  In this paper we provide a general condition for the reducibility of the
Reshetikhin-Turaev quantum representations of the mapping class groups. Namely,
for any modular tensor category with a special symmetric Frobenius algebra with
a non-trivial genus one partition function, we prove that the quantum
representations of all the mapping class groups built from the modular tensor
category are reducible. In particular for SU(N) we get reducibility for certain
levels and ranks. For the quantum SU(2) Reshetikhin-Turaev theory we construct
a decomposition for all even levels. We conjecture this decomposition is a
complete decomposition into irreducible representations for high enough levels.
