Curved momentum spaces from quantum groups with cosmological constant

Abstract

We bring the concept that quantum symmetries describe theories with nontrivial momentum space properties one step further, looking at quantum symmetries of spacetime in presence of a nonvanishing cosmological constant $\Lambda$. In particular, the momentum space associated to the $\kappa$-deformation of the de Sitter algebra in (1+1) and (2+1) dimensions is explicitly constructed as a dual Poisson-Lie group manifold parametrized by $\Lambda$. Such momentum space includes both the momenta associated to spacetime translations and the `hyperbolic' momenta associated to boost transformations, and has the geometry of (half of) a de Sitter manifold. Known results for the momentum space of the $\kappa$-Poincaré algebra are smoothly recovered in the limit $\Lambda \to 0$, where hyperbolic momenta decouple from translational momenta. The approach here presented is general and can be applied to other quantum deformations of kinematical symmetries, including (3+1)-dimensional ones.

arXiv comment13 pages

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Cited by in corpus (14)

• Deformed relativistic kinematics on curved spacetime -- a geometric approach
• The $\kappa$-(A)dS noncommutative spacetime
• Curved momentum spaces from quantum (Anti-)de Sitter groups in (3+1) dimensions
• Noncommutative spaces of worldlines
• Interplay between spacetime curvature, speed of light and quantum deformations of relativistic symmetries
• Physical Constraints on Quantum Deformations of Spacetime Symmetries
• Relativistic compatibility of the interacting $\kappa$-Poincaré model and implications for the relative locality framework
• Extending general covariance: Moyal-type noncommutative manifolds
• The Poincaré group as a Drinfel'd double
• Coisotropic Lie bialgebras and complementary dual Poisson homogeneous spaces
• Noncommutative momentum and torsional regularization
• $\kappa$-Galilean and $\kappa$-Carrollian noncommutative spaces of worldlines
• A Grassmann and graded approach to coboundary Lie bialgebras, their classification, and Yang-Baxter equations
• Fuzzy worldlines with $\kappa$-Poincaré symmetries