Quantum groups and Yang-Baxter equations

This introductory review is devoted to the newest section of the theory of symmetries -- the theory of quantum groups.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 physics 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. Noncommutative differential geometry on quantum groups of special types is discussed. Trigonometric solutions of the Yang-Baxter equations associated with the quantum groups GLq(N), SOq(N), Spq(2n) and supergroups GLq(N|M), Ospq(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 (quantum Young projectors) and their quantum dimensions. Applications of the theory of quantum groups and Yang-Baxter equations in various areas of theoretical physics are briefly discussed. This is a modified version of the review paper published in 2004 as a preprint of the Max-Planck-Institut für Mathematik in Bonn.