Lectures on Quantum Groups and Yang-Baxter Equations

Abstract

The principles of the theory of quantum groups are reviewed from the point of view of the possibility of their use for deformations of symmetries in physical models. The R-matrix approach to the theory of quantum groups is discussed in detail and is taken as the basis of the quantization of classical Lie groups, as well as some Lie supergroups. We start by laying out the foundations of non-commutative and non-cocommutative Hopf algebras. Much attention has been paid to Hecke and Birman-Murakami-Wenzl (BMW) R-matrices and related quantum matrix algebras. Trigonometric solutions of the Yang-Baxter equation associated with the quantum groups GL_q(N), SO_q(N), Sp_q(2n) and supergroups GL_q(N|M), Osp_q(N|2m), as well as their rational (Yangian) limits, are presented. Rational R-matrices for exceptional Lie algebras and elliptic solutions of the Yang-Baxter equation are also considered. The basic concepts of the group algebra of the braid group and its finite dimensional quotients (such as Hecke and BMW algebras) are outlined. A sketch of the representation theories of the Hecke and BMW algebras is given (including methods for finding idempotents and their quantum dimensions). Applications of the theory of quantum groups and Yang-Baxter equations in various areas of theoretical physics are briefly discussed.