Algebras associated to intermediate subfactors

The Temperley-Lieb algebras are the fundamental symmetry associated to any inclusion of II1 factors N subset of M with finite index. We analyze in this paper the situation when there is an intermediate subfactor P of N subset of M. The additional symmetry is captured by a tower of certain algebras IA(n), associated to N subset of P subset of 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 (FCm,n)(n is an element of N) of colored generalizations of the Temperley-Lieb algebras, using a diagrammatic approach, a la Kauffman, that is independent of the subfactor context. The Fuss-Catalan numbers 1/(m+1)n+1(((m+2)(n)n)) appear as the dimensions of our algebras. We give a presentation of the FC1,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 FC1,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 greater than or equal to 4, we prove that the algebras IA(n), and FC1,n coincide. If one of the indices is < 4, IA(n) is a quotient of FC1,n, and we compute the Bratteli diagram of the tower (IA(k))(k is an element of N). Our results generalize to a chain of m intermediate subfactors.