The trigonometric Casimir connection of a simple Lie algebra

Abstract

Let g be a complex, semisimple Lie algebra, G the corresponding simply-connected Lie group and H ⊂ G a maximal torus. We construct a flat connection on H with logarithmic singularities on the root hypertori and values in the Yangian Y ( g ) of g . By analogy with the rational Casimir connection of g , we conjecture that the monodromy of this trigonometric connection is described by the quantum Weyl group operators of the quantum loop algebra U ℏ ( L g ) .