Deformed Schrödinger symmetry on noncommutative space

We construct the deformed generators of Schrödinger symmetry consistent with noncommutative space. The examples of the free particle and the harmonic oscillator, both of which admit Schrödinger symmetry, are discussed in detail. We construct a generalised Galilean algebra where the second central extension exists in all dimensions. This algebra also follows from the Inonu–Wigner contraction of a generalised Poincaré algebra in noncommuting space. The introduction of noncommuting relativistic coordinate spacetime, [x̂, x̂ ] = iθ , μ, ν = 0, i, (1) for constant θ implies, among other things, a breakdown of Lorentz invariance. From an algebraic point of view, the Jacobi identity involving the angular momentum operator and the noncommuting coordinates is violated. Recently it has been found by Wess [1] and collaborators [2, 3, 4] that, by appropriately deforming the classical Poincaré generators, consistency with (1) is achieved while preserving the original Poincaré algebra. In other words, a new representation of the Poincaré algebra that is compatible with (1) has been obtained. But the coproduct rules are modified. Also, the modified coproduct rules agree with those found [5, 6] from another (quantumgroup theoretic) approach based of the application of twist functions [7]. The extension of these ideas to field theory and possible implications for Noether symmetry are discussed in [8, 9]. Very recently, the deformed Poincaré generators for a Lie algebraic θ (rather than a constant θ) have also been analysed [10]. In this paper we consider the invariance of the Schrödinger group [11] (this contains, in addition to the centrally extended Galilei group, two conformal generators, namely dilatations and special conformal transformations or expansions) on nonrelativistic noncommutative space, [ x̂, x̂ ] = iθ . (2) There are some good reasons for pursuing such an investigation. The question of this invariance is interesting in its own right. Also, the Schrödinger group is an entirely consistent