Algebraic description of a two-dimensional system of charged particles in an external magnetic field and periodic potential

Properties of the magnetic translation operators for a charged particle mov ing in a crystalline potential and a uniform magnetic field show that it is necessary to consider all inequivalent irreducible projective representati ons of the crystal lattice translation group. These considerations lead to the concept of magnetic cells and indicate the periodicity of physical prop erties with respect to the charge. It is also proven that a direct product of such representations describes a system of two (many, in general) partic les. Therefore, they can be applied in a description of interacting electro ns in a magnetic field, for example in the fractional quantum Hall effect.