Tresses et structure entière sur la catégorie des représentations de SLN quantique
Résumé
According to a theorem of Masbaum and Wenzl [11], the Turaev-Viro invariant of a 3-manifold associated with the modular categories constructed from UqslN is a cyclotomic integer when q is of prime order. We extend this result to premodular categories (where the S-matrix need not be invertible) of 'type A of a more general kind. One defines premodular categories of type A, rank N and level K over C depending on two complex parameters a and u, with q=s2 a root of unity of order l=N+k and u a N-th root of a , in the case SLN or more generally any root of unity (a variant introduced by Blanchet, [2]). There are essentially two methods for constructing these premodular categories: through quantum groups, and through Hecke algebras. We carry out both constructions, and check that they yield the same result. A 'local handle-slide property' for a premodular category (1.6.4.) characterizes those for which the Turaev-Viro invariant TV is defined. This applies to premodular categories of type A. It turns out (3.3.6.) that TV is defined precisely when the premodular category is modularizable in the sense of [4] (assuming 5 is of order 2l) We show that the Turaev-Viro invariant associated with a such a premodular category, when denned, is a cyclotomic integer if l=N+K is prime; this holds also for PGL. More precisely, we show that such a cateory is 'defined' over k=Z[u,s,1/[l-1]^!_s]. Extending a criterion of [11] to the premodular case, we conclude that the Turaev-Viro invariant lies in k.