  We review and extend the results of [1] that gives a condition for
reducibility of quantum representations of mapping class groups constructed
from Reshetikhin-Turaev type topological quantum field theories based on
modular categories. This criterion is derived using methods developed to
describe rational conformal field theories, making use of Frobenius algebras
and their representations in modular categories. Given a modular category C, a
rational conformal field theory can be constructed from a Frobenius algebra A
in C. We show that if C contains a symmetric special Frobenius algebra A such
that the torus partition function Z(A) of the corresponding conformal field
theory is non-trivial, implying reducibility of the genus 1 representation of
the modular group, then the representation of the genus g mapping class group
constructed from C is reducible for every g\geq 1. We also extend the number of
examples where we can show reducibility significantly by establishing the
existence of algebras with the required properties using methods developed by
Fuchs, Runkel and Schweigert. As a result we show that the quantum
representations are reducible in the SU(N) case, N>2, for all levels k\in
\mathbb{N}. The SU(2) case was treated explicitly in [1], showing reducibility
for even levels k\geq 4.
