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

N "exotic" [alias non-commutative] particles with masses m(a), charges e(a) and non-commutative parameters theta(a) moving in a uniform magnetic field B, separate into center-of-mass and internal motions if Kohn's condition e(a)/m(a) = const is supplemented with e(a)theta(a) = 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 Theta. For vanishing electric field off the critical case e Theta B not equal 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 = B-c = (e Theta)(-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 e Theta B not equal 1. For B = B-c 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. (C) 2011 Elsevier B.V. All rights reserved.