  We study the physics of a rapidly-rotating gas of ultracold atomic bosons,
with an internal degree of freedom. We show that in the limit of rapid rotation
of the trap the problem exactly maps onto that of non-interacting fermions with
spin in the lowest Landau level. The spectrum of the real bosonic system is
identical to the one of the effective fermions, with the same eigenvalues and
the same density of states. When the ratio of the number of atoms to the spin
degeneracy is an integer number, the ground state for the effective fermions is
an integer quantum Hall state. The corresponding bosonic state is a fractional
quantum Hall liquid whose filling factor ranges in the sequence nu=1/2, 2/3,
3/4, ..., as the spin degeneracy increases. Anyons with 1/2,1/3,1/4, ...
statistics can be created by inserting lasers with the appropriate
polarizations. A special situation appears when the spin degeneracy equals the
number of atoms in the gas. The ground state is then the product of a
completely antisymmetric spin state and a nu=1 Laughlin state. In this case the
system exhibits fermionic excitations with fermionic statistics though the real
components are bosonic atoms.
