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Journal of the London Mathematical Society Advance Access originally published online on November 25, 2007
Journal of the London Mathematical Society 2008 77(1):83-99; doi:10.1112/jlms/jdm071
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© 2007 London Mathematical Society

Representations of the odd affine Temperley–Lieb algebra

Sarah A. Reznikoff

Department of Mathematics and Statistics
University of Victoria
PO Box 3045 STN CSC
Victoria, BC
Canada V8W 3P4

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 {delta}. 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 {delta} is of the form Formula. 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.

2000 Mathematics Subject Classification 46L.

Research conducted in part while the author was a postdoctoral fellow at the University of Victoria and supported by NSERC of Canada.

Received November 3, 2006; published online November 25, 2007.


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