Abstracts

• Shlomo Gelaki (Iowa State University), "Group scheme theoretical categories"

Abstract: Group scheme theoretical categories (GSTC) are interesting  tensor categories which are defined in group scheme theoretical terms. In my talk I will explain the construction of GSTC, and then discuss their structure and properties, focusing on the finite case in positive characteristic.

• Thorsten Heidersdorf (Bonn University), "Koszulity for representations of Deligne categories"

Abstract: I will explain why representations of the Deligne categories Rep(GL_t) and Rep(O_t) (t a complex number) are Koszul (joint with Nehme and Stroppel).

• Victor Kac (MIT), "Duality of representations of linearly compact Lie superalgebras"

Abstract: I will discuss duality in the category of continuous representations of linearly compact Lie superalgebras and the closely related conformal representations of multivariable Lie conformal superalgebras. 

Slides

• Vera Serganova (UC Berkeley) , "Localization of integrals on CS manifolds with application to representation theory of supergroups"

Abstract:  We prove an analogue of the Schwarz-Zaboronsky localization formula for complex smooth supermanifolds.

Let X be a compact CS manifold and  Q  an odd vector field on X such that Q^2 is compact. Assume that Q has isolated singular points on X and preserves a volume form w.  Then the integral of w over X equals the sum of local contribution at all singular points. We apply the localization formula in the case of homogeneous superspace X=G/K which admits a G invariant volume form. For specific choices of G and K we show that the integral of w over X is not zero. This allows us to use the unitary trick and show that K is a splitting subgroup of G. In particular, we prove that a defect subgroup is splitting in the case when Lie G is any basic classical or exceptional superalgebra. Finally we show that the DS associated variety detects projectivity in the category of finite-dimensional G-modules.

The talk is based on joint work with D. Vaintrob.

• Alexander Sherman (University of Sydney) , "A queer Kac-Moody construction"

Abstract: We introduce a new Kac-Moody construction for Lie superalgebras, with the aim of putting the queer Lie superalgebra into a Kac-Moody class.  The idea is to start with a non-commutative 'super torus'  \h  as the Cartan subalgebra, and to take irreducible representations of \h as the replacements for Chevalley generators.  The results of Chapter I in Kac's book carry over nearly verbatim in our setup, and so we obtain a new way to produce many superalgebras with well-behaved triangular decompositions.  For more interesting structure, we will impose certain local conditions on our construction along with a global integrability condition.  We will then ask how such superalgebras can be classified, and what we obtain.  Remarkably we find that our construction is very rigid, and we will explain a conjecture on just how much so, along with evidence.  

Part of a joint work with Lior-David Silberberg.

Organizers

Inna Entova-Aizenbud (BGU), Maria Gorelik (WIS).

Sponsored by 

The Arthur and Rochelle Belfer Institute of Mathematics and Computer Science.