  Hall conductivity for the intrinsic quantum Hall effect in homogeneous
systems is given by the topological invariant composed of the Green function
depending on momentum of quasiparticle. This expression reveals correspondence
with the mathematical notion of the degree of mapping. A more involved
situation takes place for the Hall effect in the presence of external magnetic
field. In this case the mentioned expression remains valid if the Green
function is replaced by its Wigner transformation while ordinary products are
replaced by the Moyal products. Such an expression, unfortunately, is much more
complicated and might be useless for the practical calculations. Here we
represent the alternative representation for the Hall conductivity of a uniform
system in the presence of constant magnetic field. The Hall conductivity is
expressed through the Green function taken in Harper representation, when its
nonhomogeneity is attributed to the matrix structure while functional
dependence is on one momentum that belongs to magnetic Brillouin zone. Our
results were obtained for the non - interacting systems. But we expect that
they remain valid for the interacting systems as well. We, therefore, propose
the hypothesis that the obtained expression may be used for the topological
description of fractional quantum Hall effect.
