Spring, 2013

Spring, 2013

  • Speaker Siddhartha Sahi, Rutgers University
    • Title Intertwining operators for Grassmannians
    • Time/place 2/22/2013, Friday, 12:00 in Hill 705
    • Abstract Let G be the group GL(n,F) where F is a local field, let Gr(k,n) be the Grassmannian of k-dimensional subspaces of F^n, and let L be a G-equivariant line bundle on Gr(k,n). Then we get a (principal series) representation \pi of G on the space of smooth sections of L.

      Let \pi_1, \pi_2 be two such representations corresponding to line bundles L_1 and L_2 on two Grassmannians Gr(k_1,n) and Gr(k_2,n). We give an explicit description of the space of G-intertwiners from \pi_1 to \pi_2, and show that it has dimension at most 1.

      This is joint work with Dmitry Gourevitch.

  • Speaker Jim Lepowsky, Rutgers University
    • Title Braided tensor categories and extensions of vertex operator algebras
    • Time/place 3/1/2013, Friday, 12:00 in Hill 705
    • Abstract The notion of commutative associative algebra in a braided tensor category is analogous to the classical notion of commutative associative algebra, but based on the braided tensor category structure. Under suitable conditions, the module category of a vertex operator algebra has a natural vertex tensor category structure, in the sense of Huang-Lepowsky. This is a subtle enhancement, requiring complex variables, of braided tensor category structure. Using this structure, we discuss theorems of Huang-Kirillov-Lepowsky relating commutative associative algebras in the braided tensor category of modules for a suitable vertex operator algebra V to vertex operator algebras containing V as a subalgebra (extensions of V).

  • Speaker Yi-Zhi Huang, Rutgers University
    • Title Modular tensor categories in mathematics and physics
    • Time/place 3/8/2013, Friday, 12:00 in Hill 705
    • Abstract Modular tensor categories were first discovered in physics by Moore and Seiberg in 1988. Later, they were formulated and constructed mathematically and found applications in many different branches of mathematics and physics. This is an expository talk on modular tensor categories and their applications in, in particular, representation theory, knot theory, string theory and quantum computation. The talk will be accessible to advanced undergraduates and beginning graduate students.

  • Speaker Yi-Zhi Huang, Rutgers University
    • Title The Verlinde formula, rigidity and modular transformations
    • Time/place 3/15/2013, Friday, 12:00 in Hill 705
    • Abstract Let g be a finite-dimensional simple Lie algebra and \hat{g} the corresponding affine Lie algebra. The pioneering work of Moore and Seiberg on conformal field theory in 1988 led to the conjecture that the category of \hat{g}-modules generated by the integrable highest weight \hat{g}-modules of a fixed positive integral level k could be given a structure of modular tensor category. For a long time (at least twenty years), many mathematicians have believed that these modular tensor categories must have been constructed either by using the works of Tsuchiya-Ueno-Yamada, Beilinson-Feigin-Mazur and Bakalov-Kirillov, or by using the works of Kazhdan-Lusztig and Finkelberg. In particular, the famous Verlinde formula would be an easy consequence of such a construction.

      The present talk is on a recent discovery showing that this belief has been wrong. It has been known for a while that, despite a statement in the book of Bakalov-Kirillov, the works of Tsuchiya-Ueno-Yamada, Beilinson-Feigin-Mazur and Bakalov-Kirillov cannot be used to prove the rigidity of these tensor categories or to identify the S-matrices for these tensor categories with the modular transformations associated to \tau \mapsto -1/\tau on the space spanned by the characters of the integrable highest weight \hat{g}-modules of level k. Most recently, it has been discovered, and graciously acknowledged by Finkelberg, that the works of Kazhdan-Lusztig and Finkelberg alone also did not prove the rigidity of these tensor categories and thus also did not identify these S-matrices; for such a proof and for such an identification, one in fact needs the Verlinde formula proved by Faltings, Teleman and me. Moreover, in the cases (i) g=E_6, k=1, (ii) g=E_7, k=1 and (iii) g=E_8, k=1 or 2, even the Verlinde formula proved by Faltings and Teleman does not help because, as has long been known, the works of Kazhdan-Lusztig and Finkelberg simply do not apply to these cases and were never claimed to apply to these cases. In these cases, especially in the deep case g=E_8, k=2, the only proof of the rigidity and the only identification of the S-matrices mentioned above were given by me based on (i) the general vertex-algebraic tensor category theory constructed by Lepowsky and me and (ii) my vertex-algebraic theorems on modular invariance for compositions of intertwining operators and on the Verlinde conjecture, which gave, in particular, a much stronger version of the Verlinde formula relating the fusion rules, modular transformations, and braiding and fusing matrices. These general vertex-algebraic theorems applied to all k and all affine Lie algebras, as well as to the minimal models and many other families of conformal-field-theoretic structures.

  • Speaker Lilit Martirosyan, University of California, Berkeley
    • Title Representation theory of exceptional Lie superalgebras F(4) and G(3)
    • Time/place 3/29/2013, Friday, 12:00 in Hill 705
    • Abstract In this talk, I will talk about the problem of classifying all indecomposable representations of exceptional Lie superalgebras F(4) and G(3) and finding (super)-character or (super)-dimension formulae for the simple modules. I will discuss the problem of constructing the "superanalogue" of Borel-Weil-Bott theorem for these Lie superalgebras. I will start with basic definitions and explain the ideas in the proof.

  • Speaker Haisheng Li, Rutgers University - Camden
    • Title On quasi modules at infinity for vertex algebras
    • Time/place 4/19/2013, Friday, 12:00 in Hill 705
    • Abstract A notion of quasi module at infinity for quantum vertex algebras was previously introduced, in order to associate quantum vertex algebras to certain algebras including double Yangians. In this talk, we shall discuss quasi modules at infinity for vertex $\Gamma$-algebras with $\Gamma$ a group and we shall present a commutator formula for the vertex operators for quasi modules at infinity for a vertex $\Gamma$-algebra. We then show how to use this commutator formula to establish an equivalence between the category of lowest weight type modules for a certain family of Lie algebras and that of quasi modules at infinity for certain vertex $\Gamma$-algebras. This talk is partially based on joint work with Qiang Mu.

  • Speaker Gestur Olafsson, Louisiana State University
    • Title The Cos^\lambda transform and intertwining operators
    • Time/place 4/26/2013, Friday, 12:00 in Hill 705
    • Abstract Cos^\lambda transform has been widely studied during the last few years because of it's connection to convex geometry and to some classical integral transforms, like the Funk and Radon transform on the sphere and their generalizations to Grassmann manifolds.

      For the sphere S^n, the Cos^\lambda transform is defined on L^2(\rS^n) by

      C^\lambda (f)(\omega )=\int_{S^n} |(x,\omega )|^{\lambda - (n+1)/2}\, f(x)\, d\sigma (x)

      where $d\sigma$ is the rotational invariant probability measure on S^n and (\cdot,\cdot) stands for the usual inner product on \mathbb{R}^{n+1}. The factor (n+1)/2 is included so that C^\lambda agrees with a standard intertwining operator between certain principal series representations of SL(n+1,\mathcal{R}).

      The name cosine transform was first introduced by E. Ludwag for the case \lambda-(n+1)/2=1 the integral kernel is a power of the cosine of the angle between x and \omega.

      We will in this talk start by discussing the Cos^\lambda-transform on the sphere and it's connection to the Funk transform. Then we discuss the generalization to the Grassmanians of p-planes in (n+1)-dimensional space and its connection to intertwining operators. We then use the spectrum generating operator introduced by Branson-{\'O}lafsson-{\O}rsted in 1996 to determine the spectrum of the Cos^\lambda-transform.

  • Speaker Antun Milas, SUNY Albany
    • Title Beyond C_2-cofinite vertex algebras
    • Time/place 5/3/2013, Friday, 12:00 in Hill 705
    • Abstract