Ben Webster

Knot invariants and higher representation theory

We construct knot invariants categorifying the quantum knot variants for all representations of quantum groups. We show that these invariants coincide with previous invariants defined by Khovanov for sl_2 and sl_3 and by Mazorchuk-Stroppel and Sussan for sl_n. Our technique is to study 2-representations of 2-quantum groups (in the sense of Rouquier and Khovanov-Lauda) categorifying tensor products of irreducible representations. These are the representation categories of certain finite dimensional algebras with an explicit diagrammatic presentation, generalizing the cyclotomic quotient of the KLR algebra. When the Lie algebra under consideration is

2-block Springer fibers: convolution algebras and coherent sheaves

\mathfrak{sl}_n

Canonical bases and higher representation theory

, we show that these categories agree with certain subcategories of parabolic category O for gl_k. We also investigate the finer structure of these categories: they are standardly stratified and satisfy a double centralizer property with respect to their self-dual modules. The standard modules of the stratification play an important role as test objects for functors, as Vermas do in more classical representation theory. The existence of these representations has consequences for the structure of previously studied categorifications. It allows us to prove the non-degeneracy of Khovanov and Laudas 2-category (that its Hom spaces have the expected dimension) in all symmetrizable types, and that the cyclotomic quiver Hecke algebras are symmetric Frobenius. In work of Reshetikhin and Turaev, the braiding and (co)evaluation maps between representations of quantum groups are used to define polynomial knot invariants. We show that the categorifications of tensor products are related by functors categorifying these maps, which allow the construction of bigraded knot homologies whose graded Euler characteristics are the original polynomial knot invariants.

Mirković–Vilonen polytopes and Khovanov–Lauda–Rouquier algebras

For a fixed 2-block Springer fiber, we describe the structure of its irreducible components and their relation to the Bialynicki-Birula paving, following work of Fung. That is, we consider the space of complete flags in C^n preserved by a fixed nilpotent matrix with 2 Jordan blocks, and study the action of diagonal matrices commuting with our fixed nilpotent. In particular, we describe the structure of each component, its set of torus fixed points, and prove a conjecture of Fung describing the intersection of any pair. Then we define a convolution algebra structure on the direct sum of the cohomologies of pairwise intersections of irreducible components and closures of C^*-attracting sets (that is, Bialynicki-Birula cells), and show this is isomorphic to a generalization of the arc algebra of Khovanov defined by the first author. We investigate the connection of this algebra to Cautis & Kamnitzers recent work on link homology via coherent sheaves and suggest directions for future research.

Yangians and quantizations of slices in the affine Grassmannian

This paper develops a general theory of canonical bases, and how they arise naturally in the context of categorification. As an application, we show that Lusztigs canonical basis in the whole quantized universal enveloping algebra is given by the classes of the indecomposable 1-morphisms in a categorification when the associated Lie algebra is finite type and simply laced. We also introduce natural categories whose Grothendieck groups correspond to the tensor products of lowest and highest weight integrable representations. This generalizes past work of the authors in the highest weight case.

A geometric model for Hochschild homology of Soergel bimodules

We describe how Mirkovic–Vilonen (MV) polytopes arise naturally from the categorification of Lie algebras using Khovanov–Lauda–Rouquier (KLR) algebras. This gives an explicit description of the unique crystal isomorphism between simple representations of KLR algebras and MV polytopes. MV polytopes, as defined from the geometry of the affine Grassmannian, only make sense in finite type. Our construction on the other hand gives a map from the infinity crystal to polytopes for all symmetrizable Kac–Moody algebras. However, to make the map injective and have well-defined crystal operators on the image, we must in general decorate the polytopes with some extra information. We suggest that the resulting ‘KLR polytopes’ are the general-type analogues of MV polytopes. We give a combinatorial description of the resulting decorated polytopes in all affine cases, and show that this recovers the affine MV polytopes recently defined by Baumann, Kamnitzer, and the first author in symmetric affine types. We also briefly discuss the situation beyond affine type.

Khovanov-Rozansky homology via a canopolis formalism

We study quantizations of transverse slices to Schubert varieties in the affine Grassmannian. The quantization is constructed using quantum groups called shifted Yangians --- these are subalgebras of the Yangian we introduce which generalize the Brundan-Kleshchev shifted Yangian to arbitrary type. Building on ideas of Gerasimov-Kharchev-Lebedev-Oblezin, we prove that a quotient of the shifted Yangian quantizes a scheme supported on the transverse slices, and we formulate a conjectural description of the defining ideal of these slices which implies that the scheme is reduced. This conjecture also implies the conjectural quantization of the Zastava spaces for PGL(n) of Finkelberg-Rybnykov.

A categorical action on quantized quiver varieties

An important step in the calculation of the triply graded link homology of Khovanov and Rozansky is the determination of the Hochschild homology of Soergel bimodules for SL.n/. We present a geometric model for this Hochschild homology for any simple group G , as B ‐equivariant intersection cohomology of B B ‐orbit closures in G . We show that, in type A, these orbit closures are equivariantly formal for the conjugation B ‐action. We use this fact to show that, in the case where the corresponding orbit closure is smooth, this Hochschild homology is an exterior algebra over a polynomial ring on generators whose degree is explicitly determined by the geometry of the orbit closure, and to describe its Hilbert series, proving a conjecture of Jacob Rasmussen. 17B10; 57T10

Current algebras and categorified quantum groups

In this paper, we describe a canopolis (ie categorified planar algebra) formalism for Khovanov and Rozansky’s link homology theory. We show how this allows us to organize simplifications in the matrix factorizations appearing in their theory. In particular, it will put the equivalence of the original definition of Khovanov‐Rozansky homology and the definition using Soergel bimodules in a more general context, allow us to give a new proof of the invariance of triply graded homology and give a new analysis of the behavior of triply graded homology under the Reidemeister IIb move. 57M27; 13D02 In [9; 10], Khovanov and Rozansky introduced a series of homology theories for links. These theories categorify the quantum invariants for sln , and the HOMFLYPT polynomial. Unfortunately, they remain very difficult to calculate, not least because of the complicated matrix factorizations used in their original combinatorial definition. Later work of I Frenkel, Khovanov and Stroppel [6; 8; 14; 15] has suggested a more systematic definition of these invariants and a connection between these theories and the structure of the BGG category O for the Lie algebra gln , but progress toward computational simplifications along these lines has been slow. In this paper, we will show that these invariants can be understood, computed and in fact, defined in the context of canopolises. We hope that this approach will both lead to computational benefits and help the reader to understand the definition of Khovanov‐ Rozansky homology better. A canopolis 1 is a categorification of the notion of a planar

On generalized category \mathcal {O} for a quiver variety

In this paper, we describe a categorical action of any symmetric Kac–Moody algebra on a category of quantized coherent sheaves on Nakajima quiver varieties. By “quantized coherent sheaves,” we mean a category of sheaves of modules over a deformation quantization of the natural symplectic structure on quiver varieties. This action is a direct categorification of the geometric construction of universal enveloping algebras by Nakajima.