# Representations and Lie Theory Seminar

## Schedule of talks:

 TIME SPEAKER TITLE January 17 Diego Penta Decomposition of a rank 3 hyperbolic Kac-Moody Lie algebra with respect to the hyperbolic Fibonacci' KM subalgebra January 24 Staff Isomorphism between quantum and classical sl(n) February 14 No Seminar Colloquium Ivan Losev February 21 Shotaro Makisumi Perverse sheaves on moment graphs and Soergel category O February 28 Matthew Harper The Family of Induced Representations for Quantum Groups at a 4th Root of Unity March 7 Daniele Rosso Irreducible components of exotic Springer fibers and the Robinson-Schensted correspondence March 14 No seminar Spring break April 11 Florencia Orosz Hunziker Equivalence of tensor categories between rep(sl(2,C)) and the fusion category of the modules for the vertex algebras L(1,0) April 18 No seminar Zassenhaus series by S. Fomin. April 25 Matt Szczesny Hall algebra of coherent sheaves on toric varieties over F_1

## Abstracts

January 17: In 1983, Feingold-Frenkel found important structural results about the rank 3 hyperbolic Kac-Moody Lie algebra F from its decomposition with respect to the affine subalgebra A_1^{(1)}. In this talk, we will discuss an alternative decomposition of F with respect to the rank 2 hyperbolic KM algebra Fib, the so-called Fibonacci' algebra (Feingold, 1980). We find that F has a grading by Fib-level, and show that each graded piece Fib(m) is an integrable Fib-module which contains infinitely many irreducible components. We also discuss the existence of non-standard modules on levels |m| less than 3, whose weight multiplicities do not appear to satisfy the recursion formulas given by Racah-Speiser and Kac-Peterson.

January 24: Quantum groups, introduced by Drinfeld and Jimbo, are Hopf algebras naturally associated to simple Lie algebras. For a simple Lie algebra g, the quantum group U_h(g) is known to be isomorphic, as an algebra, to the enveloping algebra U(g)[[h]] (where h is a formal variable). This isomorphism exists due to a non-trivial cohomological argument and has only been written down for g=sl(2) by Drinfeld and Chari-Pressley. In this talk, I will give an explicit formula resulting in such an isomorphism between U_h(sl(n)) and U(sl(n))[[h]] for every n greater than or equal to 2. This is a recent joint work with Andrea Appel (arxiv:1712.03601).

February 21: Moment graphs are labeled graphs associated to varieties with a nice torus action and were used by Braden and MacPherson to give a combinatorial algorithm to compute torus-equivariant intersection cohomology. I will explain an analogue of the "mixed derived category" formalism of Achar and Riche in this setting, which leads to a notion of "mixed perverse sheaves" on moment graphs, even those that do not arise from torus-equivariant geometry. I will discuss applications of this construction to the study of the Hecke category.

February 28: For any non-zero complex number, we can construct an induced representation of $U_\xi(\mathfrak{sl}_2)$ for $\xi$ a root of unity. As shown by Ohtsuki, these representations can be used to define a knot invariant, in this case it is Alexander polynomial. We will go through the steps of this construction and examine how the representation theory imposes the skein relation. From the rank one case, we build the analogous rank two invariant, and classify the reducibility of these representations. Finally, we examine the tensor product structure of these representations, the resulting "$\mathfrak{sl}_3$" skein relation, and an approach on how to generalize further to higher rank quantum groups.

March 7: The Springer resolution is a resolution of singularities of the variety of nilpotent elements in a reductive Lie algebra. It is an important geometric construction in representation theory, but some of its features are not as nice if we are working in Type C (Symplectic group). To make the symplectic case look more like the Type A case, Kato introduced the exotic nilpotent cone and its resolution, whose fibers are called the exotic Springer fibers. We give a combinatorial description of the irreducible components of these fibers in terms of standard Young bitableaux and obtain an exotic version of the Robinson-Schensted correspondence. This is joint work with Vinoth Nandakumar and Neil Saunders.

April 11: K. Styrkas stated in his thesis that there is an equivalence of tensor categories between finite dimensional irreducible representations of sl(2, C) and the fusion category of irreducible representations of the Virasoro vertex algebra with central charge c=1. In this talk we will discuss the proof of this statement and some natural extensions of this result.

April 25: Hall algebras of categories of quiver representations and coherent sheaves on smooth projective curves over F_q recover interesting representation-theoretic objects such as quantum groups and their generalizations. I will define and describe the structure of the Hall algebra of coherent sheaves on a projective variety over F_1, with P^2 as the main example. Examples suggest that it should be viewed as a degenerate q->1 limit of its counterpart over F_q.