Quasi-Coxeter Algebras, Dynkin Diagram Cohomology, and Quantum Weyl Groups

Abstract

The author, and independently, De Concini, conjectured that the monodromy of the Casimir connection of a simple Lie algebra is described by the quantum Weyl group operators of the quantum group ⁠. The aim of this article, and of its sequel [47], is to prove this conjecture. The proof relies upon the use of quasi-Coxeter algebras, which are to generalized braid groups what Drinfeld's quasitriangular quasibialgebras are to the Artin braid groups Bn. Using an appropriate deformation cohomology, we reduce the conjecture to the existence of a quasi-Coxeter, quasitriangular quasibialgebra structure on the enveloping algebra which interpolates between the quasi-Coxeter algebra structure underlying the Casimir connection, and the quasitriangular quasibialgebra structure underlying the Knizhnik–Zamolodchikov equations. The existence of this structure will be proved in [47].