Monodromy of the trigonometric Casimir connection for sl_2

Details Monodromy of the trigonometric Casimir connection for sl_2 Monodromy of the trigonometric Casimir connection for sl_2

2011-09-12

arxiv.org/abs/1109.2367v1

Mathematics

Quantum Algebra

44 pages

Scientific paper

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

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