## 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.
## Fall, 2023In this semester, the seminar will be mostly in person. Occasionally there might be online talks using zoom. 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**Lisa Carbone, Rutgers University**Title**Integrality of Kac-Moody groups**Time/place**9/15/2023, Friday, 12:10 pm (Eastern Time), Hill 705 (in person)**Abstract**Let A be a symmetrizable generalized Cartan matrix with corresponding Kac-Moody algebra g=g(A) over Q. Let V be an integrable highest weight g-module and let V_Z be a Z-form of V. Let G=G(Q) be an associated minimal representation-theoretic Kac--Moody group and let G(Z) be its integral subgroup. By analogy with the finite dimensional case, the integrality question for G is to determine if G(Z)=Stab_G(V_Z), that is, to determine if G(Z) coincides with the subgroup of G that stabilizes the integral lattice V_Z. Integrality of semi-simple algebraic groups G over Q was established by Chevalley in the 1950’s as part of his work on associating an affine group scheme to G(Q) and V_Z. We discuss our approach to this question for Kac-Moody groups and we prove integrality for inversion subgroups U_w of G. Here, for w in W, U_w is the group generated by positive real root groups that are flipped to negative root groups by w^{-1}. There are various applications of this result, including integrality of subgroups of the unipotent subgroup U of G that are generated by commuting real root groups. This is joint work with Abid Ali, Dongwen Liu and Scott H. Murray.
**Speaker**Vidya Venkateswaran, Center for Communications Research at Princeton**Title**Quasi-polynomials, partial symmetry, and metaplectic Whittaker functions**Time/place**9/22/2023, Friday, 12:10 pm (Eastern Time), Hill 705 (in person)**Abstract**In joint work with Siddhartha Sahi and Jasper Stokman, we introduced quasi-polynomial generalizations of (nonsymmetric and symmetric) Macdonald polynomials for arbitrary root systems. In this talk, we discuss some recent results on (anti-) symmetric quasi-polynomials. For GL_r, we show that there is a bijection between (anti-)symmetric quasi-polynomials and partially (anti-)symmetric polynomials. Moreover, under this bijection, the q \rightarrow \infty limit of quasi-polynomial generalizations of (anti-)symmetric Macdonald polynomials map to Macdonald polynomials with prescribed symmetry (as introduced in 2000 by Baker-Dunkl-Forrester), in the same limit. Translating our results to the metaplectic context, we provide a precise statement and proof of a conjecture of Brubaker-Buciumas-Bump-Gustafsson on a ``duality" between metaplectic spherical Whittaker functions and non-metaplectic parahoric Whittaker functions. These results extend to other root systems as well if we restrict to a certain subspace of the quasi-polynomials.**Slides**pdf file.
**Speaker**Robert McRae, Yau Mathematical Sciences Center at Tsinghua University**Title**Non-rigid tensor categories for affine sl_2 at admissible levels**Time/place**9/29/2023, Friday, 12:10 pm (Eastern Time), Hill 705 (in person)**Abstract**The Kazhdan-Lusztig category KL^k(sl_2) is the category of finite-length modules for affine sl_2 at level k whose composition factors are irreducible highest-weight modules whose highest weights are dominant integral for the finite-dimensional subalgebra sl_2. In this talk, we show that for admissible levels k = −2 + p/q, where p > 1 and q > 0 are relatively prime integers, KL^k(sl_2) admits the vertex algebraic braided tensor category structure of Huang-Lepowsky-Zhang, but that it is not rigid, that is, not every object has a dual. Instead, an object of KL^k(sl_2) is rigid if and only if it is projective and, moreover, KL^k(sl_2) has enough projectives. Most of the indecomposable projective objects are logarithmic modules, which means that the Virasoro L(0) operator acts non-semisimply. We show also that the monoidal subcategory of rigid and projective objects is tensor equivalent to tilting modules for quantum sl_2 at the root of unity e^{pi i/(k+2)}. This leads to a universal property for KL^k(sl_2), which allows us to construct an essentially surjective (but not fully faithful) exact tensor functor from KL^k(sl2) to the category of finite-dimensional weight modules for quantum sl_2 at e^{pi i/(k+2)}. This is joint work with Jinwei Yang.
**Speaker**Angela Gibney, University of Pennsylvania**Title**Mode Transition Algebras**Time/place**10/6/2023, Friday, 12:10 pm (Eastern Time), Hill 705 (in person)**Abstract**This talk is about recent work with Damiolini and Krashen, where we define a new series of associative algebras, which we call mode transition algebras, that give insight into geometry of moduli of curves, and representations of the vertex operator algebras from which they are derived. In this talk I will focus mainly on the VOA side of the story. I'll define the mode transition algebras, and relate our main theorem in this direction, which characterizes how mode transition algebras give information about higher level Zhu algebras. As an application, as I will illustrate, one can give an explicit description of all higher level Zhu algebras for the Heisenberg VOA, proving a conjecture of Addabbo-Barron.
**Speaker**Song Gao, University of Notre Dame**Title**Coisotropicity of fixed points under torus action on the variety of Lagrangian subalgebras**Time/place**10/13/2023, Friday, 12:10 pm (Eastern Time), Hill 705 (in person)**Abstract**I will talk about my recent study of coisotropic subalgebras of Lie bialgebras. Given a complex semisimple Lie algebra \frak{g} with adjoint group G, the set of coisotropic subalgebras of \frak{g} form an algebraic variety, which is called the variety of coisotropic subalgebras. Let H be a fixed maximal torus of G. I will introduce my results on fixed points of H-action on the variety of coisotropic subalgebras. Approaches of toric varieties and algebraic groups will be used.
**Speaker**Darlayne Addabbo, University of Arizona**Title****Time/place**11/10/2023, Friday, 12:10 pm (Eastern Time), Hill 705 and online**Abstract**
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