tensor categories from quantum groups

In a recent paper [Ki], Kirillov, Jr., defined a *-operation on the morphisms of the representation category of a quantum group Uqg, with lql = 1, using constructions going back to Andersen, Kashiwara and Lusztig. He then conjectured that for any two morphisms f,g: V -* W, the form (f, g) = Trq(g*f) is positive semidefinite for certain roots of unity q, where Trq is the categorical q-trace. As a consequence, the morphisms in a certain quotient category, usually referred to as fusion category, become Hilbert spaces, and one obtains C* categories. We prove this conjecture in this paper for all Lie types. In particular, one obtains for any object in the fusion category a subfactor whose index is the square of its categorical dimension. If the object is simple, the corresponding subfactor is irreducible. We consider the tensor category T of tilting modules of a quantum group Uqg, where g is a simple Lie algebra and q = eri/Id , with d c {1, 2, 3} being the ratio of the square lengths of a long and a short root and 1 being larger than the dual Coxeter number of g. Here, and in the following, we use the notations of Lusztig's book [Lu]. It is shown in [A], [AP] that T contains a semisimple quotient category F whose simple objects are labelled by the dominant weights A which satisfy 0 (A + p) 0 for any morphism f in F. Kirillov defined a *-operation which is functorial; the remaining problem is to prove positivity. The approach in this paper can be sketched in the following way: Assume that we have bases of W1 and W2 with respect to which the generators of the quantum group act via matrices whose coefficients are polynomials in q. Moreover, assume that we can find an idempotent p whose coefficients depend continuously on q such that p(q) commutes with Uqg for q in a neighborhood of 1 containing qo = eri/dl and such that p(qo) projects onto CW1?,W2, with kernel C' oW *. Moreover, assume we have defined a Hermitian form on W = WI X4 W2 also depending continuously on q which is nondegenerate on p(q)W for all q near 1, and such that it is positive definite on p(1)W. Then it follows by continuity that it has to be positive definite