Projection Operator Method for Quantum Groups

Abstract

At present there can be no doubt to say that the quantum group theory has been one of the most important, modern and rapidly developing directions of mathematics and mathematical physics in the end of the twentieth century. Although initially (in the early 1980’s) the quantum group theory was formulated for solving problems in the theory of integrable systems and statistical physics, later the surprising connection of this theory with many branches of mathematics, and the theoretical and mathematical physics was discovered. Today the quantum group theory is connected with such mathematical fields as special functions (especially, with (q-orthogonal polynomials and basic hypergeometric series), the theory of difference and differential equations, combinatorial analysis and representation theory, matrix and operator algebras, noncommutative geometry, knot theory, topology, category theory and so on. From point of view of the mathematical physics there exists interconnection of the quantum group theory with the quantum inverse scattering method, conformai and quantum fields theory and so on. It is expected that the quantum groups will provide deeper understanding of concept of symmetry in physics.KeywordsQuantum GroupSimple RootBasic Hypergeometric SeriesReduction ChainGeneral Projection OperatorThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.