Quasi-Coxeter algebras, Dynkin diagram cohomology, and quantum Weyl groups. (English) Zbl 1169.17010

Toledano Laredo, Valerio

IMRP, Int. Math. Res. Pap. 2008, Article ID rpn009, 167 p. (2008).

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.).

Reviewer: Ye Jiachen (Shanghai)

MSC:

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

Keywords:

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