Perturbative Symmetries on Noncommutative Spaces

Abstract

Perturbative deformations of symmetry structures on noncommutative spaces are studied in view of noncommutative quantum field theories. The rigidity of enveloping algebras of semi-simple Lie algebras with respect to formal deformations is reviewed in the context of star products. It is shown that rigidity of symmetry algebras extends to rigidity of the action of the symmetry on the space. This implies that the noncommutative spaces considered can be realized as star products by particular ordering prescriptions which are compatible with the symmetry. These symmetry preserving ordering prescriptions are calculated for the quantum plane and four-dimensional quantum Euclidean space. Using these ordering prescriptions greatly facilitates the construction of invariant Lagrangians for quantum field theory on noncommutative spaces with a deformed symmetry.