Evans, David E.; Pugh, Mathew \(A_{2}\)-planar algebras I. (English) Zbl 1213.46058 Quantum Topol. 1, No. 4, 321-377 (2010). 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. Cited in 1 ReviewCited in 8 Documents 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 Keywords:subfactors; planar algebras; SU(3); \(A_{2}\)-Temperley-Lieb algebra; subfactor basic construction; Wenzl subfactor PDFBibTeX XMLCite \textit{D. E. Evans} and \textit{M. Pugh}, Quantum Topol. 1, No. 4, 321--377 (2010; Zbl 1213.46058) Full Text: DOI arXiv References: [1] F. A. Bais J. K. and Slingerland, Condensate-induced transitions between topologically ordered phases. Phys. Rev. B 79 (2009), 045316. This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.