  Motivated by the search for chiral spin liquids (CSL), we consider a simple
model defined on the kagome lattice of interacting SU(3) spins (in the
fundamental representation) including two-site and three-site permutations
between nearest neighbor sites and on triangles, respectively. By combining
analytical developments and various numerical techniques, namely exact Lanczos
diagonalizations and tensor network variational approaches, we find a rich
phase diagram with non-topological (``trivial") and topological (possibly
chiral) gapped spin liquids (SLs). Trivial spin liquids include an
Affleck-Kennedy-Lieb-Tasaki (AKLT)-like phase and a trimerized phase, the
latter breaking the inversion center between the up and down triangles of the
kagome lattice. A topological SL is stabilized in a restricted part of the
phase diagram by the time-reversal symmetry breaking (complex) 3-site
permutation term. Analyzing the chiral edge modes of this topological SL on
long cylinders or on finite disks, we have come up with two competing
scenarios, either a CSL or a double Chern-Simon SL characterized by a single or
by two counter-propagating Wess-Zumino-Witten SU(3)$_1$ chiral mode(s),
respectively. In the vicinity of the extended ferromagnetic region we have
found a magnetic phase corresponding either to a modulated canted ferromagnet
or to a uniform partially magnetized ferromagnet.
