Universal $κ$-Poincaré covariant differential calculus over $κ$-Minkowski space

Unified graded differential algebra, generated by $κ$-Minkowski noncommutative (NC) coordinates, Lorentz generators and anticommuting one-forms, is constructed. It is compatible with $κ$-Poincaré-Hopf algebra. For time- and space-like deformations, the super-Jacobi identities are not satisfied. By introducing additional generator, interpreted as exterior derivative, we find a new unique algebra that satisfies all super-Jacobi identities. It is universal and valid for all type of deformations (time-, space-, and light-like). For time-like deformations this algebra coincides with the one in \cite{sitarz}. Different realizations of our algebra in terms of super-Heisenberg algebra are presented. For light-like deformations we get 4D bicovariant calculus, with $κ$-Poincaré-Hopf algebra and present the corresponding twist, which is written in a new covariant way, using Poincaré generators only. In the time- and space-like case this twist leads to $κ$-Snyder space. Our results might lead to applications in NC quantum field theories (especially electrodynamics and gauge theories), quantum gravity models, and Planck scale physics.

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