  For various two dimensional lattices such as honeycomb, kagome, and
square-octagon, gauge conventions (string gauge) realizing minimum magnetic
fluxes that are consistent with the lattice periodicity are explicitly given.
Then many-body interactions of lattice fermions are projected into the
Hofstadter bands to form pseudopotentials. By using these pseudopotentials,
degenerate many-body ground states are numerically obtained. We further
formulate a scheme to calculate the Chern number of the ground state multiplet
by the pseudopotentials. For the filling factor of the lowest Landau level,
$\nu=1/3$, a simple scaling form of the energy gap are numerially obtained and
the ground state is unique except the three-fold topological degeneracy. This
is a quantum liquid, which can be lattice analogue of the Laughlin state. For
the $\nu=1/2$ case, validity of the composite fermion picture is discussed in
relation to the existence of the Fermi surface. Effects of disorder are also
described.
