Coxeter categories and quantum groups

We define the notion of braided Coxeter category, which is informally a monoidal category carrying compatible, commuting actions of a generalised braid group B_W and Artin’s braid groups B_n on the tensor powers of its objects. The data which defines the action of B_W bears a formal similarity to the associativity constraints in a monoidal category, but is related to the coherence of a family of fiber functors. We show that the quantum Weyl group operators of a quantised Kac–Moody algebra U_ħ𝔤 , together with the universal R -matrices of its Levi subalgebras, give rise to a braided Coxeter category structure on integrable, category 𝒪 -modules for U_ħ𝔤 . By relying on the 2-categorical extension of Etingof–Kazhdan quantisation obtained in Appel and Toledano Laredo (Selecta Math NS 24:3529–3617, 2018 ), we then prove that this structure can be transferred to integrable, category 𝒪 -representations of 𝔤 . These results are used in Appel and Toledano Laredo ( arXiv:1512.03041 , p 48, 2015 ) to give a monodromic description of the quantum Weyl group operators of U_ħ𝔤 , which extends the one obtained by the second author for a semisimple Lie algebra.