This publication offers with previous mathematical difficulties. the 1st is the matter of creating an analog of a Lie workforce for normal nonlinear Poisson brackets. the second one is the quantization challenge for such brackets within the semiclassical approximation (which is the matter of actual quantization for the easiest periods of brackets). those difficulties are gradually coming to the fore within the smooth conception of differential equations and quantum thought, because the process in line with buildings of algebras and Lie teams turns out, in a undeniable experience, to be exhausted. The authors' major objective is to explain intimately the recent gadgets that seem within the resolution of those difficulties. Many principles of algebra, smooth differential geometry, algebraic topology, and operator conception are synthesized the following. The authors turn out all statements intimately, therefore making the ebook available to graduate scholars.

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