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 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 An (n 3), Dn (n 4), E6, E7, and E8. © 2005 Elsevier Inc. All rights reserved.