Coxeter categories and quantum 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.
Cite
@article{arxiv.1610.09741,
title = {Coxeter categories and quantum groups},
author = {Andrea Appel and Valerio Toledano-Laredo},
journal= {arXiv preprint arXiv:1610.09741},
year = {2019}
}
Comments
Substantial revision. Changes include 1) greatly expanded introduction 2) Section 8 split into the new sections 8 (Universal Algebras) and 9 (Universal pre-Coxeter Structures) 3) more thorough account of the PROPic nature of transferred structure (Sect. 10) 4) notion of Upsilon and a-strict pre-Coxeter structures (Sect. 3, 9 and 10). Final version. To appear in Selecta Math. 81 pages