Mathematics Department - Lie Groups Quantum Mathematics Seminar - Spring 2017

Lie Groups Quantum Mathematics Seminar - Spring 2017



Organizer(s)

Lisa Carbone, Yi-Zhi Huang, James Lepowsky, Siddhartha Sahi

Archive

Website

http://www.rci.rutgers.edu/~yzhuang/math/lie-quantum.html



Upcoming Talks


Friday, March 31st

Yi-Zhi Huang, Rutgers University

"Intertwining operators among twisted modules associated to not-necessarily-commutative automorphisms"

Time: 12:00 PM
Location: Hill 705
Abstract: Intertwining operators among twisted modules or twisted intertwining operators for a vertex operator algebra are the main objects of interest in the construction and study of orbifold conformal field theories. However, there has not been even a definition of such operators among twisted modules associated to noncommutative automorphisms in the literature. In a recent paper (arXiv:1702.05845), I have introduced such intertwining operators and have proved their basic properties that are necessary for a future proof of a conjecture and a construction of the associated crossed tensor categories. The proofs of these results involve careful analysis of the analytic extensions corresponding to the actions of the not-necessarily-commutative automorphisms of the vertex operator algebra. In this talk, I will discuss these operators and their properties.

Archive paper arXiv:1702.05845.


Friday, April 7th

Yi Sun, Columbia University

"Affine Macdonald conjectures and special values of Felder-Varchenko functions"

Time: 12:00 PM
Location: Hill 705
Abstract: I will explain how to refine the statement of the denominator and evaluation conjectures for affine Macdonald polynomials proposed by Etingof-Kirillov Jr. and to prove the first non-trivial cases of these conjectures. Our method applies recent work of the speaker to relate these conjectures for U_q(sl_2 hat) to evaluations of certain theta hypergeometric integrals defined by Felder-Varchenko. We then evaluate the resulting integrals, which may be of independent interest, by well-chosen applications of the elliptic beta integral of Spiridonov.

These results are joint work with E. Rains and A. Varchenko.





Past Talks


Friday, March 10th

Eveliina Peltola, University of Geneva

"Hidden quantum group structure on solution spaces of BPZ PDEs of CFT"

Time: 12:00 PM
Location: Hill 705
Abstract: I describe a systematic method for solving PDEs of conformal field theory, known as Belavin-Polyakov-Zamolodchikov (BPZ) equations. These PDEs also arise in connections with statistical physics, in the theory of Schramm-Loewner evolutions (SLEs). Our method is a correspondence associating vectors in a tensor product representation of a quantum group to Coulomb gas type integral functions, in which properties of the functions are encoded in natural, representation theoretical properties of the vectors. In particular, this hidden quantum group structure on the solution space of such PDEs enables explicit calculation of the asymptotics and monodromy properties of the solutions. This also leads us to a generalization of the Temperley-Lieb algebra, defined in terms of a diagrammatic representation, which is nothing but the commutant algebra of the quantum group in the setup of the quantum Schur-Weyl duality.

Joint work with Kalle Kytölä (Aalto University) and Steven Flores (University of Helsinki).


Friday, March 3rd

You Qi, Yale University

"Categorification at prime roots of unity"

Time: 12:00 PM
Location: Hill 705
Abstract: We sketch an algebraic approach to categorification of quantum groups at a prime root of unity, with the outlook towards eventually categorifying Witten-Reshetikhin-Turaev 3-manifold invariants.

This is based on joint work of the speaker with B. Elias, M. Khovanov and J. Sussan.


Friday, February 24th

Dmitry Vaintrob, IAS

"Mirror symmetry and the K theory of p-adic groups"

Time: 12:00 PM
Location: Hill 705
Abstract: We study the category of (complex-valued) finitely-generated smooth representations of a p-adic group G and its K theory. We show that every representation has a resolution by representations induced from finitely-generated representations of open compact subgroups. We do this by studying another category, the compactified representation category recently defined by Bezrukavnikov and Kazhdan, and using techniques from toric mirror symmetry to send it functorially into a geometric category of equivariant constructible sheaves on the Bruhat-Tits building.


Friday, February 17th

Siddhartha Sahi, Rutgers University

"Schur Q-functions and the Capelli eigenvalue problem for the Lie superalgebra q(n)"

Time: 12:00 PM
Location: Hill 705
Abstract: Let V be the associative superalgebra of type Q(n). Then V carries a two sided action of the queer Lie superalgebra q(n) and hence an action of l:= q(n)× q(n). We consider a distinguished basis {D_λ} of the algebra of l-invariant polynomial super-differential operators on V, which is indexed by strict partitions of length at most n. We show that the spectrum of D_λ, on the algebra P(V) of super-polynomials on V, is given by the factorial Schur Q-function of Okounkov and Ivanov.

This generalizes a result of Nazarov. As a further application we show that the radial projections of the spherical polynomials of the symmetric pair (q(n) x q(n),q(n)) are the classical Schur Q-functions.

This is joint work with Alexander Alldridge and Hadi Salmasian and available at https://arxiv.org/abs/1701.03401


Friday, February 10th

Wilfried Schmid, Harvard University

"B-functions and Hodge Modules"

Time: 12:00 PM
Location: Hill 705
Abstract: After describing how b-functions come up in analysis and geometry, I shall discuss a class of examples that arose in my joint work with Kari Vilonen on unitary representation of reductive groups.


Friday, February 3rd

Robert Laugwitz, Rutgers University

"The Relative Hopf and Drinfeld center of a Monoidal Category"

Time: 12:00 PM
Location: Hill 705
Abstract: The Drinfeld (or quantum) center of a monoidal category is a well-known categorical construction with applications to quantum field theory. From the point of view of representation theory, this construction gives the Drinfeld double of a Hopf algebra. Another related construction is the Heisenberg double of a Hopf algebra, giving the Weyl algebra as an example. In this talk, I will discuss how these constructions can be adapted to the setting of a monoidal category relative to a braided monoidal category. In this generality, the double bosonization of S. Majid is recovered on the algebraic level, and a purely categorical definition of a generalization of the Heisenberg double can be given, called the relative Hopf center. Further, generalizing work of V. Ostrik, a Morita dual of the relative Drinfeld center can be identified, giving a classification of categorical modules in the case of a finite tensor category. The relative Hopf center obtains a categorical action of the Drinfeld center this way, generalizing a result of J.-H. Lu.


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