June 19, 2018

Young Researchers Workshop on
Higher Algebraic and Geometric Structures: Modern Methods i
n Representation Theory

May 7-9, 2012


Sabin Cautis
The higher structure of vertex operators

We will sketch a 2-category which captures the higher structure of the Heisenberg algebra associated to an affine Dynkin diagram. One can then define complexes of 1-morphisms in this 2-category which satisfy the relations in the corresponding quantum affine Lie algebra. This lifts the "vertex operator" construction of Frenkel-Kac-Segal from vector spaces to categories. (joint work with Tony Licata)

Ben Cooper

Generalized Hecke Algebras

:(Work in preparation). Hecke algebras have been closely tied to invariants of knots and links since the foundational work of Jones. I will describe new, geometrically defined, families of these algebras and explain how they can be used to enrich the story surrounding quantum invariants.

Ben Elias
Categorical actions of Coxeter groups and Hecke algebras

Actions of Coxeter groups and their braid groups by functorial equivalences are quite common in geometry and representation theory. Such an action is "strict" if it can be equipped with a compatible system of natural transformations corresponding to composition of functors. We present here an alternative way to view/check strictness, by constructing a presentation of the Coxeter group or braid group (as a monoidal category!) by generators and relations. This should be thought of as a "higher presentation" of a Coxeter group, and it is related to the topology of the (dual) Coxeter complex. We then continue this approach to discuss categorical actions of Hecke algebras, and a presentation of the Hecke category by generators and relations. This is joint work with Geordie Williamson.

David Hill
Categorification of Kac-Moody Superalgebras

We categorify one half of a quantum Kac-Moody superalgebra with non-isotropic odd roots.

David Jordan
Quantized multiplicative quiver varieties

We introduce a new class of algebras $D_q(Mat_d(Q))$ associated to a quiver $Q$ and dimension vector $d$, which yield a flat (PBW) $q$-deformation of the algebra of differential operators on the space of matrices associated to $Q$. This algebra admits a $q$-deformed moment map from the quantum group $U_q(gl_d)$, acting by base change at each vertex. The quantum Hamiltonian reduction, $A^\xi_d(Q)$, of $D_q$ by $\mu_q$ at the character $\xi$, is simultaneously a quantization of the Crawley-Boevey and Shaw's multiplicative quiver variety, and a $q$-deformation of Gan and Ginzburg's quantized quiver variety.

Specific instances of the data $(Q,d,\xi)$ yield $q$-deformations of familiar algebras in representation theory: for example, the spherical DAHA's of type $A$ arise from Calogero-Moser quivers, quantizations of parabolic character varieties (Deligne-Simpson moduli spaces) arise from comet-shaped quivers, and algebras of difference operators on Kleinian singularities arise from affine Dynkin quivers.

Carl Mautner
An Auto-Equivalence of the Equivariant Derived Category of a Nilpotent Cone

Let G be a complex reductive group and N the nilpotent cone of its Lie algebra. For any field k, one can consider the G-equivariant derived category of sheaves of k-vector spaces. This category is known to encode much information about the representation theory of the Weyl group and the finite groups of Lie type. The subject of this talk will be joint work with P. Achar in which we exhibit an interesting auto-equivalence of this equivariant derived category and in the case of the general linear group recover a type of Ringel duality.

Jaime Thind
Quantum McKay Correspondence and Equivariant Sheaves on $\mathbb{P}^{1}_{q}$

The McKay correspondence gives a bijection between finite subgroups of SU(2) and affine A,D,E Dynkin diagrams. There is a quantum version of this statement (due to Kirillov Jr and Ostrik) which relates "finite subgroups" of $U_{q} (sl_{2})$ and finite A,D,E Dynkin diagrams. We use this correspondence to construct the category of "equivariant" coherent sheaves on $\mathbb{P}^{1}_{q}$ - the quantum projective line. This is done by defining analogues of the symmetric algebra and the structure sheaf, and using them to define a triangulated category which is a natural analogue of the derived category of equivariant sheaves on $\mathbb{P}^{1}$. We then produce natural objects in this triangulated category, which provide tilting objects relating our category to the derived category of representations of the corresponding A,D,E quiver. This can be thought of as a quantum analogue of the projective McKay correspondence of Kirillov Jr.

Ben Webster
3 talks on symplectic singularities and their representation theory:

Talk 1: The flag variety: why is this cotangent bundle different from all other cotangent bundles?
Abstract: I'll give an introduction to the structures that interest me in the case of the flag variety. This includes the classical theory of category O, Harish-Chandra bimodules, the geometric construction of the Hecke algebra, left and right cells, Soergel bimodules, primitive ideals and all that jazz.

Talk 2:Flavors of conic symplectic resolutions: waffle, sugar or cake?
Abstract: The next step is to try to reconstruct this representation theory in other contexts, provided by the geometry of symplectic singularities such as quiver varieties, Hilbert schemes, hypertoric varieties and Slodowy slices. I'll introduce this cast of characters and what is known about generalizing classical representation theory to these contexts, with a light dusting of symplectic reflection algebras, and sprinkling of the theory of categorical representations.

Talk 3: How one (might) take the dual of a symplectic singularity: did you try reversing the polarity?
Abstract: I'll try to lay out the evidence that there is a duality operation on symplectic singularities. At the moment there is no definition as such for this operation, but there are a lot of surprising coincidences pointing in that direction, coming from Koszul duality in representation theory, mirror symmetry in physics and the geometry of the varieties themselves.