In the past decade there has been an extemely rapid growth in the interest and development of quantum group theory.This book provides students and researchers with a practical introduction to the principal ideas of quantum groups theory and its applications to quantum mechanical and modern field theory problems. It begins with a review of, and introduction to, the mathematical aspects of quantum deformation of classical groups, Lie algebras and related objects (algebras of functions on spaces, differential and integral calculi). In the subsequent chapters the richness of mathematical structure and power of the quantum deformation methods and non-commutative geometry is illustrated on the different examples starting from the simplest quantum mechanical system - harmonic oscillator and ending with actual problems of modern field theory, such as the attempts to construct lattice-like regularization consistent with space-time Poincare symmetry and to incorporate Higgs fields in the general geometrical frame of gauge theories. Graduate students and researchers studying the problems of quantum field theory, particle physics and mathematical aspects of quantum symmetries will find the book of interest.
Part 1 Mathematical aspects of quantum groups theory and non-commutative geometry: Hopf algebra and Poisson structure of classical Lie groups and algebras; quantum groups, algebras and their duality; non-commutative spaces and quantum groups invariant differential calculi; elements of quantum groups representations theory; tensor products of representations; q-tensors, q-vectors, q-scalars. Part 2 Deformation of harmonic oscillators: q-deformation of single-harmonic oscillator; different forms of commutation relations; representations; q real and roots of unity cases; algebraic maps from non-deformed to deformed oscillators; path integral quantization of q-oscillator. Part 3 Q-deformation of space-time symmetries: classical relativistic space-time symmetries; Poincare group as a "classical" deformation of Galilei group; its representations; multiparametric q-deformation of linear groups, twisted groups and algebras; q-Poincare group as a q-subgroup of q-conformal one and its induced representations. Part 4 Non-commutative geometry and unification models: unified models, the problem of natural introduction of Higgs fields; unified models of Higgs fields in the frame of non-commutative geometry; and others.