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Quasi-Coxeter algebras, Dynkin diagram cohomology, and quantum Weyl groups. (English) Zbl 1169.17010
Author’s summary: The author, and independently, De Concini, conjectured that the monodromy of the Casimir connection of a simple Lie algebra $𝔤$ is described by the quantum Weyl group operators of the quantum group ${U}_{\hslash }𝔤$. 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 ${B}_{n}$. Using an appropriate deformation cohomology, we reduce the conjecture to the existence of a quasi-Coxeter, quasitriangular quasibialgebra structure on the enveloping algebra ${u}_{𝔤}$ 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.).

##### MSC:
 17B37 Quantum groups and related deformations 17B56 Cohomology of Lie (super)algebras 20J06 Cohomology of groups