Kappa Snyder deformations of Minkowski spacetime, realizations and Hopf algebra

Abstract

We present Lie-algebraic deformations of Minkowski space with undeformed Poincaré algebra. These deformations interpolate between Snyder and $\kappa$-Minkowski space. We find realizations of noncommutative coordinates in terms of commutative coordinates and derivatives. By introducing modules, it is shown that although deformed and undeformed structures are not isomorphic at the level of vector spaces, they are however isomorphic at the level of Hopf algebraic action on corresponding modules. Invariants and tensors with respect to Lorentz algebra are discussed. A general mapping from $\kappa$-deformed Snyder to Snyder space is constructed. Deformed Leibniz rule, the Hopf structure and star product are found. Special cases, particularly Snyder and $\kappa$-Minkowski in Maggiore-type realizations are discussed. The same generalized Hopf algebraic structures are as well considered in the case of an arbitrary allowable kind of realisation and results are given perturbatively up to second order in deformation parameters.

38 pages, LaTeX2e class file

References in corpus (27)

• Prospects for constraining quantum gravity dispersion with near term observations
• Kappa-Minkowski space-time and the star product realizations
• New realizations of Lie algebra kappa-deformed Euclidean space
• Twisted Statistics in kappa-Minkowski Spacetime
• Covariant realizations of kappa-deformed space
• Deformed Oscillator Algebras and QFT in $\kappa$-Minkowski Spacetime
• kappa-Minkowski spacetime as the result of Jordanian twist deformation
• Towards Quantum Noncommutative $\kappa$-deformed Field Theory
• From noncommutative kappa-Minkowski to Minkowski space-time
• Bounds on an energy-dependent and observer-independent speed of light from violations of locality
• A Lagrangian for DSR Particle and the Role of Noncommutativity
• Covariant particle statistics and intertwiners of the kappa-deformed Poincare algebra
• kappa-deformed Spacetime From Twist
• Yang's Model as Triply Special Relativity and the Snyder's Model--de Sitter Special Relativity Duality
• $\kappa$-Deformed Statistics and Classical Fourmomentum Addition Law
• On kappa-deformation and triangular quasibialgebra structure
• Covariant particle exchange for kappa-deformed theories in 1+1 dimensions
• Differential structure on the $\kappa$-Minkowski spacetime from twist
• Differential Structure on kappa-Minkowski Spacetime Realized as Module of Twisted Weyl Algebra
• Kappa-deformed oscillators, the choice of star product and free kappa-deformed quantum fields
• On limitations of the extent of inertial frames in non-commutative relativistic spacetimes
• Jordanian Twist Quantization of D=4 Lorentz and Poincare Algebras and D=3 Contraction Limit
• Reply to arXiv:1006.2126 by Giovanni Amelino-Camelia et al
• Comments on Nonlocality in Deformed Special Relativity, in reply to arXiv:1004.0664 by Lee Smolin and arXiv:1004.0575 by Jacob et al
• Snyder Space-Time: K-Loop and Lie Triple System
• $\kappa$-Minkowski and Snyder algebra from reparametrisation symmetry
• Comment on arXiv:1007.0718 by Lee Smolin

Cited by in corpus (17)

• 30 years in: Quo vadis generalized uncertainty principle?
• Heisenberg doubles for Snyder type models
• Generalizations of Snyder model to curved spaces
• Unification of $\kappa$-Minkowski and extended Snyder spaces
• Noncommutative Yang model and its generalizations
• Associative realizations of $\kappa$-deformed extended Snyder model
• Quantum perturbative solutions of extended Snyder and Yang models with spontaneous symmetry breaking
• Deformed Quantum Phase Spaces, Realizations, Star Products and Twists
• Generalized quantum phase spaces for the $\kappa$-deformed extended Snyder model
• From Snyder space-times to doubly $\kappa$-dependent Yang quantum phase spaces and their generalizations
• Symmetric ordering and Weyl realizations for quantum Minkowski spaces
• Exponential Formulas, Normal Ordering and the Weyl-Heisenberg Algebra
• Poisson electrodynamics on $\kappa$-Minkowski space-time
• Families of vector-like deformed relativistic quantum phase spaces, twists and symmetries
• Towards new relativistic doubly $\kappa$-deformed D=4 quantum phase spaces
• Generalized Heisenberg algebra, realizations of the $\mathfrak{gl}(n)$ algebra and applications
• Realizations of the Extended Snyder Model