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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 Poincaré 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.
Sample Chapter(s)
Chapter 1: Introduction (253 KB)
Contents:
- Mathematical Aspects of Quantum Group Theory and Non-Commutative Geometry:
- Hopf Algebra and Poisson Structure of Classical Lie Groups and Algebras
- Quantum Matrix Groups
- Elements of Quantum Group Representations
- q-Deformation of Harmonic Oscillators, Quantum Symmetry and All That:
- q-Deformation of Single Harmonic Oscillator
- q-Oscillators and Representations of QUEA
- Quantum Symmetries and q-Deformed Algebras in Physical Systems
- q-Deformation of Space-Time Symmetries:
- Multidimensional Jackson Calculus and Particle on Two-Dimensional Quantum Space
- Twisted Poincaré Group and Geometry of q-Deformed Minkowski Space
- Elements of General Theory of q-Inhomogeneous Groups and Classification of q-Poincaré Groups and q-Minkowski Spaces
- Non-Commutative Geometry and Internal Symmetries of Field Theoretical Models:
- Non-Commutative Geometry of Yang–Mills–Higgs Models
- Posets, Discrete Differential Calculus and Connes-Lott-Like Models
- Basic Elements of Quantum Fibre Bundle Theory
- and other papers
Readership: Theoretical physicists, mathematical physicists and mathematicians.
