April 23, 2014

October 14-16, 2011
Workshop on Category Theoretic Methods in Representation Theory
University of Ottawa


Sabin Cautis
Categorification of vertex operators

The basic representation of affine Lie algebras can be constructed via vertex operators from the Fock space representation of the Heisenberg Lie algebra. I will review and then explain how to categorify this construction. In particular, this gives a categorical action of affine Lie algebras on (derived) categories of coherent sheaves on Hilbert schemes of points on ALE spaces (joint with Anthony Licata).

James Dolan
Projective geometry = dimensional analysis

We establish a dictionary between these two forms of the study of homogeneous quantities, illuminating their roles within the tannakian philosophy of algebraic geometry.

Ben Elias
Soergel bimodules and Kazhdan-Lusztig theory I

The category of Soergel bimodules provides an algebraic categorification of the Hecke algebra H, closely linked to other geometric and representation-theoretic categorifications. We give an introduction to this category and to Kazhdan-Lusztig theory, which is roughly the study of categorifications of H and which bases of H arise from these categorifications. Then we begin to make Soergel bimodules explicit using diagrammatics (using work of Khovanov-Elias and of Elias). In this lecture we focus our attention on the Hecke algebra of the general dihedral group, where the Temperley-Lieb algebra at roots of unity makes a surprising appearance. Geordie Williamson will continue with part II.

Joel Kamnitzer
Buildings and Spiders

Certain graphs, called webs, can be used to produce invariant vectors in tensor products of standard representations of sl(n). I will explain a geometric viewpoint on these webs, using the geometric Satake correspondence. From this perspective, we see why basis webs for sl(3) are non-elliptic. This method also leads to basis webs for any sl(n). This is joint work with Bruce Fontaine and Greg Kuperberg.

Aaron Lauda
The odd nilHecke algebra and categorified quantum sl(2)

We will discuss an odd analog of the nilHecke algebra together with its applications to symmetric functions, cohomology of flag varieties, and categorification of quantum sl(2).
This is a joint work with Alexander Ellis and Mikhail Khovanov.

Anthony Licata
Heisenberg categorification and Affine Lie algebras

Given a Dynkin diagram of type ADE, we explain how to categorify the corresponding lattice Heisenberg algebra and its Fock space representation. We'll also explain the role of these categorifications in the higher representation theory of the associated affine Lie algebra. This is a joint work with Sabin Cautis.

Marco MacKaay
The extended Khovanov-Lauda calculus and the colored Rouquier complex

I will explain the extended Khovanov-Lauda (KL) calculus, which can be used to simplify calculations involving categorified divided powers. I will then explain how the extended calculus can be used to associate a homology complex, in the Schur quotient of the categorification of quantum sln (explained by Vaz in his talk), to a colored braid. The latter construction generalizes the Rouquier complex for ordinary braids. Finally, I will comment on how conjugation by the colored Rouquier complex can be lifted to give a categorification of Lusztig's braid group action on quantum sln.
These results are part of various (ongoing) collaborations; with Stosic and Vaz, with Khovanov, Lauda and Stosic and with Lauda, Stosic and Vaz.

Volodymyr Mazorchuk
2-representations of finitary 2-categories

In the talk I will try to describe how one constructs and compares principal and cell 2-representations of finitary 2-categories. Based on a joint work with Vanessa Miemietz.

Kevin McGerty
A geometric construction of the quantum Frobenius morphism

We will show how one can lift Lusztig's quantum version of the Frobenius morphism to the level of perverse sheaves on the moduli space of quiver representations, and discuss its action on the canonical basis.

Scott Morrison
Invariants of 4-manifolds from Khovanov homology

I'll explain how to define invariants of 4-manifolds (possibly with boundary, possibly with a link in the boundary) from Khovanov homology. The construction relies on a new property of Khovanov homology, roughly 'S^3 functoriality'. I'll try to indicate where the difficulties may lie establishing this property for the variations of Khovanov homology based on categorified quantum groups other than SU(2). Finally, I'll explain how this construction is best viewed as a partial categorification of the 3+1 dimensional TQFT of which the usual Witten-Reshetikhin-Turaev theories are the boundary, rather than a direct categorification of a WRT theory.

Raphael Rouquier
Adjoint braid group actions

We show that 2-representations of Kac-Moody algebras admit an action of the associated braid group (on homotopy categories). This is based on the theory of perverse equivalences.

This is a joint work with Joe Chuang.

Pedro Vaz
A diagrammatic categorification of the q-Schur algebra

In this talk I will explain how to categorify the q-Schur algebra as a quotient of Khovanov and Lauda's categorified sl(n). I will also explain the connection with (singular) Soergel bimodules.
This is a joint work with Mackaay and Stosic.

Ben Webster
Categorification of Reshetikhin-Turaev invariants

I'll describe one schema for categorifying the tensor products of representations of quantum groups and how functors between these categorify the ribbon structure on representations of quantum groups.

Geordie Williamson
Soergel bimodules and Kazhdan-Lusztig theory II

Following Ben's talk, I will continue the discussion of our presentation of Soergel bimodules via generators and relations (building on earlier work of Libedinsky and Elias-Khovanov). I will explain why an important role in the story is played by actions of Coxeter groups on categories, and how this leads naturally to certain generalisations of the Zamolodchikov equations related to the platonic solids. If time permits I will discuss some calculations of the "p-canonical basis" for Hecke algebras, and try to explain why this is important in representation theory.

Oded Yacobi
Polynomial functors and categorification

The category P of strict polynomial functors was originally introduced to study the finite generation of affine group schemes, and it has since been applied widely in algebraic topology. In this talk we describe actions of affine special linear Lie algebras and the Heisenberg algebra on P, and relate these to Schur-Weyl duality.

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