  We propose 1D and 2D lattice wave functions constructed from the SU(n)_1
Wess-Zumino-Witten (WZW) model and derive their parent Hamiltonians. When all
spins in the lattice transform under SU(n) fundamental representations, we
obtain a two-body Hamiltonian in 1D, including the SU(n) Haldane-Shastry model
as a special case. In 2D, we show that the wave function converges to a class
of Halperin's multilayer fractional quantum Hall states and belongs to chiral
spin liquids. Our result reveals a hidden SU(n) symmetry for this class of
Halperin states. When the spins sit on bipartite lattices with alternating
fundamental and conjugate representations, we provide numerical evidence that
the state in 1D exhibits quantum criticality deviating from the expected
behaviors of the SU(n)_1 WZW model, while in 2D they are chiral spin liquids
being consistent with the prediction of the SU(n)_1 WZW model.
