Colloquium abstracts

Pavel Etingof*
Massachusetts Institute of Technology, USA
November 26, 2020

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 (super)group 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. 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 one can do algebra in any STC, and in categories other than Rep(G) this would be a new kind of algebra. The answer turns out to be ?yes?, and beautiful examples in characteristic zero were provided by Deligne-Milne in 1981. These very interesting categories are interpolations of representation categories of classical groups GL(n), O(n), Sp(n) to arbitrary values of n in k. Deligne later