Abstract.A class function on the braid group is derived from the Kauffman link invariant. This function is used to construct representations of the braid groups depending on 2 parameters. The decomposition of the corresponding algebras into irreducible components is given and it is shown how they are related to Jones' algebras and to Brauer's centralizer algebras.In [J,3] Vaughan Jones announced the discovery of a new polynomial invariant of knots and links, which bore many similarities to the classical Alexander polynomial, but was seen to detect properties of a link which could not be detected by the Alexander invariants. The discovery was a real surprise, one of those exciting moments in mathematics when two seemingly unrelated disciplines turn out to have deep interconnections. The discovery came about in the following way. Jones' earlier contributions in the area of Operator Algebras had produced, in [J,l], a family of algebras An(t), t G C, indexed by the natural numbers «=1,2,3,..., and equipped with a trace function t : An(t) -* C. His algebra An(t) was a quotient of the well-known Hecke algebra of the symmetric group, which we denote by %?n(l,m) to delineate our particular 2-parameter version of it. Jones had discovered, in [J,2], that there were representations of Artin's braid group Bn in the algebra An(t), in fact there were maps Bn±.S?n(l,m)^An(t) from Bn into the multiplicative group of An(t) which factored through %,(!>**).Links enter the picture via braids. Each oriented link L in oriented S can be represented by a (nonunique) element ß in some braid group Bn . There is an equivalence relation on P^ = LI^I, P" , known as Markov equivalence, which determines a 1-1 correspondence between equivalence classes [ß] G B and isotopy types of the associated oriented links L". Jones' discovery was that with a small renormalization his trace function on A^t) = ]X?=XA (t) could be made into a function which lifted to an invariant on Markov classesReceived by the editors December 9, 1987. 1980 Mathematics Subject Classification (1985. Primary 57M25; Secondary 20F29, 20C07.The work of the first author was supported in part by NSF grant #DMS-8503758. The work of the second author was supported in part by NSF grant #DMS-8510816. in P^ . That modified trace, described in [J,3] and in more detail in [J,4], is the Jones polynomial VL(t). (It becomes a polynomial when the parameter t is regarded as an indeterminate.)The polynomial VL(t) was quickly generalized in a six-author paper [FYHLMO], to a 2-variable polynomial PL(l,m).One of the authors was A. Ocneanu. Ocneanu's interpretation of Pjj, m), as described in [FYHLMO] and [J,4] The purpose of this note is to reverse the process begun by Jones. We will use the existence of KL(l ,m), and apply the methods used to construct it in [K,l] to construct a new two-parameter family of finite-dimensional algebras, {Wn(l,m); n = 1,2,3, ...}, complete with trace, such that KL(l,m) is, after appropriate renormalization, that trace, just as PL(l,m), renormalized, was s...

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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...