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Quasi-Coxeter algebras, Dynkin diagram cohomology and quantum Weyl groups

V. Toledano-Laredo
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Abstract

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_h(g). The aim of this paper, and of its sequel [TL4], is to prove this conjecture. The proof relies upon the use of quasi-Coxeter algebras, which are to generalised 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 Ug which interpolates between the quasi-Coxeter structure underlying the Casimir connection and the quasitriangular quasibialgebra underlying the KZ equations. The existence of this structure will be proved in [TL4].

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V. Toledano-Laredo. 2008-12-21. Quasi-Coxeter algebras, Dynkin diagram cohomology and quantum Weyl groups. https://arxiv.org/abs/math/0506529

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