MIT Infinite Dimensional Algebra Seminar (Spring 2008)

Drinfeld defined a unitarized R-matrix for the quantum group U_q(g). This gives a commutor for the category of U_q(g) representations, making it into a coboundary category. When g is of finite type, Henriques and Kamnitzer defined another commutor which also gives U_q(g) representations the structure of a coboundary category. We show that a particular case of Henriques and Kamnitzer's construction agrees with Drinfeld's commutor. Next we describe the action of Drinfeld's commutor on a tensor product of two crystal bases, and explain the relation to the crystal commutor (a coboundary structure on g crystals). We then discuss how some of these results generalize to non-finite type g. This is joint work with Joel Kamnitzer.

In a series of papers Schechtman and Varchenko showed how to construct U^+_q (the positive part of the quantum enveloping algebra) as cohomology of certain sheaves over configuration spaces. This was upgraded by Bezrukavnikov-Finkelber-Schechtman into an equivalence of "factorizable sheaves". In this talk I will explain, following an idea of Jacob Lurie, how the notion of factorizable sheaf fits into the framework of modules over a chiral algebra, and how the constructions mentioned above follow from Koszul duality.

Quantum W-algebras have been studied intensively by the mathematicians and physicists over the last two decades. These "algebras" are not Lie algebras in the classical sense, but rather vertex algebras, and are usually defined via the process of quantum reduction from vertex algebras associated to affine Lie algebras or from lattice vertex algebras.

In this talk we will discuss the structure and representations of W-algebras obtained by reduction from rank one lattice vertex algebras. Interestingly, for certain values of the central charge these W-algebras are quasi-rational (i.e.,irrational and of finite-representation type). Moreover, these vertex algebras also admit logarithmic representations and exhibit a version of modular invariance for generalized characters. Some of our results have applications in logarithmic conformal field theory. (Joint work with D. Adamovic).

Braided fusion categories can be thought of as quantum analogues of metric Lie algebras (i.e.,Lie algebras having a non-degenerate invariant symmetric bilinear form). We will explain how Muger's theory of centralizers in braided categories allows to extend some classical linear algebra constructions to a categorical setting. As an application, we describe non-degenerate nilpotent braided fusion categories in group-theoretical terms. This description is based on the new notions of a quantum Manin pair and Lagrangian subcategory. This is a joint work with V.Drinfeld, S.Gelaki, and V.Ostrik.

(N.B. :second lecture March 11, Noncommutative algebra seminar, 5-7 PM, Room 2-136)

The semisimple representations of (double) affine Hecke algebras are defined w.r.t. the Bernstein-Zelevinsky commutative subalgebra. In the affine GL-case, the irreducible ones are described by blocks of skew Young diagrams (I.Ch., M.Nazarov) and have important relations to the classical representation theory and quantum groups.

We will begin with the general theory of the irreducible representations of affine Hecke algebra for GL (due to Zelevinsky) and then describe the semisimple ones. They are all pseudo-unitary, however the corresponding anti-involution of the affine Hecke algebra has little to do with the one that naturally comes from the p-adic theory. It may explain why the geometric interpretation of the irreducible semisimple representations via the general classification due to Kazhdan-Lusztig and Ginzburg remains an open question for arbitrary root systems.

In the daha theory, the pseudo-unitary representations are of much greater importance than in the affine case. The examples of the pseudo-unitary structures are a) the Macdonald inner product, b)the Verlinde pairing for conformal blocks, c)the Harish-Chandra pairing in the theory of spherical functions. The GL-classification is given in terms of the blocks of infinite periodic skew Young diagrams (I.Ch, continued by T.Suzuki, M.Vazirani). Such class of diagrams is a natural challenge for the specialists in combinatorics.

In this talk I will describe the connections between quantum groups and derived algebraic geometry. I will mainly focus on derived loop spaces and derived group schemes and how they are equivalent to quantum objects. I will give a short introduction to derived algebraic geometry so no prior knowledge is assumed.

The symmetric algebra S(g) of a Lie algebra g carries a natural Poisson bracket. Shift of argument subalgebras (introduced by Fomenko and Mishchenko in 1978) form a family of maximal Poisson-commutative subalgebras in S(\fg) for semisimple \fg. This family is parametrized by regular elements of the dual space \fg*. I will discuss the quantization problem for shift of argument subalgebras, namely, how to lift these subalgebras to commutative subalgebras in the universal enveloping algebra U(\fg), and how to describe the spectra of the ``quantum shift of argument subalgebras'' of U(\fg) on (finite-dimensional) \fg-modules. These questions are related to the classical representation theory, in particular, it was observed by Vinberg, that the Gelfand-Tsetlin subalgebra in U(gl_n) is a certain limit of quantum shift of argument subalgebras, and hence the spectra of quantum shift of argument subalgebras on a finite-dimensional \fg-module can be regarded as a deformation of the corresponding Gelfand--Tsetlin polytope. The construction of the quantum shift of argument subalgebras is a version of the Feigin-Frenkel-Reshetikhin construction of higher hamiltonians for the Gaudin model. The universal enveloping algebra of the corresponding affine Lie algebra \hat{\fg} at the critical level by an appropriate quantum Hamiltonian reduction. The center at the critical level is naturally identified with the algebra of polynomial functions on the space of opers on the formal punctured disk with respect to the Langlands dual group G^L (roughly, the space of gauge equivalence classes of connections in a principal G^L-bundle with some transversality condition). This allows us to treat the spectra of the quantum shift of argument subalgebras on \fg-modules as some subsets in the space of opers. I will give a precise description of these subsets for irreducible finite-dimensional \fg-modules.

Le \hat{g} be an affine Kac-Moody algebra associated with a simple, simply laced finite dimensional Lie algebra. For each conjugacy class of its Weyl group, Kac-Peterson gave a vertex operator construction of its basic representation. The conjugacy class of the unit element gives the homogeneous realization, andthe conjugacy class of the Coxeter element gives the principle realization. I will give ageometric interpretation of this construction by using the geometry of the affine Springer fibers. I will also discuss its implication to the still conjectural Langlands duality for Kac-Moody groups.

Let X be a resolution of the ADE singularity C^2/\Gamma. Together with D. Maulik we computed the operators of divisor multiplication in the ring of quantum equivariant cohomology of Hilb_n(X). The answer is given in terms of the loop algebra of the corresponding type and the structure of the formulas is reminiscent of the Casimir operators. Conjecturally, these operators generate the whole ring of quantum cohomology. In my talk I will mostly discuss the case of A_1 singularity. All necessary geometric definitions unfamiliar to the audience will be reminded.

We will present a current attempt towards (part of) a proof of the 'Anti de Sitter - Conformal Field Theory' conjecture arising in the context of string theory, based on the discovery of a certain (infinite dimensional) Hopf algebra underlying the integrability of the problem.

D. Joyce recently defined invariants counting semistable objects in an abelian category A with a given class in K(A). He obtained wall-crossing formulae with respect to a change of stability condition for these invariants, constructed holomorphic generating functions for these and showed that they satisfy an intriguing non-linear PDE.

I will explain how Joyce's wall-crossing formulae may be understood as Stokes phenomena for a connection on the Riemann sphere taking value in the Ringel-Hall Lie algebra of the category A. This allows one in particular to interpret his generating functions as defining an isomonodromic family of such connections parametrised by the space of stability conditions of A.

This is joint work with T. Bridgeland (arXiv:0801.3974).