Representations and Lie Theory Seminar

Schedule of talks:

Abstracts

August 23: In this talk, I will speak about shifted quantum affine algebras and shifted Yangians, as well as their incarnations through geometry of parabolic Laumon spaces, affine Grassmannians, and Toda lattice.
The shifted Yangians were originally introduced by Brundan-Kleshchev in the gl(n) case with a dominant shift and were later generalized by Kamnitzer-Webster-Weekes-Yacobi to any simple Lie algebra with an arbitrary shift. These algebras attracted recently a new interest due to their interplay with the Coulomb branches.
In the first half of the talk, I will remind those results about shifted Yangians, while the second part will be devoted to the multiplicative analogue of this story. On the algebraic side this leads to the notion of shifted quantum affine algebras, while on the geometric side we replace cohomology by K-theory.
This is a joint work with M. Finkelberg.

August 30: We will work out the derivation of the Yang--Baxter equation as a sufficient condition for solvability of a lattice model of Statistical Mechanics. No prior knowledge of physics will be assumed. We will go over what lattice models are, and what 'solvability' means. Time permitting, we will also discuss Bethe ansatz - a method of actually 'solving' when 'solvability' holds.

September 22: Building on the work of Calaque-Enriquez-Etingof, Lyubashenko-Majid, and Arakawa-Suzuki, Jordan constructed a functor from quantum D-modules on general linear groups to representations of the double affine Hecke algebra (DAHA) in type A. When we input quantum functions on GL(N) the output is L(k^N), the irreducible DAHA representation indexed by an N by k rectangle. For the specified parameters, L(k^N) is Y-semisimple, i.e. one can diagonalize the Dunkl operators. We give an explicit combinatorial description of this module via its Y-weight basis. This is joint work with David Jordan.

October 18 Generalizing work of Maulik and Okounkov, we produce a solution to the Yang-Baxter equation from the Hilbert scheme of points on a general surface using an intertwiner of certain highest weight Virasoro modules. We also explain how to modify this construction to produce formulas for multiplication by Chern classes of tautological bundles on the Hilbert scheme.

November 1 This is a joint work-in-progress with Valerio Toledano Laredo. We study the monodromy of the trigonometric KZ connection associated to the covariant representation J _{\theta} of the trigonometric Cherednik algebra H_c. This is motivated by Finkelberg and Ginzburg's study of a mirabolic version G_{\theta,c} of the Harish-Chandra D-module defined by Hotta and Kashiwara. Through Hamiltonian reduction and the KZ functor, G_{\theta,c} can be understood by the above monodromy problem. The monodromy of J_{\theta} was computed by Opdam, Heckman, and Cherednik for values of \theta and c when the connection is non-resonant. Our aim is to compute monodromy for all values of the parameters. Our tools are Opdam's shift operators in c as well as shift operators in \theta arising from a commuting copy of the difference Cherednik algebra. In this talk, I will focus on the example of the rank one case, which is easier to illustrate and fully proved.

November 29 In this talk I will try to shed new light on a storied topic: the R-matrix formalism for Yangians. The Yangian of a simple Lie algebra g is a Hopf algebra which canonically deforms the universal enveloping algebra of the polynomial current algebra g[z]. Its R-matrix presentation is a realization given entirely in terms of data obtained from a single finite-dimensional module V. In the first part of my talk, I will give an overview of its construction and explain why it gives an equivalent definition of the Yangian. In the second part, I will sketch a related construction which allows one to explicitly recover the universal R-matrix of the Yangian.

Previous seminar (Spring 2017)

Previous seminars (2015-16)