Subfactors Constructed from Quantum Groups

Abstract

Introduction. In [Jon83], Jones constructs, from any III subfactor pair M,N with finite index [M, N] = q + q-I + 2, a canonical hyperfinite subfactor pair of the same index. The pair is constructed from the completion of the TemperleyLieb algebra by a Markov trace. By showing that such a completion yields a II, subfactor pair exactly when q is a positive real or of the form e27ri/k, he obtained his index of subfactors result. The fact that the set of admissible values of q divides into a discrete piece and a continuous piece is puzzling and is indicative of qualitative differences between the subfactors themselves. For one thing, at roots of unity the subfactors are of finite depth. That is, after a certain point in the Bratteli diagram the centers of the finite-dimensional approximating algebras are periodically isomorphic. On the contrary, in the positive real case each finite-dimensional algebra in the Bratteli diagram has a strictly larger center than the previous one. Furthermore, the relative commutant of the smaller algebra in the larger algebra consists only of multiples of the identity in the root of unity case. For positive real q, not only is there a relative commutant, but in fact the subfactor is locally trivial. That is, the subfactor pair, restricted to a minimal projection in the relative commutant, is the trivial subfactor pair. In addition to its direct relevance to the study of subfactors, Jones' work had indirect but important ramifications. The Temperley-Lieb algebra in the construction forms a representation of the braid group, and the Markov trace as a functional on the braid group is, with proper normalization, a polynomial link invariant. This discovery triggered an explosion of mathematics which eventually came to see the algebra, trace, and invariant as a special case of a general construction which associates, via quantum groups, an example of all three to each irreducible representation of each. simple Lie algebra. The original Jones polynomial comes from the fundamental representation of s12. The quantum groups construction can be done entirely without reference to subfactors, or even analysis. Nevertheless, there are echos of Jones' work, and in particular both the roots of unity and the positive real q continue to play a distinct and special role. For example, when q is a root of unity, the quantum group