Monodromy of the Casimir connection of a symmetrisable Kac-Moody algebra

TL;DR

This paper demonstrates that the monodromy of the Casimir connection for symmetrisable Kac-Moody algebras induces a braid group action on modules, which aligns with the quantum Weyl group action upon quantum deformation.

Contribution

It extends the known correspondence between monodromy and quantum Weyl group actions from semisimple to symmetrisable Kac-Moody algebras.

Findings

Monodromy defines a braid group action on integrable modules.

The braid group action is equivalent to the quantum Weyl group action.

Extension of previous results to symmetrisable Kac-Moody algebras.

Abstract

Let g be a symmetrisable Kac-Moody algebra and V an integrable g-module in category O. We show that the monodromy of the (normally ordered) rational Casimir connection on V can be made equivariant with respect to the Weyl group W of g, and therefore defines an action of the braid group B_W of W on V. We then prove that this action is canonically equivalent to the quantum Weyl group action of B_W on a quantum deformation of V, that is an integrable, category O-module V_h over the quantum group U_h(g) such that V_h/hV_h is isomorphic to V. This extends a result of the second author which is valid for g semisimple.