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

TL;DR

This paper constructs deformed Poincaré algebras acting on noncommutative Minkowski space, introducing Casimir operators and invariants, and characterizes their algebraic structure as quantum groups with nontrivial antipodes.

Contribution

It presents a method to deform Poincaré algebra into Hopf algebras on noncommutative spacetime, defining invariants and Casimir operators for these deformations.

Findings

Deformations form Hopf algebras of quantum universal enveloping type.

Casimir operators and invariants are constructed for all deformations.

Deformed algebras act on a noncommutative Minkowski space $ heta$-deformation.

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.