, Volume 128, Issue 1, pp 89-157

Algebras associated to intermediate subfactors

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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.

Oblatum 1-XII-1995 & 1-VII-1996