Construction of $\theta$-Poincar\'e Algebras and their Invariants on $\mathcal{M}_\theta$

Abstract

In the present paper we construct deformations of the Poincar\'e algebra as representations on a noncommutative spacetime with canonical commutation relations. These deformations are obtained by solving a set of conditions by an appropriate ansatz for the deformed Lorentz generator. They turn out to be Hopf algebras of quantum universal enveloping algebra type with nontrivial antipodes. In order to present a notion of $\theta$-deformed Minkowski space $\mathcal{M}_\theta$, we introduce Casimir operators and spacetime invariants for all deformations obtained.