Galilean symmetry in noncommutative field theory

When the interaction potential is suitably reordered, the Moyal field theory admits two types of Galilean symmetries, namely the conventional mass-parameter-centrally-extended one with commuting boosts, but also the two-fold centrally extended “exotic” Galilean symmetry, where the commutator of the boosts yields the noncommutative parameter. In the free case, one gets an “exotic” two-parameter central extension of the Schrödinger group. The conformal symmetry is, however, broken by the interaction.