Monodromy of the trigonometric Casimir connection for sl2
Abstract
We show that the monodromy of the trigonometric Casimir connection on the tensor product of evaluation modules of the Yangian Ysl2 is described by the quantum Weyl group operators of the quantum loop algebra Uh(Lsl2). The proof is patterned on the second author's computation of the monodromy of the rational Casimir connection for sln via the dual pair (glk,gln), 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 Uh(slk) and the quantum Weyl group element of Uh(sl2), and a formula expressing the quantum Weyl group action of the coroot lattice of SL2 in terms of the commuting generators of Uh(Lsl2). Using this formula, we define quantum Weyl group operators for the quantum loop algebra Uh(Lgl2) and show that they describe the monodromy of the trigonometric Casimir connection on a tensor product of evaluation modules of the Yangian Ygl2
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