Hecke algebras, modular categories and 3-manifolds quantum invariants

We construct modular categories from Hecke algebras at roots of unity. For a special choice of the framing parameter, we recover the Reshetikhin-Turae v invariants of closed 3-manifolds constructed from the quantum groups U(q) sl(N) by Reshetikhin-Turaev and Turaev-Wenzl, and from skein theory by Yoko ta. The possibility of such a construction was suggested by Turaev, as a co nsequence of Schur-Weil duality. We then discuss the choice of the framing parameter. This leads, for any rank N and level K, to a modular category (H ) over tilde(N,K) and a reduced invariant <(tau)over tilde>(N,K). If N and K are coprime, then this invariant coincides with the known invariant tau(P SU(N)) at level K. If gcd(N, K) = d > 1, then we show that the reduced inva riant admits spin or cohomological refinements, with a nice decomposition f ormula which extends a theorem of H. Murakami. (C) 1999 Elsevier Science Lt d. All rights reserved.