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It is demonstrated that all observed fractions at moderate Landau level fillings for the quantum Hall effect can be obtained without recourse to the phenomenological concept of composite fermions. The necessary additional flux is supplied by the vortex lattice, which allows us to consider all fractions in a unified frame. The group classification predicts the electron density of the ground state and the existence of a gap that separates it from excited states. This gap was calculated for some lattices in a simplified model.

We solve two-dimensional model of $N$-component dense electron gas in the limit of large $N$ and in a range of the Coulomb interaction parameter: $N^{-3/2}\ll r_s\ll 1$. The quasiparticle interaction on the Fermi circle vanishes as 1/N. The ground state energy and the effective mass are found as series in powers of $r_s^{2/3}$. In the quantum Hall state on the lowest Landau level at integer filling: $1\ll\nu

The Jahn-Teller effect in a bivalley Si͑100͒ heterostructure under conditions of the quantum Hall effect at integer filling factors ϭ1, 2, and 3 is studied. This system is described by an SU͑4͒ hidden symmetry. At ϭ2 static and dynamic lattice deformations give rise to an easy-plane anisotropy and an antiferromagnetic exchange, and lift the valley degeneracy. At ϭ1, and 3 Coulomb interaction is essential to produce a weak easy-plane anisotropy. At ϭ2 three phases, ferromagnetic, canted antiferromagnetic, and spin-singlet, have been found. The anisotropy energy of charged skyrmion excitation in every phase is calculated.

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