Diagrammatic methods for quantum symmetries

Scott Morrison
Australian National University

Webpage of the Course

Abstract: We will discuss diagrammatic approaches to the study of quantum symmetries, using planar algebras and pivotal categories. I will emphasize throughout a “generators modulo local relations” approach to describing quantum symmetries.

Most of the lectures will be focused on examples:

• we will “discover” the Fibonacci categories as a toy example of a classification problem,
• we will use this to motivate the definitions of planar algebras and of pivotal categories, and explain the relationship between these definitions and planar diagrams,
• we will continue Eric’s introduction to the Temperley-Lieb-Jones categories, discovering the $D_{2n}$ categories via their skein theory,
• we will use this example to motivate a diagrammatic approach to algebras and their modules,
• we will begin studying diagrams on manifolds other than the disc, discovering the Drinfeld center and explaining the relationship with the algebraic definition,
• time permitting we will learn how to constrain possible “generators mod relations” presentations of a fusion category based on combinatorial properties of the fusion ring learn how to constrain possible “generators mod relations” presentations of a fusion category based on combinatorial properties of the fusion ring.