Representations of noncommutative quantum mechanics and symmetries

We present a unified approach to representations of quantum mechanics on noncommutative spaces with general constant commutators of phase-space variables. Generally, we find two phases and duality relations among them. Conditions for physical equivalence of different representations of a given system are analysed. Symmetries and classification of phase spaces are discussed. Specially, the dynamical symmetry of a physical system is investigated and a new mechanism for symmetry breaking, originating from (phase) space structure is proposed. Finally, we apply our analyses to the two-dimesional harmonic oscillator and the Landau problem