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 \(\mathfrak g\) is described by the quantum Weyl group operators of the quantum group \(U_{\hbar}\mathfrak g\). 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 \(u_{\mathfrak g}\) 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.).


17B37 Quantum groups (quantized enveloping algebras) and related deformations
17B56 Cohomology of Lie (super)algebras
20J06 Cohomology of groups
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