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

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

2005-06-26

arxiv.org/abs/math/0506529v3

International Mathematics Research Papers 2008, article ID rpn009, 167 pages

Mathematics

Quantum Algebra

Final version. Coproduct axiom for quasi Coxeter quasi triangular quasi bialgebra in Section 6.9 changed to an equivalent form

Scientific paper

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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