Monodromy of the trigonometric Casimir connection for sl_2

TL;DR

This paper demonstrates that the monodromy of the trigonometric Casimir connection for sl_2 is governed by quantum Weyl group operators of the quantum loop algebra, extending previous rational case results and introducing new algebraic formulas.

Contribution

It introduces an affine extension of duality and a new formula for quantum Weyl group actions, linking monodromy of the Casimir connection to quantum loop algebra operators.

Findings

Monodromy described by quantum Weyl group operators of U_h(Lsl_2)

New formula for quantum Weyl group action of the coroot lattice

Extension of duality between R-matrix and Weyl group elements

Abstract

We show that the monodromy of the trigonometric Casimir connection on the tensor product of evaluation modules of the Yangian Ysl_2 is described by the quantum Weyl group operators of the quantum loop algebra U_h(Lsl_2). The proof is patterned on the second author's computation of the monodromy of the rational Casimir connection for sl_n via the dual pair (gl_k,gl_n), and rests ultimately on the Etingof-Geer-Schiffmann computation of the monodromy of the trigonometric KZ connection. It relies on two new ingredients: an affine extension of the duality between the R-matrix of U_h(sl_k) and the quantum Weyl group element of U_h(sl_2), and a formula expressing the quantum Weyl group action of the coroot lattice of SL_2 in terms of the commuting generators of U_h(Lsl_2). Using this formula, we define quantum Weyl group operators for the quantum loop algebra U_h(Lgl_2) and show that they describe the monodromy of the trigonometric Casimir connection on a tensor product of evaluation modules of the Yangian Ygl_2