The trigonometric Casimir connection of a simple Lie algebra

TL;DR

This paper introduces a new flat connection on a maximal torus of a complex semisimple Lie algebra with potential implications for understanding monodromy via quantum group operators.

Contribution

It constructs a novel trigonometric Casimir connection valued in the Yangian of g, extending the rational case and conjecturing its monodromy relates to quantum Weyl group operators.

Findings

Constructed a flat connection with logarithmic singularities on root hypertori.

Conjectured the monodromy is described by quantum Weyl group operators.

Provides a new perspective on the interplay between Lie algebra, Yangians, and quantum groups.

Abstract

Let g be a complex, semisimple Lie algebra, G the corresponding simply-connected Lie group and H a maximal torus in G. 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_h(Lg).