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 A. Sitarz, Phys. Lett. B349, 42 (1995), arXiv:hep-th/9409014. 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.
DIFFERENTIAL CALCULUS ON A THREE-PARAMETER OSCILLATOR ALGEBRA
