Resumen de Coxeter categories and quantum groups

Andrea Appel, Valerio Toledano Laredo

• We define the notion of braided Coxeter category, which is informally a monoidal category carrying compatible, commuting actions of a generalised braid group BW and Artin’s braid groups Bn on the tensor powers of its objects. The data which defines the action of BW 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 Uℏg , together with the universal R-matrices of its Levi subalgebras, give rise to a braided Coxeter category structure on integrable, category O -modules for Uℏg . 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 Uℏg , which extends the one obtained by the second author for a semisimple Lie algebra.