Topology and Quantization

Abstract

A simple algebraic model for charged particle moving in two dimensional space under influence of singular magnetic field is given. The fundamental assumption for the model is that every charged particle coupled to the magnetic field is transformed into a system of quasiparticles. It is also assumed for simplicity that there is in average N fluxes per particle. If the number N of fluxes is even, then the system can be identified as composite fermions. If N is odd, then we obtain composite bosons. Quantization is described as gradation by certain abelian group G. Statistics is determined by a commutation factor on the grading group G. It is schown that for N = 2 there is a state for which the magnetic field is completely compensatad by the system of quasiparticles and for N = 3 the Landau levells are fractionally filled like in the fractional quantum Hall effect.