Abstract

We define the odd affine Temperley–Lieb algebra (OATLA) to be the category defined by planar diagrams in an annulus with an odd number of marked points on each boundary connected in pairs by disjoint strings and a modulus δ . This algebra is the odd part of the annularization of the Temperley–Lieb planar algebra. A positivity result is proved, which allows us to completely characterize the Hilbert space representations of OATLA when the parameter δ is of the form 2 cos π N . The results finish the project of describing the irreducible Hilbert representations of the affine Temperley–Lieb algebra, which naturally arises in the study of subfactors.