MIT Infinite Dimensional Algebra Seminar (Fall 2012)

This talk will be based on three papers all pertaining to the study of finite-dimensional Hopf algebras on (variants of) Artin-Schelter regular algebras. The first half of the talk will be dedicated to classification quantum analogues of finite subgroups of SL(2,k), and their actions on Artin-Schelter regular algebras of global dimension two ("Quantum binary polyhedral groups and their actions on quantum planes" joint with K. Chan, E. Kirkman, and J. Zhang). Moreover, I will discuss the remaining two papers during the last half of the talk: "Hopf actions and Nakayama automorphisms" joint with K. Chan and J. Zhang) and "Hopf actions on filtered regular algebras" joint with K. Chan, Y. Wang, and J. Zhang.

We study the exotic t-structure on the derived category of coherent sheaves on the Springer fibre for a two-block nilpotent in type $A$. The exotic t-structure has been defined by Bezrukavnikov and Mirkovic for any exact base change of the Springer resolution. Using work of Cautis and Kamnitzer, we construct functors indexed by affine tangles, between categories of coherent sheaves on different two-block Springer fibres. After checking some exactness properties of these functors, we describe the irreducible objects in the heart of the exotic t-structure for the nilpotent of type $(m+n,n)$ and enumerate them by crossingless $(m,m+2n)$ matchings. This is based on work of Anno, where the $m=0$ case was handled.

Joint with M.Finkelberg, B.Feigin, A.Negut, A. Braverman.

Laumon moduli spaces are certain smooth closures of the moduli spaces of based maps from the projective line to the flag variety of $GL_n$. They are similar (but different) to symplectic resolutions. Namely, Laumon spaces are small resolutions of singular affine algebraic varieties (called Zastava spaces) and there is a natural Poisson structure on Laumon spaces which is symplectic at generic point. We construct the action of the Yangian of $sl_n$ in the (localized equivariant) cohomology of Laumon spaces by certain natural correspondences. In fact they are analogous to the correspondences used by M. Varagnolo to construct the action of Yangians in the equivariant cohomology of quiver varieties. We compute cohomology rings and (conjecturally) quantum cohomology rings of Laumon spaces in terms of this Yangian action. The affine analogs of Laumon spaces give a geometric construction of representations of the toroidal Yangian of type A (similar representations were studied by D.Uglov). Using this, we describe the cohomology ring of the Gieseker spase of framed rank n torsion-free sheaves on the projective plane in Yangian terms. I will also discuss the generalization of this construction to partial flag varieties -- this leads to shifted Yangians and W-algebras.

We propose an algebraic approach to categorify some small quantum groups at prime roots of unity, using p-differential graded algebras. The talk will focus on the example of the sl(2) case.

The odd symmetric functions are a Hopf superalgebra that, while neither commutative nor cocommutative, admits combinatorial structure similar to that of the usual symmetric functions. We introduce "odd" analogues of the the nilHecke algebra, the cohomology of Grassmannians, and related algebras. These algebras are used in constructing an odd categorification of quantum sl(2) and, conjecturally, odd Khovanov homology.

The purpose of the talk is to explain that formal Frobenius manifolds (also known under the name of hypercommutative algebras) are in one-to-one correspondence with BV-algebras where the action of BV-operator is trivialized. The equivalence of categories is explained on the level of operads where Frobenius manifolds are considered as algebras over the operad of compactified moduli space of curves of genus 0 and trivialization of a BV-operator means a particular homotopy with the trivial operator. In addition, this identification provides an explanation of the nature of the Givental group action on Frobenius manifolds.

Joint with N.Markarian, S.Shadrin http://arxiv.org/abs/1206.3749

This is part of a mini-course given by Prof. Schneps during the week Nov. 5th-Nov. 9th in the framework of the "Noncommutative algebra seminar".

This lecture continues directly from lecture 2 by detailing some of the most important theorems and proofs of recent years, in particular Furusho's proof that grt needs only one defining relation and his remarkable use of polylogs to prove that grt injects into the formal multiple zeta Lie algebra, and Francis Brown's proof that the fundamental Lie algebra of the category of mixed Tate motives over Z is free, yielding an injection of this canonical free Lie algebra into grt. These algebras are conjecturally isomorphic; if true this would imply a Lie version of the conjecture that the absolute Galois group is isomorphic to GT.

I will explain how the conjectural relationship between hilbert schemes of points on a singular curve and the HOMFLY homology of its link may be restated by replacing the Hilbert scheme by a (type A) Hitchin fibre with the given curve as its spectral curve. In particular, the recent proof of the Euler characteristic version of this conjecture by Maulik gives a formula for the Euler characteristic of a nil-elliptic affine springer fibre of type A: it is a certain coefficient of the HOMFLY polynomial of the link of the singularity of the spectral curve.

In the special case of torus knots, the corresponding Hitchin fibre furnishes a representation of the rational DAHA, and the cohomological information in the Hilbert schemes becomes a filtration. This filtration is hard to access, but I will describe two conjecturally equivalent filtrations of a more explicit nature. Time permitting, I will discuss the relationship of the above story to the conjectures of Aganagic, Shakirov, and Cherednik on the HOMFLY homology of torus knots.

Following Dennis Gaitsgory, we give a conceptual proof of a classical result of Atiyah and Bott about the cohomology ring of a moduli space Bun_G of G-principal bundles on an algebraic curve. We write down its cohomology as a chiral homology of a certain very natural chiral algebra on the curve. And then compute this chiral homology using various methods developed by Gaitsgory, Beilinson and Drinfeld.

We give an analog of this result for the two-dimensional case, where we use a Hilbert scheme of a surface in place of Bun_G. That is, we compute the cohomology of the Hilbert scheme in a very similar manner as we did for curves. We notice that the Hilbert scheme factorizes similarly to Beilinson-Drinfeld Grassmanian. Using this factorization structure, we find a chiral algebra on our surface, whose chiral homology is the cohomology ring of the Hilbert scheme.