Spring, 2017
- Speaker Robert Laugwitz, Rutgers University
- Title The Relative Hopf and Drinfeld center of
a Monoidal Category
- Time/place 2/3/2017, Friday, 12:00 in 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.
- Speaker Wilfried Schmid, Harvard University
- Title B-functions and Hodge Modules
- Time/place 2/10/2017, Friday, 12:00 in 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.
- Speaker Siddhartha Sahi, Rutgers University
- Title Schur Q-functions and the Capelli eigenvalue problem for the Lie superalgebra q(n)
- Time/place 2/17/2017, Friday, 12:00 in 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_?} 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_?, 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.
- Archive paper
arXiv:1701.03401.
- Speaker Dmitry Vaintrob, IAS
- Title Mirror symmetry and the K theory of
p-adic groups
- Time/place 2/24/2017, Friday, 12:00 in 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.
- Speaker You Qi, Yale University
- Title Categorification at prime roots of unity
- Time/place 3/3/2017, Friday, 12:00 in 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.
- Speaker Eveliina Peltola, University of Geneva
- Title Hidden quantum group structure on
solution spaces of BPZ PDEs of CFT
- Time/place 3/10/2017, Friday, 12:00 in 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).
- Speaker Yi-Zhi Huang, Rutgers University
- Title Intertwining operators among twisted
modules associated to not-necessarily-commutative
automorphisms
- Time/place 3/31/2017, Friday, 12:00 in 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.
- Speaker Yi Sun, Columbia University
- Title Affine Macdonald conjectures and special
values of Felder-Varchenko functions
- Time/place 4/7/2017, Friday, 12:00 in 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.
- Speaker Stephen Griffeth, Universidad de Talca, Chile
- Speaker Haijun Tan, Changchun University of Science and Technology and Rutgers University
- Title Some non-graded modules for the Virasoro algebra and affine Lie algebras
- Time/place 4/21/2017, Friday, 12:00 in Hill 705
- Abstract In this talk, we will introduce several classes of non-graded
modules for the Virasoro algebra and affine Lie algebras. Non-graded modules include
some restricted modules, which are in some sense the most important modules for the
representation theory of vertex algebras, and non-restricted modules. We will describe
the structures and the irreducible conditions of these modules constructed by us.
- Speaker Anatoly Vershik, St. Petersburg Department of Steklov Institute of Mathematics
- Title Standard and non-standard filtrations and tensor-like products of algebras
- Note Joint Logic/Lie Group/Quantum Mathematics Seminar
- Time/place 4/28/2017, Friday, 12:00 in Hill 705
- Abstract The problem of the classification of the decreasing sequences of algebras
(or sigma-fileds) leads to the fundmental notion of standardness. The main partial case is classification of dyadic filtrations or classification of infinite tensor-like product of algebras $M_2(C)$. The tool of classifiction is criteria of standardness; the invariants in general case is "highest Kolmogorov 0-1 laws" which tell us about
behavior of filtrations in infinity".
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