## Lie Group/Quantum Mathematics SeminarOrganizers Lisa Carbone, Yi-Zhi
Huang, Jim
Lepowsky and Siddhartha Sahi.
Starting from Spring, 2008, the Lie Group Seminar and Quantum Mathematics Seminar have merged together to a single seminar called the Lie Group/Quantum Mathematics Seminar. This seminar also has a page Lie Groups Quantum Mathematics Seminar, maintained by webmaster@math.rutgers.edu. For the Lie Group/Quantum Mathematics seminar in previous semesters, see this page. For talks in the Quantum Mathematics Seminar from Spring, 1998 to Fall, 2007, see this page. For the last few years, the Quantum Mathematics Seminar shared the time and place with the Algebra Seminar. For talks in both the Algebra and Quantum Mathematics Seminars in the last few semesters, see the page for the Algebra Seminar. For all the seminars and colloquia in the department, see the Seminars and Colloquia page.
## Fall, 2017**Speaker**Siddhartha Sahi, Rutgers University**Title**The Capelli eigenvalue problem for Lie superalgebras**Time/place**9/8/2017, Friday, 12:00 in Hill 705**Abstract**The Tits-Kantor-Koecher (TKK) construction attaches a simple Lie algebra to a simple Jordan algebra. In this setting one has a Jordan "norm" that generalizes the determinant, and a family of invariant differential operators generalizing the Capelli operators of classical invariant theory. In the early 1990s Bert Kostant and I studied the eigenvalues of these generalized Capelli operators, and a few years later Friedrich Knop and I discovered a surprising connection to Macdonald polynomials.It turns out that these ideas have analogs for Lie superalgebras, although there are several subtle issues and new phenomena. I will describe a number of recent results in this direction, which have been obtained in joint work with Hadi Salmasian, Alexander Alldridge, and Vera Serganova.
**Speaker**Sven Moeller, Rutgers University**Title**Dimension Formulae in Genus Zero and Uniqueness of Vertex Operator Algebras**Time/place**9/15/2017, Friday, 12:00 in Hill 705**Abstract**We prove a dimension formula for orbifold vertex operator algebras of central charge 24 by automorphisms of order $n$ such that $\Gamma_0(n)$ is a genus zero group. We then use this formula together with the inverse orbifold construction for automorphisms of orders 2, 4, 5, 6 and 8 to establish that each of the following fifteen Lie algebras is the weight-one space $V_1$ of exactly one holomorphic, $C_2$-cofinite vertex operator algebra $V$ of CFT-type of central charge 24: $A_5C_5E_{6,2}$, $A_3A_{7,2}C_3^2$, $A_{8,2}F_{4,2}$, $B_8E_{8,2}$, $A_2^2A_{5,2}^2B_2$, $C_8F_4^2$, $A_{4,2}^2C_{4,2}$, $A_{2,2}^4D_{4,4}$, $B_5E_{7,2}F_4$, $B_4C_6^2$, $A_{4,5}^2$, $A_4A_{9,2}B_3$, $B_6C_{10}$, $A_1C_{5,3}G_{2,2}$ and $A_{1,2}A_{3,4}^3$.This is joint work with Nils Scheithauer (Darmstadt) and Jethro van Ekeren (IMPA, Rio de Janeiro).
**Speaker**Bin Gui, Vanderbilt University**Title**A unitary tensor product theory for unitary representations of unitary vertex operator algebras**Time/place**9/22/2017, Friday, 12:00 in Hill 705**Abstract**A formal definition of unitary vertex operator algebras was introduced by Dong, Lin. For many examples of unitary VOAs (unitary minimal models, affine Lie algebras at non-negative integer levels), all representations are unitarizable. It is natural to ask whether their tensor product theories are unitary. In this talk, we try to answer this question. Let V be a unitary vertex operator algebra. We define a sesquilinear form on the tensor product of two unitary V-modules. We show that, when these sesquilinear forms are positive definite (i.e., when they are inner products), the modular tensor category for V is unitary. The positive definiteness of these sesquilinear forms, especially the positivity, is much harder to prove. We explain the main idea of the proof if time permitted.
**Speaker**Fei Qi, Rutgers University**Title**A cohomological criterion for the reductivity for modules for vertex algebras**Time/place**10/6/2017, Friday, 12:00 in Hill 705**Abstract**We use the cohomology theory of meromorphic open-string vertex algebras (MOSVA) to obtain a sufficient condition for the reductivity of left modules for such an algebra. In particular, this result gives a sufficient condition for vertex algebras, which are special MOSVAs. In this talk I will start from the theory of MOSVAs and its representations, discuss the cohomology theory of MOSVAs and present our main theorem. Many technical issues arise in the proof of this main theorem. I will address some of such issues.This is a joint work with Y.-Z. Huang.
**Speaker**Lisa Carbone, Rutgers University**Title**Groups for Borcherds algebras**Time/place**10/13/2017, Friday, 12:00 in Hill 705**Abstract**Borcherds algebras are generalizations of Kac-Moody algebras and have wide applications in physical theories and the study of automorphic forms. We discuss the problem of associating the analog of a Lie a group to a Borcherds algebra and we present some examples, including the Monster Lie algebra.
**Speaker**Jinwei Yang, Yale University**Title**Braided tensor categories of admissible modules for affine Lie algebras**Time/place**10/20/2017, Friday, 12:00 in Hill 705**Abstract**We construct a braided tensor category structure with a twist on a semisimple category of modules for an affine Lie algebra at an admissible level. We also prove the rigidity and modularity of this tensor category in the case of sl_2^. This is a joint work with T. Creutzig and Y.-Z. Huang.
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