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International Mathematics Research Papers (2008) Vol. 2008 : article ID rpn009, 167 pages, doi:10.1093/imrp/rpn009 published on December 12, 2008
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© The Author 2008. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: journals.permissions@oxfordjournals.org.

Quasi-Coxeter Algebras, Dynkin Diagram Cohomology, and Quantum Weyl Groups

Valerio Toledano Laredo

Université Pierre et Marie Curie-Paris 6, Institut de Mathématiques de Jussieu, UMR 7586, Case 191, 16 rue Clisson, F–75013, Paris

Correspondence: Correspondence to be sent to: V.ToledanoLaredo{at}neu.edu

The author, and independently, De Concini, conjectured that the monodromy of the Casimir connection of a simple Lie algebra Formula is described by the quantum Weyl group operators of the quantum group Formula . The aim of this article, and of its sequel [47], 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 Bn. Using an appropriate deformation cohomology, we reduce the conjecture to the existence of a quasi-Coxeter, quasitriangular quasibialgebra structure on the enveloping algebra Formula 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 [47].


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