Hilbert space representations of the annular Temperley-Lieb algebra. (English) Zbl 1131.46042

Jones, Vaughan F. R.; Reznikoff, Sarah A.

Pac. J. Math. 228, No. 2, 219-249 (2006).

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.

MSC:

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)

Keywords:

planar algebras; subfactors; annular Temperley-Lieb; category; affine Hecke