Representation Theory and Tensor Categories
Representation Theory and Tensor Categories
Fall 2025
Tuesdays 2:10pm - 3:30pm, Evans 740
Fall 2025
Tuesdays 2:10pm - 3:30pm, Evans 740
Organizers: Vera Serganova and Ilia Nekrasov
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I will review Joseph's proof of the following theorem: the annihilator of a Verma module in the enveloping algebra of a semisimple Lie algebra is generated by its intersection with the center of the enveloping algebra. This result leads to Bernstein-Gelfand equivalence and the Dulfo theorem: the primitive spectrum of the enveloping algebra consists of the annihilators of highest weight modules.
The classical proof contains some complicated algebro-geometric arguments, so the generalization to the quantum case required another approach. Joseph's approach is based on the fact that PRV (Parthasarathy – Ranga-Rao – Varadarajan) determinants and Shapovalov determinants have the same set of linear factors.
By viewing the type-$A$ Grassmannian $Gr(k,n)$ as a coadjoint orbit, the momentum map image of $Gr(k,n)$ and its toric subvarieties are convex polytopes in $\mathbb{R}^n$. The polytopes obtained from the toric varieties are the ones we should hope for, in the sense that they are dual to the corresponding polyhedral fans. I will describe this phenomenon, as well as a parallel that arises in the super setting with the type-$Q$ super-Grassmannian (as an odd adjoint orbit) and its toric subsupervarieties.
Invariants of Lie pseudogroup actions are important in solving the equivalence problem for geometric structures. Differential invariants arise as those for the lifted action to the space of jets, and relative invariants describe singular orbits (differential equations). Scalar relative invariants correspond to invariant divisors, while their weights to equivariant line bundles. These stabilize in jets and we discuss the basic question of finite generation of the algebra of relative differential invariants.
The work is joint with Eivind Schneider.
Contact manifolds are a natural geometric setting to describe classical dynamical systems. These structures are quantized by assigning a Schrodinger equation to every path on the manifold. This is achieved by constructing a bundle of Hilbert spaces over the manifold along with a flat connection. Fedosov’s deformation procedure guarantees formal existence of such “quantum connections”. However, exact quantum connections are in general difficult to construct. We show that the standard contact seven sphere, realized as a contact homogeneous space of the quaternionic unitary group U(2,H), canonically determines a symplectic spinor bundle of Hilbert spaces.
In addition, we obtain a sequence of exact flat quantum connections on certain finite-dimensional Hilbert subbundles of the symplectic spinor bundle carrying certain highest weight unitary U(2,H)-representations. A key ingredient of our construction is a formal embedding of u(2,H) in the universal enveloping algebra of the Heisenberg algebra U(heis_3) that mimics the Holstein-Primakoff mechansim for spin representations.
We study one-sided representations of finite-dimensional Jordan algebra $J$ with $Rad^2 J=0$. For each Jordan algebra $J$ of this class we consider its Tits-Kantor-Koecher construction $TKK(J)$ and then associate to the latter a quiver with relations $Q(J)$ such that the category of representations of $Q(J)$ is isomorphic to the category of one-sided representations of $J$. Finally we discuss techniques to determine when $Q(J)$ has finite or tame representation type.
In this talk, I will describe a definitive solution to a puzzle that has gathered occasional attention: reconciling the Reshetikhin-Turaev (RT) $3$-dimensional TQFTs, defined from modular tensor categories (as well as their spin generalizations) with Lurie’s `Cobordism Hypothesis’. RT theories are defined on the bordism $2$-category of manifolds of dimensions $1,2,3$, while Lurie’s result classifies TQFTs on the full $0,1,2,3$ bordism $3$-category in terms of the value on a point.
The special case of Turaev-Viro TQFTs, generated by spherical fusion categories, is known to fit within Lurie’s framework. Proposed generalizations to RT theories (using conformal nets or vertex algebras) have been less than conclusive. I will present a solution to the problem by constructing a target $3$-category, extending that of fusion categories, in which every Reshetikhin-Turaev theory has a `point generator’ in Lurie’s sense. The construction requires spelling out symmetric monoidality in a higher category, and clarifies some inconsistencies about the `central charge.'
This slightly `retro’ talk is based on with Dan Freed and Claudia Scheimbauer, dating some time back, but whose recent write-up clarified some sticking points. It relies on an idea of K. Walker and work of Douglas, Schommer-Pries, Snyder, Brochier and Jordan.
Microformal or thick morphisms, introduced by Ted Voronov, are a generalisation of smooth maps between manifolds that still give rise to pullbacks on functions. These pullbacks are in general nonlinear and formal, and in special cases they define L-infinity morphisms between the algebras of functions on homotopy Poisson or homotopy Schouten manifolds. In this talk, I will give a brief introduction to thick morphisms (of which there are so called classical and quantum versions), and describe a graphical calculus which calculates all terms in the formal power series that result from their pullbacks. The method heavily resembles the use of Feynman diagrams in perturbative quantum field theory, and the relationship between the calculi for classical and quantum thick morphisms is exactly the relationship between the tree-level and full perturbative expansions in QFT.
Differential operators of infinite order (DOI) are infinite series in derivatives with holomorphic coefficients decaying so fast that the action on holomorphic functions converges and preserves the domain of definition. Thus exp(d/dx) (shift operator) is not a DOI but cos(\sqrt{d/dx}) is.
In 1973 M. Sato gave a characterization of theta-zerovalues by a manifestly modular invariant system of DOI in the modular variable alone, thus deducing modularity from local conditions. This has been developed by several authors since.
I will present a "supersymmetric" approach to Sato's theory based on two observations:
The exponential of any odd supersymmetry generator is a DOI.
In some cases such odd generators, acting "on-shell" (in the space of solutions of equations of motion) satisfy even-style commutation relations.
In this talk I will introduce how to construct the algebraic quantum difference equations for a large class of weighted modules on shuffle algebras. This can be thought of as the generalization of Etingof-Varchenko's construction of the difference trigonometric Casimir connection and the algebraic generalization of the Okounkov-Smirnov's geometric quantum difference equations. As applications, we will show that the large set of examples contain the critical K-theory for quiver varieties. Specifically, in this case of the affine type A quiver varieties (including instanton moduli space), we show that the algebraic quantum difference equation is the same as the Okounkov-Smirnov's geometric quantum difference equations.
This is based on the work 2408.00615, 2405.02473, 2408.15560, 2308.00550 and a work in progress.
In the late 90’s Kuperberg developed a web basis for the invariant space of tensor products of irreducible modules for SL(2) and SL(3), providing a diagrammatic calculus for homomorphism spaces in the representation category. Since then, webs have found applications to cluster algebras, dimer models, and quantum topology. In 2025, a web basis for tensor products of fundamental representations for SL(4) was constructed by Gaetz, Pechenik, Pfannerer, Striker, and Swanson using hourglass plabic graphs. I will talk about joint work with Christian Gaetz, in which we extend Kuperberg’s ideas to describe clasped web bases for invariants of tensor products of arbitrary irreducible SL(4) representations.
Using Schur-Weyl duality between S_N and the general linear group the multiplicities in the tensor product of two irreducible S_N modules can be expressed in terms of branching multiplicities that occur when an irreducible gl_n module is restricted to gl_k inside gl_n. After this the asymptotic of Kronecker symbols can be expressed in terms of geometric asymptotic of gl_n multiplicities.
If time permits, it will be shown how these problems are related to classical and quantum integrable systems.