  Chiral Haldane phases are examples of one-dimensional topological states of
matter which are protected by projective SU($N$) group (or its subgroup
$\mathbb{Z}_N \times \mathbb{Z}_N$) with $N>2$. The unique feature of these
symmetry protected topological (SPT) phases is that they are accompanied by
inversion-symmetry breaking and the emergence of different left and right edge
states which transform, for instance, respectively in the fundamental
($\boldsymbol{N}$) and anti-fundamental ($\overline{\boldsymbol{N}}$)
representations of SU($N$). We show, by means of complementary analytical and
numerical approaches, that these chiral SPT phases as well as the non-chiral
ones are realized as the ground states of a generalized two-leg SU($N$) spin
ladder in which the spins in the first chain transform in $\boldsymbol{N}$ and
the second in $\overline{\boldsymbol{N}}$. In particular, we map out the phase
diagram for $N=3$ and $4$ to show that {\em all} the possible
symmetry-protected topological phases with projective SU($N$)-symmetry appear
in this simple ladder model.
