Séance Séminaire

Séminaire Algèbre Géométrie Algébrique Topologie Algébrique

Some applications of fusion categories as (quantum) symmetries

It is widely accepted that (classical) symmetries are captured by groups. However, quantum theory has suggested a wider class of symmetries, which are instead captured by generalisations of groups known as fusion categories (or their decategorified version: fusion rings). In the first talk, I will apply this concept to the study of root systems, where group actions (foldings) were classically used to study root systems associated to non-simply-laced Dynkin diagrams (BCFG). I will show that root systems from non-crystallographic Coxeter-Dynkin diagrams (HI) can also be obtained — we just have to consider foldings by fusion rings as well. In the second talk, I will move to the categorical setting instead, which is where the actions of fusion rings from the first talk really came from. These are related to categorical actions of Artin—Tits groups, and via Bridgeland stability conditions, are related to the K(pi,1) conjecture. I will also discuss how actions of fusion categories (which aren’t just groups) arise naturally in the categorical setting; main examples include Coh(X/G), A#G-mod.