Kappa-Minkowski spacetime, Kappa-Poincaré Hopf algebra and realizations

Abstract

We unify kappa-Minkowki spacetime and Lorentz algebra in unique Lie algebra. Introducing commutative momenta, a family of kappa-deformed Heisenberg algebras and kappa-deformed Poincare algebras are defined. They are specified by the matrix depending on momenta. We construct all such matrices. Realizations and star product are defined and analyzed in general and specially, their relation to coproduct of momenta is pointed out. Hopf algebra of the Poincare algebra, related to the covariant realization, is presented in unified covariant form. Left-right dual realizations and dual algebra are introduced and considered. The generalized involution and the star inner product are analyzed and their properties are discussed. Partial integration and deformed trace property are obtained in general. The translation invariance of the star product is pointed out. Finally, perturbative approach up to the first order in $a$ is presented in Appendix.

references added, typos corrected, acceped in J. Phys. A

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• Deformed Covariant Quantum Phase Spaces as Hopf Algebroids
• Unified description for kappa-deformations of orthogonal groups
• Quantum field theory in generalised Snyder spaces
• Relative locality in a quantum spacetime and the pregeometry of $\kappa$-Minkowski
• Twisting and kappa-Poincare
• Lie-deformed quantum Minkowski spaces from twists: Hopf-algebraic versus Hopf-algebroid approach
• Remarks on simple interpolation between Jordanian twists
• Realizations of $\kappa$-Minkowski space, Drinfeld twists and related symmetry algebras
• $\kappa$-Minkowski star product in any dimension from symplectic realization
• Snyder-type spaces, twisted Poincaré algebra and addition of momenta
• Unification of $\kappa$-Minkowski and extended Snyder spaces
• Differential forms and k-Minkowski spacetime from extended twist
• Universal $\kappa$-Poincaré covariant differential calculus over $\kappa$-Minkowski space
• Light-like $\kappa$-deformations and scalar field theory via Drinfeld twist
• Hawking radiation in the kappa-spacetime
• $\kappa$-deformed phase spaces, Jordanian twists, Lorentz-Weyl algebra and dispersion relations
• Hermitian realizations of kappa-Minkowski spacetime
• Time-space noncommutativity and Casimir effect
• Lie-Poisson gauge theories and $\kappa$-Minkowski electrodynamics
• Vacuum energy from noncommutative models
• Twisted conformal algebra related to $\kappa$-Minkowski space
• The Weyl realizations of Lie algebras and left-right duality
• Symmetries of $\kappa$ Minkowski space-time: A possibility of exotic momentum space geometry?
• Directly from $H$-flux to the family of three nonlocal $R$-flux theories
• Superdense star in a space-time with minimal length
• How does Casimir energy fall in $\kappa$-deformed space-time?
• Centripetal force on Casimir energies in $\kappa$-deformed rotating frame
• Maxwell's equations and Lorentz force in doubly special relativity
• The universe is not a Lie, but actually an Hopf, algebra