Noncommutative Translations and $\star$-Product Formalism

We consider the noncommutative space-times with Lie-algebraic noncommutativity (e.g. $κ$-deformed Minkowski space). In the framework with classical fields we extend the $\star$-product in order to represent the noncommutative translations in terms of commutative ones. We show the translational invariance of noncommutative bilinear action with local product of noncommutative fields. The quadratic noncommutativity is also briefly discussed.

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