Quasi-Hopf Algebras and Knizhnik-Zamolodchikov Equations

Abstract

This paper is a brief exposition of [6]. In §1 we remind the notion of quasitriangular Hopf algebra which is an abstract version of the notion of R-matrix. In §2 the notion of quasitriangular quasi-Hopf algebra is introduced (coassociativity is replaced by a weaker axiom). In §3 we construct a class of quasitriangular quasi-Hopf algebras using the differential equations for n-point functions in the WZW theory introduced by V.G.Knizhnik and A.B.Zamolodchikov. Theorem 1 asserts that within perturbation theory with respect to Planck’s constant essentially all quasitriangular quazi-Hopf algebras belong to this class. A natural proof of Kohno’s theorem on the equivalence of two kinds of braid group representations is given. In §4 we discuss applications to knot invariants. In §5 the classical limit of various quantum notions is discussed.