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\(A_{2}\)-planar algebras I. (English) Zbl 1213.46058

Summary: 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)\(\mathcal {ADE}\) 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.

MSC:

46L37 Subfactors and their classification
46L60 Applications of selfadjoint operator algebras to physics
81T40 Two-dimensional field theories, conformal field theories, etc. in quantum mechanics
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References:

[1] F. A. Bais J. K. and Slingerland, Condensate-induced transitions between topologically ordered phases. Phys. Rev. B 79 (2009), 045316.
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