  We treat elementary excitations, the spin-liquid state, and the anomalous
Hall effect (including the quantum one in purely 2D situation) in layered
highly correlated systems. The mechanisms of the formation of a topological
state associated with bare flat energy bands, correlations, and spin-orbit
interactions, including the appearance of correlated Chern bands, are analyzed.
A two-band picture of the spectrum in metallic kagome lattices is proposed,
which involves a transition from the ferromagnetic state, a flat strongly
correlated band, and a band of light Dirac electrons. In this case, the effect
of separation of the spin and charge degrees of freedom turns out to be
significant. The application of the representations of the Kotliar-Rukenstein
auxiliary bosons and the Ribeiro-Wen dopons to this problem is discussed.
