Temperley–Lieb planar algebra modules arising from the ADE planar algebras

A Hilbert module over a planar algebra P is essentially a Hilbert module over a canonically defined algebra spanned by the annular tangles in P. It follows that any planar algebra Q containing P is a module over P, and in particular, any subfactor planar algebra is a module over the Temperley–Lieb p...

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Bibliographic Details
Published in:Journal of functional analysis 2005-11, Vol.228 (2), p.445-468
Main Author: Reznikoff, Sarah A.
Format: Article
Language:English
Subjects:
Subfactors and their classification
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Summary:A Hilbert module over a planar algebra P is essentially a Hilbert module over a canonically defined algebra spanned by the annular tangles in P. It follows that any planar algebra Q containing P is a module over P, and in particular, any subfactor planar algebra is a module over the Temperley–Lieb planar algebra with the same modulus. We describe a positivity result that allows us to describe irreducible Temperley–Lieb planar algebra modules, and apply the result to decompose the planar algebras determined by the Coxeter graphs A n ( n ⩾ 3 ), D n ( n ⩾ 4 ), E 6 , E 7 , and E 8 .
ISSN:0022-1236
1096-0783
DOI:10.1016/j.jfa.2005.07.006