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

Abstract

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 .

A2-planar algebras II: Planar modules

“…We believe that we in fact have equality here, so that P A (n) = (l 1 ,l 2 ) H β (l 1 ,l 2 ) . In the A 1 -case this was achieved by a dimension count of the left-and right-hand sides [33,Theorem 15]. However, we have not yet been able to determine a similar result in our A 2 -setting.…”

“…, r + , denote the eigenvalues of Λ G Λ T G . Then the following result is given in [33,Proposition 13]: The irreducible weight-zero submodules of P G are H μ j , j = 1, . .…”

“…Reznikoff [33] computed the decomposition of P G as a TL-module into irreducible TLmodules for the ADE Dynkin diagrams. For the graphs A m , m 3,…”

“…As in [33], in order to make the resulting submodules orthogonal we take an orthogonal set of eigenvectors. 2…”

Representations of the odd affine Temperley-Lieb algebra

“…The first related results appeared in [6], where Jones re-worked Graham and Lehrer's results in the subfactor context and described an algorithm for decomposing a Temperley-Lieb planar algebra representation into irreducibles. In [12] the present author used these techniques and combinatorial analysis of the spaces to decompose the planar algebras determined (from the method of [5]) from the ADE Coxeter graphs into irreducible ATL modules. The affine Temperley-Lieb algebra, which is a generalization of ATL, was defined and studied in [8].…”