Spring School

Infinite Analysis 13 Spring School at Nagoya

Yangians and quantum loop algebras

Lecturer: Valerio Toledano Laredo (Northeastern University)

Date：March 12 (tue) - 14 (thu), 2013

Place：Graduate School of Mathematics, Nagoya University, Sci Build. 1 Room 109

(http://www.math.nagoya-u.ac.jp/en/direction/index.html)

Schedule:

March 12 (tue) 10:00-11:30. 13:30-15:00

March 13 (wed) 10:00-11:30. 13:30-15:00

March 14 (thu) 10:00-11:30

Abstract: These lectures are based on joint work with Sachin Gautam.

They will concentrate on two closely related infinite dimensional quantum groups associated to a complex, semisimple Lie algebra g: the Yangian Y_h(g), and the quantum loop algebra U_q(Lg). The former is a deformation of the current algebra g[s], and the latter of the loop algebra g[z,z^{-1}].

It was pointed out by Drinfeld in the 80s that the quantum loop algebra degenerates to the Yangian. In the first part of the lectures, I will show that these two objects are in fact isomorphic after completion, by an appropriate quantum lift of the change of variable z=e^s.

In the second part, I will review the classification of finite-dimensional irreducible representations of Y_h(g) and U_q(Lg). Assuming that q is not a root of unity, I will then construct an equivalence of categories between finite-dimensional representations of U_q(Lg) and an appropriate subcategory of finite-dimensional representations of Y_h(g). Unlike the isomorphism alluded to above, this equivalence is transcendental, and governed by the monodromy of an additive, abelian difference equation.

contact: Tomoki Nakanishi (Nagoya) nakanisi_at_math.nagoya-u.ac.jp