Hecke algebras of type A_n and subfactors

Abstract

In his paper [J-l] V. Jones introduced an index, which 'measures' the size of a subfactor in a II1 factor. The main result of that paper is that the index of a subfactor has to be either greater or equal than 4 or it has to be equal to 4cosZ(x//) for some l~N, I>3 and that there exist subfactors for all these index values. Similarly as for subgroups, the index alone does not characterize the subfactor up to conjugacy by automorphisms. The fact that there are only countably many possible index values < 4 seems to be related to another invariant. Subfactors with index less than 4 always have trivial centralizers, (or relative commutants), i.e. the only elements of the factor which commute with every element of the subfactor are multiples of the identity. On the other hand, the examples given in I-J-l] for subfactors with index greater than 4 all have nontrivial centralizers. Furthermore, all known examples of subfactors with trivial relative commutants have as index an algebraic integer. At the current state of knowledge, it is still unknown whether there are only countably many values possible for the index of subfactors with trivial centralizers. Note however, that the set of all possible index values of a subfactor with trivial centralizer in an arbitrary II~ factor has to be a closed subset of R (see [HW] In more detail, our results are as follows. In the first section, we determine the structure of an extension which is a generalization of Jones' basic construetion. We obtain from this general statements about the relative commutants of the subfactor in extensions of the factor coming from iterations of Jones' construction. To construct examples of subfactors, we consider ascending sequences of finite dimensional C* algebras (A,) and (B,) with A, cB,. We assume that there is a factor trace on the inductive limit B~o of these algebras such that also the weak limit of the union of the A,'s gives a subfactor A of the resulting factor B. To be able to compute the index of A, we need the following crucial condition:Let CA., eB, and ~A,+~ be the conditional expectations onto A,, B. and A,+~ respectively. Then we require that This implies in the limit that ~AtBn=~An (see also [PP1]). The main difficulty of this approach consists of finding examples which satisfy the condition.We will furthermore assume that the inclusion matrices for A. c B., A. c A. + and for B.~B.+I become periodic for all n greater some no. In this case, we show that the index [B:A] of the resulting subfactor is the square of the norm of the inclusion matrix for A. ~ B. for any n > no. Furthermore, we prove that the dimension of the relative commutant A' n B is less or equal than the dimension of p A'. n p B. p for all n> no and all nonzero projections peA.. This upper bound for the size of the relative commutant has proven to be sharp in all known (periodic) cases.In the second section, we construct representations of Hecke algebras of type A. If q is not a root of unity, they already appear in Hoefsmit's thesis ([-HI, unpublished), and were lat...

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