Realizations of $\kappa$-Minkowski space, Drinfeld twists and related symmetry algebras

Abstract

Realizations of $\kappa$-Minkowski space linear in momenta are studied for time-, space- and light-like deformations. We construct and classify all such linear realizations and express them in terms of $\mathfrak{gl}(n)$ generators. There are three one-parameter families of linear realizations for time-like and space-like deformations, while for light-like deformations, there are only four linear realizations. The relation between deformed Heisenberg algebra, star product, coproduct of momenta and twist operator is presented. It is proved that for each linear realization there exists Drinfeld twist satisfying normalization and cocycle conditions. $\kappa$-deformed $\mathfrak{igl}(n)$-Hopf algebras are presented for all cases. The $\kappa$-Poincaré-Weyl and $\kappa$-Poincaré-Hopf algebras are discussed. Left-right dual $\kappa$-Minkowski algebra is constructed from the transposed twists. The corresponding realizations are nonlinear. All known Drinfeld twists related to $\kappa$-Minkowski space are obtained from our construction. Finally, some physical applications are discussed.

35 pages, improved version accepted for publication in EPJC

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• Quantum field theory in generalised Snyder spaces
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• $\kappa$-Poincaré-comodules, Braided Tensor Products and Noncommutative Quantum Field Theory
• Snyder-type spaces, twisted Poincaré algebra and addition of momenta
• Vector-like deformations of relativistic quantum phase-space and relativistic kinematics
• Minimal and maximal lengths from position-dependent noncommutativity
• $\kappa$-deformed phase spaces, Jordanian twists, Lorentz-Weyl algebra and dispersion relations
• Twisted statistics and the structure of Lie-deformed Minkowski spaces
• Noncommutative correction to the entropy of charged BTZ black hole
• Interpolations between Jordanian Twists Induced by Coboundary Twists
• Deformed Quantum Phase Spaces, Realizations, Star Products and Twists
• Noncommutativity and the Weak Cosmic Censorship
• Twist for Snyder space
• One Parameter Family of Jordanian Twists
• On Light-like Deformations of the Poincaré Algebra
• Maxwell's equations and Lorentz force in doubly special relativity
• Twist deformations of Newtonian Schwarzschild-(Anti-)de Sitter classical system