# <var>A</var><sub>2</sub>-planar algebras I

### David E. Evans

Cardiff University, UK### Mathew Pugh

Cardiff University, UK

## Abstract

We give a diagrammatic presentation of the A2-Temperley–Lieb algebra. Generalizing Jones’ notion of a planar algebra, we formulate an A2-planar algebra motivated by Kuperberg’s A2-spider. This A2-planar algebra contains a subfamily of vector spaces which will capture the double complex structure pertaining to the subfactor for a finite SU(3) ADE graph with a flat cell system, including both the periodicity three coming from the A2-Temperley–Lieb algebra as well as the periodicity two coming from the subfactor basic construction. We use an A2-planar algebra to obtain a description of the (Jones) planar algebra for the Wenzl subfactor in terms of generators and relations.

## Cite this article

David E. Evans, Mathew Pugh, <var>A</var><sub>2</sub>-planar algebras I. Quantum Topol. 1 (2010), no. 4, pp. 321–377

DOI 10.4171/QT/8