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Zbl 1169.17010
Toledano Laredo, Valerio
Quasi-Coxeter algebras, Dynkin diagram cohomology, and quantum Weyl groups. (English)
[J] IMRP, Int. Math. Res. Pap. 2008, Article ID rpn009, 167 p. (2008). ISSN 1072-7928

Author's summary: The author, and independently, De Concini, conjectured that the monodromy of the Casimir connection of a simple Lie algebra $\frak g$ is described by the quantum Weyl group operators of the quantum group $U_{\hbar}\frak g$. 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 $u_{\frak g}$ 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.).
[Ye Jiachen (Shanghai)]

MSC 2000:: *17B37 Quantum groups and related deformations
17B56 Cohomology of Lie algebras
20J06 Cohomology of finite groups

Keywords: Quasi-Coxeter algebra; Casimir connection; Quantum Weyl group; Quasitriangular quasibialgebra; Dynkin diagram cohomology

Cited in: Zbl 1171.17003

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