ИСТИНА

This thesis is dedicated to the study of noncommutative algebras and (broadly speaking) their applications in physics. We introduce and study a new class of algebras, which we name quantum generalized Heisenberg algebras and denote them by H_q(f; g), related to generalized Heisenberg algebras, but allowing more parameters of freedom, so as to encompass a wider range of applications and include previously studied algebras, such as (generalized) down-up algebras and generalized Heisenberg algebras. In particular, our class now includes the enveloping algebra of the 3-dimensional Heisenberg Lie algebra and its q-deformation, neither of which can be realized as a generalized Heisenberg algebra. Here we will classify the finite-dimensional irreducible representations of these algebras, and we will determine their automorphism groups when deg f > 1 and we also will solve the isomorphism problem for this class of algebras. We will study ring-theoretical properties like Gelfand-Kirillov dimension and being Noetherian. In another direction, we will investigate Hopf quantum generalized Heisenberg algebras and their simple module theory. We will try to construct a new Poisson bracket on our simplest example sl_2 and then we will try to give a universal construction based on our universal variables and then will try to construct lattice W2 algebras which will play a key role in our other constructions on lattice W3 algebras and finally we will try to find the only nontrivial dependent generator of our lattice W4 algebras and so on for lattice Wn algebras. 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 examine several instances of algebraic torus action on the free associative algebra. We prove the free algebra analogue of a classical theorem of A. Białynicki-Birula, 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.