Hecke algebras, modular categories and 3-manifolds quantum invariants

Abstract

We construct modular categories from Hecke algebras at roots of unity. For a special choice of the framing parameter, we recover the Reshetikhin–Turaev 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 Yokota. The possibility of such a construction was suggested by Turaev, as a consequence 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 ̃ N,K and a reduced invariant τ ̃ N,K . If N and K are coprime, then this invariant coincides with the known invariant τ PSU( N) at level K. If gcd(N, K)=d>1 , then we show that the reduced invariant admits spin or cohomological refinements, with a nice decomposition formula which extends a theorem of H. Murakami.