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Hilbert space representations of the annular Temperley-Lieb algebra. (English) Zbl 1131.46042
Summary: The set of diagrams consisting of an annulus with a finite family of curves connecting some points on the boundary to each other defines a category in which a contractible closed curve counts for a certain complex number \(\delta\). For \(\delta = 2\cos(\pi/n)\), this category admits a \(C^*\)-structure and we determine all Hilbert space representations of this category for these values, at least in the case where the number of intemal boundary points is even. This result has applications to subfactors and planar algebras.

46L37 Subfactors and their classification
16D60 Simple and semisimple modules, primitive rings and ideals in associative algebras
57M27 Invariants of knots and \(3\)-manifolds (MSC2010)
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