On the Group Theory of 2-Dimensional Noncommutative Quantum Mechanics

We examine here,in some detail,the symmetry groups of a non-relativistic quantum system of 2 spatial degrees of freedom. it is argued that the (2+1) centrally extended galilei group gext gal can be considered as one of the underlying symmetry groups from the algebraic point of view. later,the phase space variables of a particle moving on a 2 dimensional plane are quantized using the coherent states arising from the unitary dual of gext gal the resulting quantized variables are found to satisfy the set of commutation relations which are postulated in the existing literatures of noncommutative quantum mechanics. we also consider a further extension resulting in the noncommutativity between the underlying momentum coordinates and explore how this is connected to the triple central extension of the abelian group of translations in r4 and its unitary irreducible representations.