• Rekha Biswal (NISER, Bhubaneswar),Ideals in enveloping algebras of affine Kac-Moody algebras

Abstract: In this talk, I will discuss about the structure of two sided ideals in the central quotient of universal enveloping algebras of affine Kac-Moody algebras. We will see that these two sided ideals are extremely large enough to cut these very big algebras down to a reasonable size which implies that the annihilator of any non-trivial integrable highest weight module over affine Lie algebras is centrally generated extending similar result of Chari for Verma modules.

• Lukas Bonfert (University of Bonn), The Weyl groupoid of sl(m|n) and osp(r|2n)

Abstract: For Lie superalgebras the Weyl group is not as powerful as for Lie algebras, for instance it fails to permute the Borel subalgebras. To fix this it has been proposed that the Weyl group should be replaced by a "Weyl groupoid", which also includes the odd reflections. 

In my talk I will explain the notion of Weyl groupoid introduced by Heckenberger and Yamane (2008), originating in the theory of Nichols algebras. In particular I will provide an explicit combinatorial description of the Weyl groupoid for sl(m|n) and osp(r|2n). I will also discuss the relation to Serganova's definition of Weyl groupoid and the root groupoid recently introduced by Gorelik, Hinich and Serganova.

This talk is based on joint work with Jonas Nehme.

• Yasmine Fittouhi (Weizmann Institute of Science), The Composition Tableau and Reconstruction of the Canonical Weierstrass Section in Type A. 

A long story, short

Abstract.

Slides

• Maria Gorelik  (Weizmann Institute of Science) , Linkage classes for Kac-Moody superalgebras

Abstract: In this talk, I will present a uniform description of the linkage classes for finite dimensional and affine Kac-Moody superalgebras. These linkage classes  can be seen as generalization of central characters. 

I will describe the interaction between these linkage classes and the Duflo-Serganova functors. The latter are homological functors from the category of representations of a Lie superalgebra to the category of representations of a Lie superalgebra of the same type, but smaller rank. 

Slides

• Dmitry Gourevitch (Weizmann Institute of Science), Finite multiplicities of restrictions to Lie subalgebras, in terms of associated varieties

Abstract: Let g be a complex Lie algebra, and h be a subalgebra. Let M be a finitely-generated g-module. Yamashita gave a geometric sufficient condition for M to be finitely generated over h. 

We will start with this result, then give another very recent twisted version, and then tell a couple of recent geometric theorems that allow to check Yamashita’s geometric condition, as well as our twisted version of it. As an application, we deduce finiteness of top degenerate Whittaker quotients. 

This is a joint work with Avraham Aizenbud, Dor Mezer, and Arieh Zimmerman.

• Max Gurevich (Technion), Positive decompositions of Kazhdan-Lusztig polynomials

Abstract: A new algorithmic approach for computation of S_n Kazhdan-Lusztig polynomials, through their restriction to lower rank Bruhat intervals, was presented by Geordie Williamson and DeepMind collaborators. 

In a joint work with Chuijia Wang we fit the hypercube decomposition into a general framework of a parabolic recursion for Weyl group Kazhdan-Lusztig polynomials. We also show how the positivity phenomena of Dyer-Lehrer and Grojnowski-Haiman come into play in such decompositions. Staying in type A, I will explain how the new approach naturally manifests through the KLR categorification of (dual) PBW and canonical bases.

• Thorsten Heidersdorf (University of Bonn), Gradings of the DS functor and indecomposable summands in tensor products

Abstract: I will discuss some applications of a graded version of the DS functor and state some conjectures and questions about indecomposable modules of GL(m|n) which are related to this grading.

Slides

• Volodymyr Mazorchuk (Uppsala University), New results on Kostant's problem

Abstract: Kostant's problem for a given Lie algebra module, as formulated by Joseph in 1980, asks whether the universal enveloping algebra surjects onto the algebra of adjointly finite linear endomorphisms. The problem is largely open even for simple higher weight modules, though a number of special cases are answered. 

In this talk I  will  describe a complete answer for simple highest weight modules indexed by fully commutative permutations (based on a joint work with Marco Mackaay and Vanessa Miemietz) and for some classes of parabolic Verma modules (based on a joint work with Shraddha Srivastava). 

Slides

• Ian Musson (University of Wisconsin-Milwaukee), On the geometry of algebras related to the Weyl groupoid

Abstract.

Slides

• Jonas Nehme (University of Bonn), A Khovanov algebra for the periplectic Lie superalgebra

Abstract: Brundan and Stroppel defined a Khovanov algebra which gives a diagrammatic description of the endomorphism ring of a projective generator for Gl(m|n). A similar construction for Osp(r|2n) is due to Ehrig and Stroppel. This gives rise to an equivalence of categories and although this equivalence is not monoidal, there exist certain bimodules such that tensoring with these corresponds to translation functors for Gl(m|n). 

In this talk I will describe the construction of a Khovanov algebra which plays a similar role for the periplectic supergroup P(n). Furthermore I will explain how translation functors for P(n) relate to tensoring with certain bimodules for the Khovanov algebra. I will use this language then to explicitly compute the action of translation functors on projective modules.

Slides

• Ivan Penkov (Constructor University Bremen, Batsheva de Rothschild visitor), Mackey Lie (super)algebras and universal tensor categories

Abstract: This is a minicourse consisting of 3 lectures, whose outline is given below. 

• Lecture 1. Some Lie (super)algebras. First universal monoidal category: tensor modules over gl(infty).

• Lecture 2. Tensor Modules over gl(infty), o(infty), sp(infty), osp(infty/infty), gl(infty/infty). Statement of results about topological categories.

Ordered Grothendieck categories.

• Lecture 3. Universal monoidal category of tensor modules over general split Mackey Lie algebra. Koszulity and explicit Ext formulas. Outlook.

Slides for last talk (based on work in progress with Valdemar Tsanov)

References:

Main reference for the "baby category":  

1. A Koszul category of representations of finitary Lie algebras, Advances of Mathematics 289 (2016), 250-278 (with Elizabeth Dan-Cohen and Vera Serganova). download

See also: 

2. Tensor representations of classical locally finite Lie algebras, in Developments and Trends in Infinite-Dimensional Lie Theory, Progress in Mathematics 288, Birkh\"auser, 2011, pp. 127-150 (with K. Styrkas). download

\widetilde{Tens} category: 

3. Categories of integrable $sl(\infty)$-, $o(\infty)$-, $sp(\infty)$-modules, in Representation Theory and Mathematical Physics, Contemporary Mathematics 557, AMS 2011, 335-357 (with V. Serganova). download

Ordered Grothendieck categories:

4. Representation categories of Mackey Lie algebras as universal monoidal categories, Pure and Applied Mathematics Quarterly 13 (2017), 77-121 (with A. Chirvasitu). download

The universal category for Aleph_0:

5. Universal tensor categories generated by dual pairs, Applied Categorical Structures 29(5), 915-950 (with A. Chirvasitu). download

Topological representations: 

6. Topological tensor representations of gl(V) for a space V of countable dimension, Algebraic Geometry and Physics, to appear, arXiv:2206.00654 (with F. Esposito). download

7. Topological semiinfinite tensor (super)modules, arXiv:2301.08921 (with F. Esposito). download

• Shifra Reif (Bar Ilan University), The Chevalley restriction theorem for super-symmetric spaces.

Abstract: Classically, the Chevalley restriction theorem on the g-invariants in S(g), for a Lie (super)algebra g, can be seen as a graded version of the Harish-Chandra theorem on the center of the universal enveloping algebra. 

We shall discuss the corresponding version for a super-symmetric space X=G/K, namely for the associated graded space of the invariant differential operators on X. Joint work with Siddhartha Sahi and Vera Serganova.

• Vadim Schechtman (University of Toulouse), Young tableaux and Hopf algebras

Abstract: It is known for some time (see a recent joint work of M.Kapranov and the speaker) that contingency tables (matrices with nonnegative integer coefficients) form a bialgebra in some universal braided tensor category.  

In this talk we shall  discuss this result in connection with the RSK correspondence, especially from the viewpoint of some classical theorem of Donald Knuth.

Notes

• Lior Silberberg (Weizmann Institute of Science), A queer Kac-Moody construction

Abstract: The queer Lie superalgebra q(n), though not a Kac-Moody superalgebra, behaves similarly to one.

We introduce a new Kac-Moody construction for Lie superalgebras that includes both q(n) and ordinary Kac-Moody superalgebras as special cases. The main idea is to start with a super-analogue Cartan subalgebra that is not necessarily even and abelian. Accordingly, the Chevalley generators will be replaced by fixed representations of the Cartan subalgebra. The results of the first chapter of Kac's book generalize to the new construction quite naturally.  We impose several conditions on our construction and investigate which superalgebras we can obtain and a way to classify them. Our construction turns out to be somewhat rigid. We obtain a nearly complete classification, a conjecture on how to finish it, and some supporting evidence thereof.

Part of a joint work with Alexander Sherman.