebras, are revisited and extended. We prove in particular a somewhat unexpected fact that the algebras of linear differential operators acting on smooth sections of two real vector bundles of rank 1 are isomorphic as Lie algebras if and only if the base manifolds are diffeomorphic, whether or not the line bundles themselves are isomorphic....
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