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- Superalgebra Theory and Representations Seminar

# 1

To any Frobenius superalgebra $A$ we associate an oriented Frobenius Brauer category and an affine oriented Frobenius Brauer category. We define natural actions of these categories on categories of supermodules for general linear Lie superalgebras $\mathfrak{gl}_{m|n}(A)$ with entries in $A$. These actions generalize those on module categories for general linear Lie superalgebras and queer Lie superalgebras, which correspond to the cases where $A$ is the ground field and the two-dimensional Clifford superalgebra, respectively.

***The mini-course will consist of 3 talks on Wednesdays (Oct. 20th, Oct. 27th, Nov. 3rd), at 19:15 Israel time.

***The mini-course will consist of 3 talks on Wednesdays (Oct. 20th, Oct. 27th, Nov. 3rd), at 19:15 Israel time.

***The mini-course will consist of 3 talks on Wednesdays (Oct. 20th, Oct. 27th, Nov. 3rd), at 19:15 Israel time.

In this talk I will describe how bigrassmannian permutations control the socle of the cokernel of

embeddings of Verma modules for sl_n. An application of this is a description of the socle of the cokernel of homomorphisms between Verma modules for the periplective Lie superalgebra. This is based on two joint works: one with Hankyung Ko and Rafael Mrden and another one with Chih-Whi Chen.

https://us02web.zoom.us/j/88189258443?pwd=S3JLcElXTUpadktqZ0VLWHNmVXdiQT09

Meeting ID: 881 8925 8443

Passcode: 40320

Dynkin introduced the notion of pi-systems in the process of classifying semi-simple subalgebras of complex semi-simple Lie algebras. In a complex simple Lie algebra, pi-systems turn out to be the simple system of regular subalgebras. One can naturally generalize the notion of pi-systems to Kac-Moody algebras. In this talk, I will talk about the classification of maximal pi-systems of affine Kac-Moody algebras. If time permits, I will also talk about the Weyl orbits of certain interesting pi-systems.

https://us02web.zoom.us/j/88189258443?pwd=S3JLcElXTUpadktqZ0VLWHNmVXdiQT09

We will discuss the classification of algebraic supergroups G for which their representation category Rep(G) is semisimple (working over an algebraically closed field of characteristic zero). The statement roughly says that OSp(1|2n) is the only 'truly super' algebraic supergroup with this property. We will discuss different proofs and related ideas, with the goal of understanding in some ways how non-semisimplicity expresses itself in a supergroup.

https://us02web.zoom.us/j/88189258443?pwd=S3JLcElXTUpadktqZ0VLWHNmVXdiQT09

In this talk, I will give an overview of the (finite dimensional) representation theory of a particular semisimple Lie superalgebra. In particular, I will explain character formulas, extension groups, block decompositions, invariant theory and Koszulity.

https://weizmann.zoom.us/j/91984558186?pwd=dzl2Z2hPRWtzaDF6R1pySXd4OGFIQT09

I will report on some progress to understand indecomposable summands in tensor products of irreducible representations of gl(m|n). I will focus on the gl(m|2) -case (m>1) which exhibits many features of the general case. The crucial tool is the Duflo-Serganova functor and some of its variants.

https://weizmann.zoom.us/j/91984558186?pwd=dzl2Z2hPRWtzaDF6R1pySXd4OGFIQT09

We will first describe some classical theory of the queer Lie superalgebras q(n), such as Clifford modules, (generic) character formula for the irreducible representations, and classification of the blocks of finite-dimensional representations. Then we will focus our attention to q(3) and provide an explicit description of the Ext-quivers of the blocks. A proof of a ``virtual'' BGG reciprocity for q(n), which then gives the radical filtrations of indecomposable projective objects for q(3), will be provided.

HTTPS://WEIZMANN.ZOOM.US/J/91984558186?PWD=DZL2Z2HPRWTZADF6R1PYSXD4OGFIQT09

I will give a series of introductory talks on tensor categories and how they give a natural setting for everything "super", such as Lie superalgebras and supergroups.No knowledge of category theory will be assumed - I will explain the categorical terms along the way.

https://weizmann.zoom.us/j/91984558186?pwd=dzl2Z2hPRWtzaDF6R1pySXd4OGFIQT09

I will give a series of introductory talks on tensor categories and how they give a natural setting for everything "super", such as Lie superalgebras and supergroups.

No knowledge of category theory will be assumed - I will explain the categorical terms along the way.

https://weizmann.zoom.us/j/91984558186?pwd=dzl2Z2hPRWtzaDF6R1pySXd4OGFIQT09