  Bose gases in rotating optical lattices combine two important topics in
quantum physics: superfluid rotation and strong correlations. In this paper, we
examine square two-dimensional systems at zero temperature comprised of
strongly repulsive bosons with filling factors of less than one atom per
lattice site. The entry of vortices into the system is characterized by jumps
of 2 pi in the phase winding of the condensate wavefunction. A lattice of size
L X L can have at most L-1 quantized vortices in the lowest Bloch band. In
contrast to homogeneous systems, angular momentum is not a good quantum number
since the continuous rotational symmetry is broken by the lattice. Instead, a
quasi-angular momentum captures the discrete rotational symmetry of the system.
Energy level crossings indicative of quantum phase transitions are observed
when the quasi-angular momentum of the ground-state changes.
