Title: Lecture 1 Algebra and representation theory without vector spaces
Speaker: Pavel Etingof (Massachusetts Institute of Technology)
Abstract: 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.