  We investigate the properties of the Lieb lattice, i.e a face-centered square
lattice, subjected to external gauge fields. We show that an Abelian gauge
field leads to a peculiar quantum Hall effect, which is a consequence of the
single Dirac cone and the flat band characterizing the energy spectrum. Then we
explore the effects of an intrinsic spin-orbit term - a non-Abelian gauge field
- and demonstrate the occurrence of the quantum spin Hall effect in this model.
Besides, we obtain the relativistic Hamiltonian describing the Lieb lattice at
low energy and derive the Landau levels in the presence of external Abelian and
non-Abelian gauge fields. Finally, we describe concrete schemes for realizing
these gauge fields with cold fermionic atoms trapped in an optical Lieb
lattice. In particular, we provide a very efficient method to reproduce the
intrinsic (Kane-Mele) spin-orbit term with assisted-tunneling schemes.
Consequently, our model could be implemented in order to produce a variety of
topological states with cold-atoms.
