The elliptic Casimir connection of a simple Lie algebra

Abstract

We construct a flat connection on the elliptic configuration space associated to any complex semisimple Lie algebra g. This elliptic Casimir connection has logarithmic singularities, and takes values in the deformed double current algebra of g defined by Guay. It degenerates to the trigonometric Casimir connection of g constructed by the first author. By analogy with the rational and trigonometric cases, we conjecture that the monodromy of the elliptic Casimir connection is described by the quantum Weyl group operators of the quantum toroidal algebra of g.