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 anal yze in this paper the situation when there is an intermediate subfacto r P of N subset of M. The additional symmetry is captured by a tower o f certain algebras IA(n), associated to N subset of P subset of M. The se algebras form a Popa system (or standard lattice) and thus, by a th eorem of Popa, arise as higher relative commutants of a subfactor. Thi s subfactor gives a free composition (or minimal product) of an A(n) a nd an A(m), subfactor. We determine the Bratteli diagram describing th eir inclusions. This is done by studying a hierarchy (FCm,n)(n is an e lement 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 emplo ying a diagrammatic method. The principal part of the Bratteli diagram describing the inclusions of the algebras FC1,n, is the Fibonacci gra ph. Our algebras have a natural trace and we compute the trace weights explicitly as products of Temperley-Lieb traces. If all indices are g reater 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 elem ent of N). Our results generalize to a chain of m intermediate subfact ors.