Hilbert space representations of the annular Temperley–Lieb algebra

Vaughan F. R. Jones and Sarah A. Reznikoff

Vol. 228 (2006), No. 2, 219–249

DOI: 10.2140/pjm.2006.228.219

Abstract

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 δ. For δ = 2cos(π∕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 internal boundary points is even. This result has applications to subfactors and planar algebras.

Keywords

planar algebras, subfactors, annular Temperley–Lieb, category, affine Hecke

Mathematical Subject Classification 2000

Primary: 46L37

Secondary: 16D60, 57M27

Milestones

Received: 12 March 2005

Accepted: 4 May 2006

Published: 1 December 2006