A2-Planar Algebras I

Details A2-Planar Algebras I A2-Planar Algebras I

2009-06-23

arxiv.org/abs/0906.4225v6

Quantum Topol.1:321-377,2010

Mathematics

Operator Algebras

53 pages; minor corrections to published version in Section 2, pages 3-4 (pages 323-324 of published version)

Scientific paper

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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