Kohn condition and exotic Newton-Hooke symmetry in the non-commutative Landau problem

N “exotic” [alias non-commutative] particles with masses ma, charges ea and non-commutative parameters θa, moving in a uniform magnetic field B, separate into center-of-mass and internal motions if Kohnʼs condition ea/ma=const is supplemented with eaθa=const. Then the center-of-mass behaves as a single exotic particle carrying the total mass and charge of the system, M and e, and a suitably defined non-commutative parameter Θ. For vanishing electric field off the critical case eΘB≠1, the particles perform the usual cyclotronic motion with modified but equal frequency. The system is symmetric under suitable time-dependent translations which span a (4+2)-parameter centrally-extended subgroup of the “exotic” [i.e., two-parameter centrally-extended] Newton-Hooke group. In the critical case B=Bc=(eΘ)−1 the system is frozen into a static “crystal” configuration. Adding a constant electric field, all particles perform, collectively, a cyclotronic motion combined with a drift perpendicular to the electric field when eΘB≠1. For B=Bc the cyclotronic motion is eliminated and all particles move, collectively, following the Hall law. Our time-dependent symmetries are reduced to the (2+1)-parameter Heisenberg group of centrally-extended translations.