Нелинейные скобки Пуассона. Геометрия и квантование

The mechanisms of occurrence of nonlinear degenerate Poisson brackets in Hamiltonian mechanics, deformations of brackets and their cohomology are considered. A geometric object that is an analogue of the Lie group for nonlinear brackets is studied in detail. The authors' proposed construction of asymptotic quantization on general symplectic and Poisson manifolds, and in particular, the quantization rule for two-dimensional films, is presented with complete proofs. In addition, the book contains an elementary introduction to the theory of semiclassical approximation, extensive reference material on the calculus of functions of non-commuting operators, and a summary of results on algebras with non-Lie commutation relations. For specialists in differential geometry, algebra, mathematical physics, asymptotic methods, for graduate students and students of mathematical faculties.