Pre talk title: Tannaka-Krein duality for symmetric tensor categories and Deligne’s theorem
Abstract: I will introduce the notion of symmetric category and explain Tannaka-Krein duality, which reconstructs an affine group scheme G from the symmetric category Rep(G). We will then go through Deligne’s classification theorems in characteristic zero and give a first idea of why it fails in positive characteristic.
Talk title: Towards group scheme theory in incompressible tensor categories
Abstract: A fundamental theorem by Deligne establishes that, over an algebraically closed field of characteristic zero, any symmetric tensor category under mild finiteness conditions admits a fiber functor to the category sVec of super vector spaces, and can thus be recovered from super group scheme theory.
The problem of obtaining analog results for fields of positive characteristic has immediate complications, but in recent years it started to enjoy a systematic development. It turns out that the theory is much richer, and the role that sVec monopolized in characteristic zero can now be played by any incompressible category. The family of incompressible categories is conjecturally classified, and includes at least a countable nested sequence, the so-called Verlinde categories.
It is presumed that each incompressible category should host its own flavor of well-behaved commutative algebra and geometry, in the same way that Vec and sVec give rise to geometry and super geometry. In particular, it should provide some interesting group scheme theory. However, such group schemes are not well understood, even through examples.
I will start the talk with an account of this recent history, following works of Benson, Coulembier, Etingof and Ostrik. Then I will discuss how to obtain group schemes from Lie algebras, and report on joint work in progress with Angiono and Plavnik where we generalize the construction of contragredient Lie algebras (those of the form g(A) for a Cartan matrix A) from sVec to any symmetric category.