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

Abstract

AbstractLet $\mathfrak {g}$ g be a symmetrisable Kac–Moody algebra and $V$ V an integrable $\mathfrak {g}$ g –module in category $\mathcal {O}$ O . We show that the monodromy of the (normally ordered) rational Casimir connection on $V$ V can be made equivariant with respect to the Weyl group $W$ W of $\mathfrak {g}$ g , and therefore defines an action of the braid group $\mathcal {B}_{W}$ B W on $V$ V . We then prove that this action is canonically equivalent to the quantum Weyl group action of $\mathcal {B}_{W}$ B W on a quantum deformation of $V$ V , that is an integrable, category $\mathcal {O}$ O module $\mathcal {V}$ V over the quantum group $U_{\hbar }\mathfrak {g}$ U ħ g such that $\mathcal {V}/\hbar \mathcal {V}$ V / ħ V is isomorphic to $V$ V . This extends a result of the second author which is valid for $\mathfrak {g}$ g semisimple.