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Star-products on symplectic manifolds. (English) Zbl 0621.58015
The paper consists of a brief introduction to the theory of formal deformations of (Lie) algebras of smooth functions on symplectic manifolds and a survey of recent results in this direction. The author describes the connection between this theory and quantum mechanics, such notions as star-product, deformed bracket, Nijenhuis-Richardson bracket, the existence theorems, etc. In particular, the author’s (in collaboration with P. Lecomte) theorems on the absence of topological obstructions in step-wise constructions of star-products and deformed brackets are formulated.
Reviewer: Yu.E.Gliklikh
37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
17B65 Infinite-dimensional Lie (super)algebras
58H15 Deformations of general structures on manifolds
81Q99 General mathematical topics and methods in quantum theory