## 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. The information on seminar talks can also be found in the Seminars & Colloquia Calendar page in the department. 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 a few years before 2008, the Quantum Mathematics Seminar shared the time and place with the Algebra Seminar. For talks in both the Algebra and Quantum Mathematics Seminars in these few semesters, see the page for the Previous Rutgers Algebra Seminars. For all the seminars and colloquia in the department, see the Seminars & Colloquia Calendar page.
## Spring, 2023In this semester, the seminar will be in the hybrid format, some of the talks will be online using zoom and some of them in person. See the information below on each talk. For online talks, here is the information for the zoom meeting: Zoom link: https://rutgers.zoom.us/j/93921465287 Meeting ID: 939 2146 5287 Passcode: 196884, the dimension of the weight 2 homogeneous subspace of the moonshine module Some of the talks will be recorded and will be placed in the YouTube Channel for the seminar. **Speaker**Hadi Salmasian, University of Ottawa**Title**Quantum Weyl algebras and a new First Fundamental Theorem of invariant theory for U_q(gl(n))**Time/place**2/10/2023, Friday, 12:10 pm (Eastern Time), Zoom link above (online)**Abstract**The First Fundamental Theorem (FFT) is one of the highlights of invariant theory of reductive groups and goes back to the works of Schur, Weyl, Brauer, etc. For the group GL(V), the FFT describes generators for polynomial invariants on direct sums of several copies of the standard module V and its dual V*. Then R. Howe pointed out that the FFT is closely related to a double centralizer statement inside a Weyl algebra (a.k.a. the algebra of polynomial-coefficient differential operators). In this talk we first construct a quantum Weyl algebra and then present a quantum analogue of the FFT. We explain the relation between the FFT and a double centralizer property inside the quantum Weyl algebra. We remark that the FFT that we obtain is different from the one proved by G. Lehrer, H. Zhang, and R.B. Zhang for U_q(gl(n)). Time permitting, I will explain the connection between this work and the Capelli Eigenvalue Problem in the context of quantum symmetric spaces. This talk is based on a joint project with Gail Letzter and Siddhartha Sahi.**Slides**pdf file.**YouTube video**Quantum Weyl algebras and a new First Fundamental Theorem of invariant theory for U_q(gl(n))
**Speaker**Guillaume Remy, Institute for Advanced Study**Title**A probabilistic approach to Liouville CFT**Time/place**3/3/2023, Friday, 12:10 pm (Eastern Time), Hill 705 (in person)**Abstract**Liouville conformal field theory (CFT) was introduced by Polyakov in 1981 as the theory governing the conformal factor in the summation over all 2d Riemannian metrics. In recent years it has undergone extensive study in the probability community as a model of random surfaces, and numerous CFT predictions have been established at a mathematical level of rigor. In this talk we will first explain how one can use probability theory to rigorously define Liouville theory in the path integral approach and then survey the main mathematical achievements of this program. In particular we will present our latest result on the modular transformation of conformal blocks and explain the connections with the algebraic approaches to Liouville CFT. No prior knowledge of probability theory will be assumed.**Slides**pdf file.**Papers**See the webpage of Guillaume Remy for the papers discussed in this talk.
**Speaker**Songhao Zhu, Rutgers University**Title**Eigenvalues of Shimura Operators for Lie Superalgebras**Time/place**3/10/2023, Friday, 12:10 pm (Eastern Time), Hill 705 (in person)**Abstract**A recent work by Siddhartha Sahi and Genkai Zhang relates the Shimura operators defined for Hermitian symmetric pairs (g, k) and symmetric interpolation polynomials. Specifically, these operators, which are realized as elements in the universal enveloping algebra, have eigenvalues equal to the Okounkov interpolation symmetric polynomials of Type BC. We obtained partial generalization to the super case. In particular, I will discuss super Shimura operators defined for a Hermitian symmetric superpair (g, k), the Harish-Chandra isomorphism, the Weyl groupoid invariance, and how these puzzle pieces are put together in the very specific Type A scenario. Of particular interest and challenge are the spherical representations which are the key ingredient in our proof. With the help of Kac modules, we propose an algebraic way to answer the question of when is an irreducible highest weight g-module spherical. No prior knowledge of Lie superalgebras will be assumed.**Slides**pdf file.
**Speaker**Roberto Volpato, University of Padova**Title**Borcherds-Kac-Moody superalgebras and superstrings**Time/place**3/24/2023, Friday, 12:10 pm (Eastern Time), Zoom link above (online)**Abstract**There are several known examples of Borcherds-Kac-Moody (BKM) (super)algebras that can be constructed as the space of physical states in a suitable vertex (super)algebra (VA) (a chiral superstring, in physics parlance). Famously, one of these examples was a key ingredient in the proof of the Monstrous moonshine conjecture. We discuss a particular case of this construction, which leads to a BKM superalgebra with a natural action of the Conway group Co0, induced by an action on the corresponding VA by vertex algebra automorphisms. On more general grounds, we describe how some of the BKM algebras constructed in this way can arise in low dimensional compactification of a (non-chiral) superstring theory. This perspective sheds a new light on the relation between BKM algebras and Moonshine conjectures. No prior knowledge of string theory is assumed.
**Speaker**Lisa Carbone, Rutgers University**Title**A Monstrous Lie group**Time/place**3/31/2023, Friday, 12:10 pm (Eastern Time), Zoom link above (online)**Abstract**
**Speaker**Jianqi Liu, University of California at Santa Cruz**Title**Yang-Baxter equations for vertex operator algebras and their operator forms**Time/place**4/7/2023, Friday, 12:10 pm (Eastern Time), Hill 705 (in person)**Abstract**The classical Yang-Baxter equation (CYBE) is the semi-classical limit of the quantum Yang-Baxter equation, named after the physicists C.N. Yang and R. Baxter. Its solution has inspired the remarkable works of Belavin, Drinfeld, and others and the important notions of Lie bialgebras and Manin triples. Motivated by the importance of the CYBE in the theory and applications of Lie algebras, various analogs of the CYBE for other algebraic structures have been extensively studied in recent years. In this talk, we will first review the tensor and operator forms of the CYBE and their connections with the Rota-Baxter operators. Then we will give a generalization of the CYBE in both tensor and operator form to the vertex operator algebras, which provides us with a parameter-independent Yang-Baxter equation whose solutions lie in an infinite-dimensional vector space. Finally, we will discuss its relations with the CYBE. No prior knowledge of Yang-Baxter equations is required. This talk is based on a joint work with Chengming Bai and Li Guo.
**Speaker**Yi-Zhi Huang, Rutgers University**Title**Modular invariance of logarithmic intertwining operators**Time/place**4/28/2023, Friday, 12:10 pm (Eastern Time), Hill 705 (in person)**Abstract**
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