Fusion Categories over Non-algebraically Closed Fields

Abstract

Much of the early work on Fusion Categories was inspired by physicists’ desire for rigorous foundations of topological quantum field theory. One effect of this was that base fields other than the complex numbers were rarely considered, if at all. The relevant features of the complex numbers that make the theory work are the fact that it is characteristic zero, and algebraically closed.

This thesis explores the many interesting new phenomena that occur in the non-algebraically closed setting. We build up the general theory, and give new examples coming from a non-split generalization of Tambara-Yamagami categories. We also adapt a theorem of Etingof, Nikshychand Ostrik in order to enable G-graded extension theory to this new setting.