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Quantum Topology

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Volume 1, Issue 4, 2010, pp. 321–377
DOI: 10.4171/QT/8

Published online: 2010-10-17

A2-planar algebras I

David E. Evans[1] and Mathew Pugh[2]

(1) Cardiff University, UK
(2) Cardiff University, UK

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.

Keywords: Subfactors, planar algebras, SU(3)

Evans David, Pugh Mathew: A2-planar algebras I. Quantum Topol. 1 (2010), 321-377. doi: 10.4171/QT/8