Representations and Lie Theory Seminar

Abstracts

April 2 Quiver Donaldson-Thomas invariants are integers determined by the geometry of moduli spaces of quiver representations. They play an important role in the description of BPS states of supersymmetric quantum field theories. I will describe a correspondence between quiver Donaldson-Thomas invariants and Gromov-Witten counts of rational curves in toric and cluster varieties. This is joint work with Hülya Argüz (arXiv:2302.02068 and arXiv:2308.07270).

April 16 The theory of Yangians arose in the 1980's as an algebraic framework for systematically producing rational solutions of the celebrated quantum Yang-Baxter equation (qYBE) from theoretical physics. Roughly speaking, the mechanism by which these solutions arise is as follows: Starting from any simple Lie algebra g, one can construct a Hopf algebra, called the Yangian of g, which quantizes the Lie bialgebra g[t] of polynomials in a single variable with coefficients in g. This Hopf algebra comes equipped with a remarkable formal series R(z), called its universal R-matrix, which provides a universal, formal solution to the qYBE. The desired rational solutions are then obtained by evaluating R(z) on the tensor product of any two finite-dimensional irreducible representations of the Yangian.
In recent joint work with Andrea Appel and Sachin Gautam, it has been shown that this construction admits a non-trivial generalization in which g is replaced by an affine Lie algebra. In this setting, the ordinary Yangian is upgraded to an "affine Yangian", which is an example of a quantum group of double affine type. In this talk, I will explain the semiclassical limit of this story, which entails constructing a toroidal analogue of the Lie bialgebra g[t] and studying its applications to the classical Yang-Baxter equation. Time permitting, I will highlight some unexpected consequences of this construction to the theory of affine Yangians. This is based on joint work with A. Weekes.

April 23 Let G be a finite group and k a field with char(k) | o(G). A map f : M \to N between kG-modules in the stable category kG-Mod is called a phantom map if for any finitely generated module A, and any map g : A \to M, the composition fg is zero. The phantom maps form an ideal in the stable category kG-Mod. In 1999, Benson and Ganacadja studied phantom maps in modular representation theory, and they made the following conjecture: does there exist a bound n(G) on the nilpotency index of the ideal of phantom maps in the stable category kG-Mod. This conjecture was solved by a theory developed by Herzog and I. In this talk, I will present the proof.