Coxeter categories and quantum groups

TL;DR

This paper introduces braided Coxeter categories, linking braid group actions to quantum groups, and demonstrates how quantum Weyl group operators induce such structures on category O-modules, extending previous monodromic descriptions.

Contribution

It defines braided Coxeter categories and shows their realization via quantum Weyl group operators and R-matrices in the context of quantum groups and category O.

Findings

Braided Coxeter structures are constructed on category O-modules.

Quantum Weyl group operators induce braided Coxeter structures.

Results extend monodromic descriptions of quantum Weyl groups.

Abstract

We define the notion of braided Coxeter category, which is informally a tensor 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_h(g), together with the universal R-matrices of its Levi subalgebras, give rise to a braided Coxeter structure on integrable, category O-modules for U_h(g). By relying on the 2-categorical extension of Etingof-Kazhdan quantisation obtained in arXiv:1610.09744, we then prove that this structure can be transferred to integrable, category O-representations of g. These results are used in arXiv:1512.03041 to give a monodromic description of the quantum Weyl group operators of U_h(g) which extends the one obtained by the second author for a semisimple Lie algebra.