ИСТИНА

This thesis is dedicated to the study of deformation quantization in noncommutative algebras and polynomial automorphisms. We find a quantization proof of Bergman’s centralizer theorem by showing that the free associative algebra has no commutative subalgebra of rank greater than or equal to two. In order to achieve this, we study the algebra of generic matrices and prove that if there is a commutative subalgebra generated by two nonsingular elements in the algebra of generic matrices, then quantized images have a non-zero commutator which contradicts the fact that they are commutative. Thereafter, we use generic matrices reduction to provide the proof of the fact that the centralizer is integrally closed. These two results supply the whole quantization proof of Bergman’s centralizer theorem. We study the problem of lifting of polynomial symplectomorphisms in characteristic zero to automorphisms of the Weyl algebra by means of approximation by tame automorphisms. We extend the result of D. Anick on approximation of polynomial automorphisms and offer some similar approximation theorems for symplectomorphisms and automorphisms of the Weyl algebra. The lifting problem has its origins in the context of deformation quantization of the affine space. This approximation method is used to prove a conjecture by A. Kanel-Belov and M. Kontsevich on the isomorphism between groups of polynomial symplectomorphisms and automorphisms of the corresponding Weyl algebra in characteristic zero. We examine several instances of algebraic torus action on the free associative algebra. We prove the free algebra analogue of a classical theorem of A. Bia lynickiBirula, which establishes linearity of maximal torus action. We also formulate and prove linearity theorems in a few specific situations, as well as provide a framework for construction of non-linearizable torus actions.