TL;DR
This paper introduces an A_2-planar algebra framework that generalizes Jones' planar algebra, capturing complex periodicities in SU(3) subfactors and providing a generator-relation description for the Wenzl subfactor.
Contribution
It formulates an A_2-planar algebra inspired by Kuperberg's A_2-spider, extending Jones' planar algebra to include SU(3) subfactor structures and periodicities.
Findings
Provides a diagrammatic presentation of the A_2-Temperley-Lieb algebra.
Describes the Jones planar algebra for the Wenzl subfactor using generators and relations.
Captures double complex structures and periodicities in SU(3) subfactors.
Abstract
We give a diagrammatic presentation of the A_2-Temperley-Lieb algebra. Generalizing Jones' notion of a planar algebra, we formulate an A_2-planar algebra motivated by Kuperberg's A_2-spider. This A_2-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 A_2-Temperley-Lieb algebra as well as the periodicity two coming from the subfactor basic construction. We use an A_2-planar algebra to obtain a description of the (Jones) planar algebra for the Wenzl subfactor in terms of generators and relations.
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