Algebraic Geometry and Representation Theory Seminar

Ben EliasUniversity of Oregon
Categorical Diagonalization

Friday, October 30, 2020 - 4:00pm

We know what it means to diagonalize an operator in linear algebra. What might it mean to diagonalize a functor?

Suppose you have an operator f and a collection of distinct scalars \kappa_i such that \prod (f - \kappa_i) = 0. Then the projection p_i to the \kappa_i-th eigenspace can be described as a polynomial in f, using a technique known as Lagrange interpolation. We think of the process of finding a complete family of orthogonal idempotents {p_i} as the diagonalization of f. After reviewing this we provide a categorical analogue: given a functor F with some additional data (akin to the collection of scalars), we construct idempotent functors P_i. The categorification of Lagrange interpolation is related to the technology of Koszul duality. Along the way we'll explain some of the basic concepts in categorification.

Diagonalization is incredibly important in every field of mathematics. I am a representation theorist, so I will briefly indicate some of the important applications of categorical diagonalization to representation theory. I'll also indicate applications to algebraic geometry.

In this talk we will follow a running example involving modules over the ring A = Z[x]/(x^2 - 1), in other words, the group algebra of the group of size 2. If you know what a complex of A-modules is (and what chain maps and homotopies are) then you have all the prerequisites needed for this talk.

This is all joint work with Matt Hogancamp.