Quasi-Coxeter quasitriangular quasibialgebras and the Casimir connection

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Let g be a complex, semisimple Lie algebra. We prove the existence of a quasi-Coxeter, quasitriangular quasibialgebra structure on the enveloping algebra of g, which binds the quasi-Coxeter structure underlying the Casimir connection of g and the quasitriangular quasibialgebra one underlying its KZ equations. This implies in particular that the monodromy of the rational Casimir connection of g is described by the quantum Weyl group operators of the quantum group Uₕ(g).