School Seminars and Colloquia

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

Geometry/Topology Seminar

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