Algebras associated to intermediate subfactors. (English) Zbl 0891.46035

Bisch, Dietmar; Jones, Vaughan

Invent. Math. 128, No. 1, 89-157 (1997).

Summary: The Temperley-Lieb algebras are the fundamental symmetry associated to any inclusion of \(\text{II}_1\) 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 \(\text{IA}_n\) 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 \((\text{FC}_{m,n})_{n\in\mathbb{N}}\) of colored generalizations of the Temperly-Lieb algebras, using a diagrammatic approach, à la Kauffman, that is independent of the subfactor context. The Fuss-Catalan numbers \({1\over(m+ 1)n+ 1}\left(\begin{smallmatrix} (m+ 2)n\\ n\end{smallmatrix}\right)\) appear as the dimensions of our algebras. We give a presentation of the \(\text{FC}_{1,n}\) 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 \(\text{FC}_{1,n}\) 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 \(\text{IA}_n\) and \(\text{FC}_{1,n}\) coincide. If one of the indices is \(<4\), \(\text{IA}_n\) is a quotient of \(\text{FC}_{1,n}\) and we compute the Bratteli diagram of the tower \((\text{IA}_k)_{k\in\mathbb{N}}\). Our results generalize to a chain of \(m\) intermediate subfactors.

MSC:

46L37

Subfactors and their classification

Keywords:

Temperley-Lieb algebras; inclusion of \(\text{II}_ 1\) factors; intermediate subfactor; Popa system; standard lattice; free composition; minimal product; Bratteli diagram; Fuss-Catalan numbers; Fibonacci graph; Temperley-Lieb traces