# Generators and relations for the representation theory of U q(sl n) as a planar algebra

*by Scott Morrison*

*Institution:*Microsoft Station Q

*Date: Thu 25th September 2008*

*Time: 1:00 PM*

*Location: Room 107, Richard Berry Bldg, The University of Melbourne*

*Abstract*: I'll begin by explaining what a planar algebra is, and then show you

how the representation theory of SU(2) and SU(3) can be given a 'finite presentation by generators and relations' as a planar algebra.

This may be familiar to some, as the Temperley-Lieb algebra for SU(2)

or Kuperberg's spider for SU(3). Next, I'll explain my work on

generalising this to all SU(n). We'll start with a category of

diagrams, generated by some trivalent vertices, and a surjective map

to the representation theory - the difficulty will be understanding the

relations amongst these diagrams. The main trick is to remember that

SU(n) sits inside SU(n 1), and conversely representations of SU(n 1)

break up (or 'branch') as representations of SU(n). I'll explain how

to understand the combinatorics of branching in terms of my diagrams,

and how to use this to 'lift' relations for diagrams from one level to

the next.

*For More Information:* Contact: Jan de Gier degier@ms.unimelb.edu.au