# Algebras associated to intermediate subfactors

DOI: 10.1007/s002220050137

- Cite this article as:
- Bisch, D. & Jones, V. Invent math (1997) 128: 89. doi:10.1007/s002220050137

## Abstract.

The Temperley-Lieb algebras are the fundamental symmetry associated to any inclusion of \({\hbox{\uppercase\expandafter {\romannumeral2}}}_1\) factors \(N \subset M\) with finite index. We analyze in this paper the situation when there is an intermediate subfactor \(P\) of \(N \subset M\). The additional symmetry is captured by a tower of certain algebras \({\rm IA}_n\) associated to \(N \subset P \subset M\). These algebras form a *Popa system* (or standard lattice) and thus, by a theorem of Popa, arise as higher relative commutants of a subfactor. This subfactor gives a *free composition* (or *minimal product*) of an \(A_n\) and an \(A_m\) subfactor. We determine the Bratteli diagram describing their inclusions. This is done by studying a hierarchy \((FC_{m,n})_{n \in {\Bbb N}}\) of colored generalizations of the Temperley-Lieb algebras, using a diagrammatic approach, à la Kauffman, that is independent of the subfactor context. The *Fuss-Catalan numbers*\(\frac{1}{(m+1)n+1}\left({(m+2)n\atop n}\right)\) appear as the dimensions of our algebras. We give a presentation of the \(FC_{1,n}\) and calculate their structure in the semisimple case employing a diagrammatic method. The principal part of the Bratteli diagram describing the inclusions of the algebras \(FC_{1,n}\) is the *Fibonacci graph*. Our algebras have a natural trace and we compute the trace weights explicitly as products of Temperley-Lieb traces. If all indices are \(\geq 4\), we prove that the algebras \({\rm IA}_n\) and \(FC_{1,n}\) coincide. If one of the indices is \(< 4\), \({\rm IA}_n\) is a quotient of \(FC_{1,n}\) and we compute the Bratteli diagram of the tower \(({\rm IA}_k)_{k \in {\Bbb N}}\). Our results generalize to a chain of \(m\) intermediate subfactors.