Author’s summary: 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 [Quasi-Coxeter quasitriangular quasibialgebras and the Casimir connection. (forthcoming)], 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
. 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 the forthcoming paper (loc. cit.).