Deformed Covariant Quantum Phase Spaces as Hopf Algebroids

Abstract

We consider the general D=4 (10+10)-dimensional kappa-deformed quantum phase space as given by Heisenberg double \mathcal{H} of D=4 kappa-deformed Poincare-Hopf algebra H. The standard (4+4) -dimensional kappa - deformed covariant quantum phase space spanned by kappa - deformed Minkowski coordinates and commuting momenta generators ({x}_{μ},{p}_{μ}) is obtained as the subalgebra of \mathcal{H}. We study further the property that Heisenberg double defines particular quantum spaces with Hopf algebroid structure. We calculate by using purely algebraic methods the explicite Hopf algebroid structure of standard kappa - deformed quantum covariant phase space in Majid-Ruegg bicrossproduct basis. The coproducts for Hopf algebroids are not unique, determined modulo the coproduct gauge freedom. Finally we consider the interpretation of the algebraic description of quantum phase spaces as Hopf algebroids.

14 pages,v4; this version appeared in Physics Letters B

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• Remarks on simple interpolation between Jordanian twists
• Noncommutative spaces of worldlines
• Interplay between spacetime curvature, speed of light and quantum deformations of relativistic symmetries
• $\kappa$-Poincaré-comodules, Braided Tensor Products and Noncommutative Quantum Field Theory
• Snyder-type spaces, twisted Poincaré algebra and addition of momenta
• $\kappa$-deformed phase spaces, Jordanian twists, Lorentz-Weyl algebra and dispersion relations
• Generalized Heisenberg algebra applied to realizations of the orthogonal, Lorentz and Poincare algebras and their dual extensions
• Twisted bialgebroids versus bialgebroids from a Drinfeld twist
• Cayley-Klein Lie bialgebras: Noncommutative spaces, Drinfel'd doubles and kinematical applications
• Quantum perturbative solutions of extended Snyder and Yang models with spontaneous symmetry breaking
• Quantum Twist-Deformed D=4 Phase Spaces with Spin Sector and Hopf Algebroid Structures
• A short introduction to $\kappa$-deformation
• Coisotropic Lie bialgebras and complementary dual Poisson homogeneous spaces
• Generalized quantum phase spaces for the $\kappa$-deformed extended Snyder model
• Hopf Algebroid Twists for Deformation Quantization of Linear Poisson Structures
• Multiparticle states in braided lightlike $\kappa$-Minkowski noncommutative QFT
• Deformed phase spaces with group valued momenta
• On Hopf algebroid structure of kappa-deformed Heisenberg algebra
• Two ${\theta }_{\mu \nu }$ -deformed covariant relativistic quantum phase spaces as Poincare-Hopf algebroids
• Hopf algebroids with balancing subalgebra