## T-Rep: A midsummer night's session on representation theory and tensor categories (July 3rd 2020)

Organised by Thorsten Heidersdorf and Catharina Stroppel.

The session will be run via Zoom. You need to register via Zoom to participate. Use the following link to register:

https://uni-bonn.zoom.us/meeting/register/tJcuce6oqz4qG9Fk9p54PxYScEDJaUMDhD58

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### Programm

(All times are in Central European Summer Time CEST, UTC +2)- 8.30pm Vera Serganova (Berkeley):
*The Jacobson-Morozov theorem for Lie superalgebras via semisimplification functor for tensor categories.*(Recorded talk) - 9:30pm Everybody:
*Discussion round. Your favorite open question? What has not received enough attention? Where is the field going?* - 10.10pm Pavel Etingof (MIT):
*New incompressible symmetric tensor categories in positive characteristic.*(Recorded talk) (Slides)

- 11.10pm Kevin Coulembier (Sydney):
*Monoidal abelian envelopes*(Recorded talk) - 12.00am Happy End/Happy Hour, time for further discussion

### Abstracts

- Kevin Coulembier

*Monoidal abelian envelopes*

Abstract: The notion of an abelian envelope of a k-linear rigid monoidal category emerged rather recently from the constructions by Entova-Hinich-Serganova and Comes-Ostrik of universal tensor categories. They subsequently reappeared in a construction by Benson-Etingof of an intriguing family of incompressible tensor categories in characteristic 2. All of this clearly demanded a thorough exploration of this concept of abelian envelopes. In this talk I will report on recent progress on the matter due to a number of authors. - Pavel Etingof

*New incompressible symmetric tensor categories in positive characteristic.*

Abstract: Let $k$ be an algebraically closed field of characteristic $p>0$. The category of tilting modules for $SL_2(k)$ has a tensor ideal $I_n$ generated by the $n$-th Steinberg module. I will explain that the quotient of the tilting category by $I_n$ admits an abelian envelope, a finite symmetric tensor category ${\rm Ver}_{p^n}$, which is not semisimple for $n>1$. This is a reduction to characteristic $p$ of the semisimplification of the category of tilting modules for the quantum group at a root of unity of order $p^n$. These categories are incompressible, i.e. do not admit fiber functors to smaller categories. For $p=1$, these categories were defined by S. Gelfand and D. Kazhdan and by G. Georgiev and O. Mathieu in early 1990s, but for $n>1$ they are new. I will describe these categories in detail and explain a conjectural formulation of Deligne's theorem in characteristic $p$ in which they appear. This is joint work with D. Benson and V. Ostrik. - Vera Serganova

*The Jacobson-Morozov theorem for Lie superalgebras via semisimplification functor for tensor categories.*

The celebrated Jacobson-Morozov theorem claims that every nilpotent element of a semisimple Lie algebra $\mathfrak{g}$ can be embedded into an $\mathfrak{sl}(2)$-triple inside $\mathfrak{g}$. Let $\mathfrak{g}$ be a Lie superalgebra with reductive even part and $x$ be an odd element of g with non-zero nilpotent $[x,x]$. We give necessary and sufficient condition when $x$ can be embedded in $\mathfrak{osp}(1|2)$ inside $\mathfrak{g}$. The proof follows the approach of Etingof and Ostrik and involves semisimplification functor for tensor categories. Next, we will show that for every odd $x$ in $\mathfrak{g}$ we can construct a symmetric monoidal functor between categories of representations of certain superalgebras. We discuss some properties of these functors and applications of them to representation theory of superalgebras with reductive even part. We also discuss possible generalization of reductive envelope of an algebraic group to the case of a supergroup. (Joint work with Inna Entova-Aizenbud).