Relativistic and Newtonian kappa spacetimes

Abstract

The deformations of the Galilei algebra and their associated noncommutative Newtonian spacetimes are investigated. This is done by analyzing the possible nonrelativistic limits of an eleven generator (pseudo)extended \kap-Poincaré algebra ${\tilde{\mathcal{P}}}_{\kappa }$ and their implications for the existence of a first order differential calculus. The additional one-form needed to achieve a consistent calculus on \kap-Minkowski space is shown to be related to the additional central generator entering in the ${\tilde{\mathcal{P}}}_{\kappa }$ Hopf algebra. In the process, deformations of the extended Galilei and Galilei algebras are introduced which have, respectively, a cocycle and a bicrossproduct structure.

Slightly shortened version with small changes to adapt the paper to JMP style

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• Newton equation for canonical, Lie-algebraic and quadratic deformation of classical space
• Deformed and extended Galilei group Hopf algebras
• Canonical and Lie-algebraic twist deformations of Galilei algebra
• Electric Chern-Simons term, enlarged exotic Galilei symmetry and noncommutative plane
• Lie Bialgebra Structures for Centrally Extended Two- Dimensional Galilei Algebra and their Lie-Poisson Counterparts
• Lorentzian Snyder spacetimes and their Galilei and Carroll limits from projective geometry
• Interplay between spacetime curvature, speed of light and quantum deformations of relativistic symmetries
• Graded contractions and bicrossproduct structure of deformed inhomogeneous algebras
• Canonical and Lie-algebraic twist deformations of $\kappa$-Poincare and contractions to $\kappa$-Galilei algebras
• Cayley-Klein Lie bialgebras: Noncommutative spaces, Drinfel'd doubles and kinematical applications
• k-deformed Poincare algebras and quantum Clifford-Hopf algebras
• Generalized twist deformations of Poincare and Galilei Hopf algebras
• Canonical, Lie-algebraic and quadratic twist deformations of Galilei group
• Twisted Conformal Algebra so(4,2)
• Induced Representations of Quantum Kinematical Algebras and Quantum Mechanics
• Projective representation of k-Galilei group
• Generalized twist deformations of Poincare and Galilei quantum groups
• Noncommutative spacetimes versus noncommutative spaces of geodesics
• Induced representations of quantum kinematical algebras