Representations and Lie Theory SeminarSpring 2018
Time: Wednesday, 16:30  17:30


TIME  SPEAKER  TITLE 
January 17  Diego Penta  Decomposition of a rank 3 hyperbolic KacMoody 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 RobinsonSchensted 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 
January 17: In 1983, FeingoldFrenkel found important structural results about the rank 3 hyperbolic KacMoody 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 socalled `Fibonacci' algebra (Feingold, 1980). We find that F has a grading by Fiblevel, and show that each graded piece Fib(m) is an integrable Fibmodule which contains infinitely many irreducible components. We also discuss the existence of nonstandard modules on levels m less than 3, whose weight multiplicities do not appear to satisfy the recursion formulas given by RacahSpeiser and KacPeterson.
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 nontrivial cohomological argument and has only been written down for g=sl(2) by Drinfeld and ChariPressley. 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 torusequivariant 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 torusequivariant geometry. I will discuss applications of this construction to the study of the Hecke category.
February 28: For any nonzero 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 RobinsonSchensted 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 representationtheoretic 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.