  Let g be a symmetrisable Kac-Moody algebra and U_h(g) its quantised
enveloping algebra. Answering a question of P. Etingof, we prove that the
quantum Weyl group operators of U_h(g) give rise to a canonical action of the
pure braid group of g on any category O (not necessarily integrable)
U_h(g)-module V. By relying on our recent results in arXiv:1512.03041, we show
that this action describes the monodromy of the rational Casimir connection on
the g-module corresponding to V under the Etingof-Kazhdan equivalence of
category O for g and U_h(g). We also extend these results to yield equivalent
quantum Weyl group and monodromic representations of parabolic pure braid
groups on parabolic category O for U_h(g) and g.
