  We investigate the algebraic structure of flat energy bands a partial filling
of which may give rise to a fractional quantum anomalous Hall effect (or a
fractional Chern insulator) and a fractional quantum spin Hall effect. Both
effects arise in the case of a sufficiently flat energy band as well as a
roughly flat and homogeneous Berry curvature, such that the global Chern
number, which is a topological invariant, may be associated with a local
non-commutative geometry. This geometry is similar to the more familiar
situation of the fractional quantum Hall effect in two-dimensional electron
systems in a strong magnetic field.
