## 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 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 and Colloquia page.
## Fall, 2018**Speaker**Johannes Flake, Rutgers University**Title**Dirac cohomology for Hopf-Hecke algebras**Time/place**9/7/2018, Friday, 12:00 in Hill 705**Abstract**Dirac operators have played an important role in representation theory, especially for real reductive Lie groups and their unitary representations. As a generalization, David Vogan suggested considering the cohomology of Dirac operators for possibly non-unitary representations. He conjectured a connection between this Dirac cohomoloy and central characters which was not only proved in the original context but more recently, versions of it for various kinds of Hecke algebras were established. We explain how these instances can be treated uniformly using superalgebras, Hopf algebras, smash products, and PBW deformations, and we prove a version of Vogan's conjecture in this generalized framework, generalizing the known results for special cases. We will discuss the concepts and techniques used in our generalized approach, results on the classification of the algebraic objects covered by our theory, new examples, and open questions.
**Speaker**Dan Ciubotaru, University of Oxford**Title**Dirac operators for rational Cherednik algebras and Calogero-Moser families**Time/place**9/14/2018, Friday, 12:00 in Hill 705**Abstract**I will first define and present the main properties of the Dirac operator for rational Cherednik algebra. Then I will explain the connection between Vogan's Dirac morphism in this setting and Gordon's partition of irreducible representations of a complex reflection group into families defined using the geometry of the generalized Calogero-Moser space. A conjecture of Gordon, Martino, and Rouquier (verified in many cases by Martino, Bellamy, Thiel) relates the Calogero-Moser families to Lusztig's families (Kazhdan-Lusztig double cells). Finally, I will explain how Dirac cohomogy can help towards verifying this conjectural relation.
**Speaker**Yi-Zhi Huang, Rutgers University**Title**Zero-mode erivations and the completely reducibility of generalized modules for a grading-restricted vertex algebra**Time/place**9/21/2018, Friday, 12:00 in Hill 705**Abstract**I will report on the recent progress jointly wirh Qi on the cohomological criterion for the complete reducibility of generalized modules of finite length for a grading-restricted vertex algebra. We found that the first cohomology of a grading-restricted vertex algebra generated by its weight-one subspace with the algebra itself as coefficients is isomorphic to its weight-one subspace. We proved that if the first cohomology of a grading-restricted vertex algebra with an arbitrary lower bounded generalized bimodule as coefficients is isomorphic to the space of what we call zero-mode derivations, then grading-restricted generalized modules satisfying the composability condition are completely reducible.
**Speaker**Robert Laugwitz, Rutgers University**Title**Noncommutative Shifted Symmetric functions**Time/place**9/28/2018, Friday, 12:00 in Hill 705**Abstract**This is joint work with Vladimir Retakh. The theory of quasi-determinants has been used to construct a ring of noncommutative symmetric functions in the 1990s. We use a similar approach to define a ring non noncommutative \emph{shifted} symmetric function. This gives a deformation of the ring of noncommutative symmetric functions in a similar way as shifted symmetric functions (in the approach of Okonkov--Olshanski) give a deformation of the ring of symmetric functions. We can provide analogues of a couple of classical formulas in this setup.
**Speaker**Eric Sommers, University of Massachusetts**Title**The defining equations for some nilpotent varieties**Time/place**10/5/2018, Friday, 12:00 in Hill 705**Abstract**In his 1963 paper "Lie group representations on polynomial rings," Kostant found the defining equations for the nilpotent cone of a simple Lie algebra and also proved it is a normal variety. Later Broer showed uniformly that the closure of the next biggest nilpotent orbit, the subregular nilpotent orbit, is a normal variety and found its defining equations. We generalize Broer's technique to the class of nilpotent orbits that are Richardson orbits for orthogonal short simple roots. The proof involves cohomological results for line bundles on cotangent bundles of generalized flag varieties and a result related to flat bases of invariant polynomials. This is joint work with Ben Johnson.
**Speaker**Simon Lentner, University of Hamburg**Title**Constructing vertex algebras with nonsemisimple category of representations**Time/place**10/12/2018, Friday, 12:00 in Hill 705**Abstract**A long-term goal is to construct vertex algebras, where the category of representations is a finite non-semisimple (modular) tensor category. The main example for such tensor categories on the algebra side are the representations of small quantum groups. In this case, there exists a conjectural construction for a corresponding vertex algebra, as kernel of certain screening operators acting on a lattice vertex algebra (free-field realization). We review some problems and results in this line of researach, in particular the general appearance of quantum group and Nichols algebras in the space of intertwining operators.
**Speaker**Haisheng Li, Rutgers University at Camden**Title**Associating quantum vertex algebras to deformed Virasoro algebras**Time/place**10/19/2018, Friday, 12:00 in Hill 705**Abstract**This talk is mainly about a theory of (weak) quantum vertex algebras and their \phi-coordinated quasi modules. In the first part, we shall review the basic concepts and results, including the definitions of weak quantum vertex algebra and \phi-coordinated quasi module, and the conceptual construction. In the second part, we shall present a particular example to show how to use the conceptual results to associate (weak) quantum vertex algebras to the deformed Virasoro algebra ${\rm Vir}_{p,q}$, which was introduced and studied by J. Shiraishi, H. Kubo, H. Awata, and S. Odake.
**Speaker**Vidya Venkateswaran, Center for Communications Research at Princeton**Title**Metaplectic representations of affine Hecke algebras and Weyl groups**Time/place**10/26/2018, Friday, 12:00 in Hill 705**Abstract**In recent work, Chinta and Gunnells discovered a surprising new action of the Weyl group W associated to an irreducible reduced root system on the space of rational functions; it is a key ingredient in their construction of Weyl group multiple Dirichlet series. Their action depends on a metaplectic parameter n, and at n=1, one recovers the standard Weyl group action. However, their formulas are complicated and showing that they actually define a W-action is non-trivial. Their proof relies on lengthy case-by-case computations with rational functions.In this talk, I'll discuss some recent work with Siddhartha Sahi and Jasper Stokman. We construct a representation of the associated affine Hecke algebra in a natural way (as a quotient of a certain induced module), and provide formulas for the action of the generators in terms of metaplectic divided-difference operators. We then show that the Chinta-Gunnells W-action can be obtained from this via a Baxterization procedure. This gives an independent and uniform proof that it is indeed an action, and allows us to generalize with extra parameters. I'll discuss applications of our work to metaplectic Whittaker functions, as well as some ongoing work on constructing metaplectic analogues of Macdonald polynomials.
**Speaker**Alejandro Ginory, Rutgers University**Title**Fusion Algebras for Twisted Affine Lie Algebras**Time/place**10/26/2018, Friday, 2:00 pm in Hill 423 (**note the special time and room**)**Abstract**In the category of integrable highest weight modules for affine Lie algebras, the usual tensor product fails to preserve the so-called level (i.e., the scalar action of the canonical central element). For untwisted affine Lie algebras, a product structure called the fusion product makes the subcategory of modules at a fixed positive integral level into a symmetric monoidal category (in particular, with non-negative integral structure constants). In this talk, I will introduce a fusion product with integral structure constants on the space of characters of twisted affine Lie algebras that generalizes the untwisted case. Surprisingly, in the A^(2)_{2n} case and for certain natural quotients for the other twisted cases, these algebras have negative structure constants "half" the time, (depending on the parity of the level). We will discuss these and other new features in the twisted cases, and their representation-theoretic meaning.
**Speaker**Bin Gui, Rutgers University**Title**Introduction to conformal nets**Time/place**11/2/2018, Friday, 12:00 in Hill 705**Abstract**Conformal nets and VOAs are two different ways to axiomatize (unitary) chiral conformal field theory. Unlike VOAs whose approach is mainly algebraic, the theory of conformal nets puts more emphasis on functional analysis and operator algebras. In this talk I will discuss the motivations behind conformal nets, and their relations with VOAs.
**Speaker**Elizabeth Jurisich, College of Charleston**Title**A Lie group analog for the Monster Lie algebra**Time/place**11/2/2018, Friday, 2:00 pm in Hill 423 (**note the special time and room**)**Abstract**The Monster Lie algebra m, which admits an action of the Monster finite simple group M, was constructed by Borcherds as part of his program to solve the Conway-Norton conjecture about the representation theory of M. We associate the analog of a Lie group G(m) to the Monster Lie algebra m by giving generators and relations. Our group G(m) contains large free subgroups and G(m) acts on a natural extension of m as derivations. This is joint work with Lisa Carbone and Ugo Moschella.
**Speaker**Chris Sadowski, Ursinus College**Title**Characters of principal subspaces for twisted affine Lie algebras and combinatorial bases**Time/place**11/16/2018, Friday, 12:00 in Hill 705**Abstract**In this talk, we discuss various recent approaches to obtaining the characters of principal subspaces of irreducible highest weight integrable representations of certain twisted affine Lie algebras. In particular, we review past results using presentations and exact sequences, a technique which has proven to be powerful at level k=1, and discuss new results using combinatorial bases for these structures for levels k>1.
**Speaker**Corina Calinescu, CUNY and New York City College of Technology**Title**Algebraic and combinatorial properties of principal subspaces of higher level standard A_2^2-modules**Time/place**11/30/2018, Friday, 12:00 in Hill 705**Abstract**In this talk we discuss presentations and graded dimensions of the principal subspaces of level k standard modules for A_2^(2). As a consequence of the presentations, we obtain a set of recursions satisfied by the graded dimensions of the principal subspaces. Although this is not a complete system to allow us to solve for the graded dimensions, we conjecture a formula for a specialized graded dimension, given by the Nahm sum of the inverse of the tadpole Cartan matrix. When k is even, this graded dimension is related to Gollnitz-Gordon-Andrews identities. This talk is based on joint work with Michael Penn and Chris Sadowski.
**Speaker**Gail Letzter, Mathematics Research Group, National Security Agency**Title**Quantum Symmetric Pairs: Structure and Applications**Time/place**12/7/2018, Friday, 12:00 in Hill 705**Abstract**There is renewed interest in the coideal subalgebras used to form quantum symmetric pairs because of recent discoveries that they play a fundamental role in the representation theory of quantized enveloping algebras. In this talk, we present highlights of this special family of quantum cordials. We start with their connection to reflection equations which parallels the relationship between quantized enveloping algebras and the Quantum Yang Baxter Equations and then illustrate how representations for these quantum cordials appear in a variety of contexts. Some observations will be related to research interests at Rutgers.
**Speaker**Henrik Gustafsson, Stanford University**Title**Vertex Operators, Solvable Lattice Models and Metaplectic Whittaker Functions**Time/place**12/7/2018, Friday, 2:00 pm in Hill 423 (**note the special time and room**)**Abstract**In this talk, based on joint work with Ben Brubaker, Valentin Buciumas and Daniel Bump, I will explain new connections relating metaplectic Whittaker functions and certain solvable lattice models with operators on a $q$-deformed fermionic Fock space. We will discuss the locality properties of these operators which agree with those of a $q$-deformed vertex operator algebra, and review the underlying quantum groups that are part of the above connections. In the process we also obtain a Fock space operator description of ribbon symmetric functions, or LLT polynomials, introduced by Lascoux, Leclerc and Thibon.
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