Nonlinear Poisson brackets. Geometry and quantization.
(Nelinejnye skobki Puassona. Geometriya i kvantovanie.)

*(Russian. English summary)*Zbl 0731.58002
Moskva: Nauka. 368 p. R. 5.80 (1991).

The first recommendation for this book is the name of the authors. The first author, Michael Karasev is a widely known specialist in non- commutative analysis, Poisson geometry and asymptotic methods, while the second author, the Academician Victor Maslov is a world-famous expert in the field of differential equations: asymptotic and operational methods, nonlinear mechanics etc. Their present monograph deserves the particular interest of the reader, indeed. It offers an extremely careful exposition of two important, old and hard common problems of mathematics and mathematical physics. The first of them, roughly speaking, is the following: what is the “right” analogue of the group for general, nonlinear, degenerated Poisson bracket? The second problem is about quantization: how to form a quantum mechanical system from the given classical system?

Chapter I is devoted to Poisson manifolds. Lie-Cartan reduction method, Dirac brackets, brackets on Lie groups and Yang-Baxter equations are considered in detail, and the deformation problem for Poisson brackets is solved in terms of deRham cohomologies. Chapter II deals with the above first problem, giving a thorough analysis of the analogue of the group operation for nonlinear Poisson bracket. Chapters III and IV treat the second problem, explaining - among many others - the technique of asymptotic quantization for symplectic manifolds and conveying a number of new results. Two large appendices and a bibliography with 287 items complete the book.

The monograph contains a wealth of original ideas as well as plenty of inspiration for further work. I am sure that it will have a strong influence on the forthcoming development of the theory of nonlinear Poisson brackets and geometric quantization.

Chapter I is devoted to Poisson manifolds. Lie-Cartan reduction method, Dirac brackets, brackets on Lie groups and Yang-Baxter equations are considered in detail, and the deformation problem for Poisson brackets is solved in terms of deRham cohomologies. Chapter II deals with the above first problem, giving a thorough analysis of the analogue of the group operation for nonlinear Poisson bracket. Chapters III and IV treat the second problem, explaining - among many others - the technique of asymptotic quantization for symplectic manifolds and conveying a number of new results. Two large appendices and a bibliography with 287 items complete the book.

The monograph contains a wealth of original ideas as well as plenty of inspiration for further work. I am sure that it will have a strong influence on the forthcoming development of the theory of nonlinear Poisson brackets and geometric quantization.

Reviewer: J.Szilasi (Debrecen)