  Hall conductivity for the intrinsic anomalous 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 quantum Hall effect in the presence of
external magnetic field. In this case the mentioned expression remains valid if
the Green function is taken in a specific representation, where it becomes the
infinite - dimensional matrix \cite{Imai:1990zz} or if it is replaced by its
Wigner transformation while ordinary products are replaced by the Moyal
products \cite{ZW2019}. Both these expressions, unfortunately, are 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
consideration for the interacting systems is non - perturbative and is based on
the Schwinger - Dyson equations truncated in a reasonable way. We demonstrate
that in this approximation the expression for the Hall conductivity in Harper
representation remains valid, where the interacting Green function is to be
used instead of the non - interacting one. We, therefore, propose that the
obtained expression may be used for the topological description of fractional
quantum Hall effect.
