Uniqueness of Coxeter structures on Kac–Moody algebras

Abstract

Let g be a symmetrizable Kac–Moody algebra, and Uħg the corresponding quantum group. We showed in [1,2] that the braided Coxeter structure on integrable, category O representations of Uħg which underlies the R-matrix actions arising from the Levi subalgebras of Uħg and the quantum Weyl group action of the generalized braid group Bg can be transferred to integrable, category O representations of g. We prove in this paper that, up to unique equivalence, there is a unique such structure on the latter category with prescribed restriction functors, R-matrices, and local monodromies. This extends, simplifies and strengthens a similar result of the second author valid when g is semisimple, and is used in [3] to describe the monodromy of the rational Casimir connection of g in terms of the quantum Weyl group operators of Uħg. Our main tool is a refinement of Enriquez's universal algebras, which is adapted to the PROP describing a Lie bialgebra graded by the non-negative roots of g.