Monodromy of the trigonometric Casimir connection for sl_2
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
Keywords
Cite
@article{arxiv.1109.2367,
title = {Monodromy of the trigonometric Casimir connection for sl_2},
author = {Sachin Gautam and Valerio Toledano-Laredo},
journal= {arXiv preprint arXiv:1109.2367},
year = {2013}
}
Comments
44 pages. Some misprints corrected in Appendix B and references were updated. Final version. To appear in the proceedings of the AMS Special Session on Noncommutative Birational Geometry and Cluster Algebras