Symmetric tensor categories I

Abstract: Lecture 1: Algebra and representation theory without vector spaces.

A modern view of representation theory is that it is a study not just of individual representations (say, finite dimensional representations of an affine group or, more generally, supergroup scheme G over an algebraically closed field k ) but also of the category Rep(G) formed by them. The properties of Rep(G) can be summarized by saying that it is a symmetric tensor category (shortly, STC) which uniquely determines G . A STC is a natural home for studying any kind of linear algebraic structures (commutative algebras, Lie algebras, Hopf algebras, modules over them, etc.); for instance, doing so in Rep(G) amounts to studying such structures with a G -symmetry. It is therefore natural to ask: does the study of STC reduce to group representation theory, or is it more general? In other words, do there exist STC other than Rep(G) ? If so, this would be interesting, since algebra in such STC would be a new kind of algebra, one “without vector spaces”. Luckily, the answer turns out to be “yes”. I will discuss examples in characteristic zero and p>0 , and also Deligne’s theorem, which puts restrictions on the kind of examples one can have.

Mathematics

Audience: researchers in the discipline

ICRA 2020

Series comments: The Workshop and International Conference on Representations of Algebras (ICRA) will take place online between 9th November and 25th November 2020.

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Deadline for submitting research snapshots: November 1st, 2020