Springfest in honor of Vera Serganova
RESEARCH WORKSHOP OF THE ISRAEL SCIENCE FOUNDATION
Abstracts
Abstracts: table of contents
The abstracts are sorted by the last name of the speaker. Some abstracts include a link to the PDF version - click on "Abstract" to view it! Handouts and notes for each talk can be found below the abstract.
Click the name of the speaker in the menu below to view their abstract.
Joseph Bernstein (Tel Aviv University)
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Remark on rigidity in Lie theory
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Abstract: Project joint with Ed Shpiz.
Let g be a finite dimensional Lie algebra over a field k of characteristic 0, i.e. it is a pair (L, b) of a vector space L and a bracket operation b. A proof of the PBW theorem is usually based on a construction of a representation p of the Lie algebra g by derivations of the algebra F=F(L) of formal functions on the space L, that lifts to a faithful representation of the universal enveloping algebra U(g). It turns out that one can choose such representation in a canonical way. I will discuss how one can describe this canonical representation.
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Jon Brundan (University of Oregon)
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Okounkov-Vershik approach to representations of the partition category
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Abstract: I will report on joint work with Max Vargas revisiting the representation theory of the partition category. Our approach is similar in spirit to the Okounkov-Vershik approach to representation theory of symmetric groups, with the Jucys-Murphy elements replaced by Enyang-Jucys-Murphy elements. Our techniques recover all of the results of Comes and Ostrik from their work on the Deligne category Rep(S_t), and we can do a little bit more besides.
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Kevin Coulembier (University of Sydney)
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Beyond abelian envelopes
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Abstract: We will give a short overview of the motivation for and main examples of abelian envelopes of rigid monoidal categories. Then we will introduce a theory which generalises abelian envelopes and is applicable to any rigid monoidal category.
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Pavel Etingof (MIT)
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New examples of incompressible symmetric tensor categories in positive characteristic
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Abstract: I'll describe generalizations Ver_{p^n}, Ver_{p^n}^+ of the incompressible abelian symmetric tensor categories defined in my joint work with D. Benson (arXiv:1807.05549) for p=2 and by Gelfand-Kazhdan and Georgiev-Mathieu in 1990s for n=1. Namely, Ver_{p^n} is the abelian envelope of the quotient of the category of tilting modules for SL_2(k) by the n-th Steinberg module, and Ver_p^n is its subcategory generated by PGL_2(k)-modules. The categories Ver_{p^n} are reductions to characteristic p of Verlinde braided tensor categories in characteristic zero, which explains the notation. I will try to describe the structure of these categories in detail, and in particular explain that they categorify the real cyclotomic rings Z[2\cos(2\pi/p^n)], and that Ver_{p^n} embeds into Ver_{p^{n+1}}. We conjecture that every symmetric tensor category of moderate growth over k admits a fiber functor to the union Ver_{p^\infty} of the nested sequence of categories Ver_{p} in Ver_{p^2} in Ver_{p^3} etc. This would provide a positive characteristic analog of Deligne's theorem in characteristic zero and a generalization of the result of arXiv:1503.01492, which shows that this conjecture holds for fusion categories (in which case the fiber functor lands in Ver_p). This is joint work with D. Benson and V. Ostrik.
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Michael Finkelberg (National Research University Higher School of Economics)
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Gaiotto conjecture on quantum geometric Satake for quantum supergroups
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Abstract: This is a joint work with Roman Travkin and Alexander Braverman. D.Gaiotto conjectured that
the category of finite dimensional representations of U_q(gl(N-1|N)) is equivalent to the category of
q-monodromic GL(N-1,C[[t]])-equivariant perverse sheaves on the determinant line bundle on the affine
Grassmannian of GL(N). I will explain an approach to this via the category of factorizable sheaves.
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Yasmine Fittouhi and Anthony Joseph (Weizmann Institute of Science)
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Components of the nilfibre in type A for Parabolic Action
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Abstract: Let G be a simple algebraic group, P a parabolic subgroup and $m$ the nilradical of its Lie algebra $p$. A theorem of Richardson says that P has a dense orbit in $m$. As a consequence the invariant algebra $\C[m]^{P'}$ is polynomial. Thus one may ask if this action admits a Weierstrass section, that is to say a linear subvariety $e+V$ of $m$ such that restriction of invariants defines an isomorphism onto $\C[e+V]$.
In type $A$, a proposal for the generators was given by Benlolo and Sanderson and verified by Joseph and Melnikov.
In previous work this was used to construct a Weierstrass section (in type A) but by heavy combinatorics.
The nilfibre N (for this action) is the zero set in $m$ of $\C[m]^{P'}_+$. It is generally not irreducible.
From the computed Weierstrass section we obtained a ``canonical'' component of N and show it to be a ``B saturation set''. Here it is suggested that the remaining components take a similar form and in turn lead to further non-equivalent Weierstrass sections. Hopefully this will be a template for constructing Weierstrass sections in general type. Preliminary computations are reported.
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Vyacheslav Futorny (IME-University of São Paulo)
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Strongly tame Gelfand-Tsetlin modules for Lie (super)algebras and vertex algebras
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Abstract: For Lie algebras and superalgebras in type A we will discuss infinite-dimensional simple modules which admit a basis of Gelfand-Tsetlin tableaux and the action via the classical Gelfand-Tsetlin formulas based on recent joint results with L.E.Ramirez, V.Serganova and J.Zhang. We will also describe the connection with the admissible representations of simple affine vertex algebras.
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Thorsten Heidersdorf (University of Bonn)
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Character formulas and the DS functor
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Abstract: I will report on joint work with Maria Gorelik on obtaining good character formulas for irreducible representations of Lie superalgebras in terms of Euler characters. We prove a formula relating the Euler characters to Kac-Wakimoto terms and determine the image of super versions of the Euler characters under the functor ds induced by the Duflo-Serganova functor DS on the supercharacter ring.
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Victor Kac (MIT)
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Cyclic elements and applications
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Abstract: Basic results on cyclic elements in simple Lie algebras will be explained, along with some applications, including normal forms of nilpotent elements and integrability of classical affine W-algebras.
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Iryna Kashuba (IME-University of São Paulo)
Joint talk with Representation Theory and Mathematical Physics Seminar
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Representation type of Jordan algebras and superalgebras
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Abstract: We will review recent and classical results on the representations of finite dimensional Jordan algebras and superalgebras. We will weigh the pros against the cons of using the Tits-Kantor-Koecher construction for this problem.
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Volodymyr Mazorchuk (Uppsala University)
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Simple modules for Lie algebras and superalgebras
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Abstract: In this talk I will try to survey the state of the art for the problem of classification of simple modules for complex Lie algebras and superalgebras. The main emphasis will be on some recent techniques and results on how to reduce the Lie superalgebra problem to the Lie algebra problem.
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Victor Ostrik (University of Oregon)
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Frobenius exact symmetric tensor categories
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Abstract: I will report on a joint work in progress with K.Coulembier and P.Etingof. We give a characterization of symmetric tensor categories over fields of positive characteristic which admit an exact tensor functor to the Verlinde category; in particular we give a characterization of Tannakian categories. A crucial
ingredient of this characterization is exactness of the Frobenius twist functor which mimics the Frobenius twist for representations of algebraic groups.
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Ivan Penkov (Jacobs University Bremen)
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Bounded weight modules at infinity
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Abstract: In this talk, I present a recent classification of simple bounded weight modules for the Lie algebras sl(\infty), o(\infty), sp(\infty) (joint work with D. Grantcharov, exploiting an unpublished idea of I. Dimitrov) and will discuss their primitive ideals (bounded primitive ideals). I will also present some results from a current joint work with D. Grantcharov and V. Serganova on the category of simple bounded weight modules for the Lie superalgebra osp(n|m) where m or n equals \infty.
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Julia Pevtsova (University of Washington)
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Support and tensor product property for integrable Hopf algebras
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Abstract: I’ll describe the hypersurface approach to the theory of support varieties for finite dimensional Hopf algebras. The idea comes from commutative algebra going back to the work of Eisenbud, Avramov-Buchweitz and Avramov-Iyengar. It has been recently implemented in two different non-commutative contexts: finite supergroup schemes in a joint project with D. Benson, S. Iyengar and H. Krause and non-braided settings such as small quantum borels in a joint project with C. Negron.
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Siddhartha Sahi (Rutgers University)
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Interpolation polynomials, Capelli operators, and Lie superalgebras
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Abstract: The interpolation polynomials are a family of inhomogeneous symmetric polynomials that are characterized by rather simple vanishing (interpolation) conditions. They were introduced by the speaker in connection with joint work with Bertram Kostant on the eigenvalues of generalized Capelli-type operators associated to Jordan algebras. Of particular interest is a one parameter subfamily, which was studied by Friedrich Knop and the speaker, and by Okunkov-Olshanski, and which is closely related to Jack polynomials and Macdonald polynomials.
We will describe two recent developments in this direction. The first set of results is joint work with Hadi Salmasian and Vera Serganova, which solves the Capelli eigenvalue problem in the setting of Lie superalgebras and Jordan superalgebras. The second is joint work with Yusra Naqvi and Emily Sergel, which proves a long-conjectured positivity property of interpolation polynomials.
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Catharina Stroppel (University of Bonn)
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Based tilting theory
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Abstract: In this talk I would like to talk about based quasihereditary algebras and generalisations thereof and describe the corresponding theory of tilting modules and Ringel duality, some kind of based tilting theory. We give a general setup in which one can study both semiinfinite highest weight categories as well as examples of algebras with triangular decompositions. Important examples arise in practise when the notion of highest weight categories is weakened to categories with (nice) stratifications. We will in particular study such categories when the strata are symmetric and describe the relevance to the representation theory.
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Arkady Vaintrob (University of Oregon)
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Mirror symmetry for invertible singularities
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Abstract: A cohomological field theory (CohFT) is an algebraic structure underlying the properties of the Gromov-Witten invariants and quantum cohomology of projective varieties. For a quasi-homogeneous polynomial W with an isolated singularity at the origin there are several known constructions of CohFTs, the so-called A- and B- Landau-Ginzburg (LG) models. The corresponding invariants play a prominent role in various mirror symmetry correspondences connecting LG models with other kinds of quantum invariants. If the polynomial W is invertible (i.e. when the number of monomials in W is equal to the number of variables), then the dual polynomial W' with the transposed matrix of exponents also has an isolated singularity, and we can talk about relations between LG models for W and W'. Correspondences of this type were first considered by physicists Berglund and Huebsch in the early 1990s, but their mathematical understanding was developed only recently. I will talk about a joint work with Weyong He, Alexander Polishchuk, and Yefeng Shen on a mirror symmetry theorem connecting a B-model of W and a A-model of W' based, respectively, on Saito's theory of primitive forms and on the CohFT constructed in my earlier work with Polishchuk using categories of matrix factorizations.
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