Quasi-Hopf algebras and Knizhnik-Zamolodchikov equations

The paper is a brief exposition of Drinfeld work. The notion of quasitriangular Hopf algebra is remained. The notion of quasitriangular quasi-Hopf algebra is introduced. A class of quasitriangular quasi-Hopf algebras using the differential equations for n-point functions in the WZW theory is constructed. Within the perturbation theory with respect to Planck's constant essentially all quasitriangular quasi-Hopf algebras belong to this class is asserted. A natural proof of Kohno's theorem on the equivalence of two kinds of braid group representations is given. Applications to knot invariants and the classical limit of various quantum notions are discussed. 16 refs