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…