Coisotropic Lie bialgebras and complementary dual Poisson homogeneous spaces

Abstract

Quantum homogeneous spaces are noncommutative spaces with quantum group covariance. Their semiclassical counterparts are Poisson homogeneous spaces, which are quotient manifolds of Lie groups $M=G/H$ equipped with an additional Poisson structure $\pi$ which is compatible with a Poisson-Lie structure $\Pi$ on $G$. Since the infinitesimal version of $\Pi$ defines a unique Lie bialgebra structure $\delta$ on the Lie algebra $\frak g=\mbox{Lie}(G)$, we exploit the idea of Lie bialgebra duality in order to study the notion of complementary dual homogeneous space $M^{\perp }=G^{\ast }/H^{\perp }$ of a given homogeneous space $M$ with respect to a coisotropic Lie bialgebra. Then, by considering the natural notions of reductive and symmetric homogeneous spaces, we extend these concepts to $M^{\perp }$ thus showing that an even richer duality framework between $M$ and $M^{\perp }$ arises from them. In order to analyse physical implications of these notions, the case of $M$ being a Minkowski or (Anti-) de Sitter Poisson homogeneous spacetime is fully studied, and the corresponding complementary dual reductive and symmetric spaces $M^{\perp }$ are explicitly constructed in the case of the well-known $\kappa$-deformation, where the cosmological constant $\Lambda$ is introduced as an explicit parameter in order to describe all Lorentzian spaces simultaneously. In particular, the fact that $M^{\perp }$ is a reductive space is shown to provide a natural condition for the representation theory of the quantum analogue of $M$ that ensures the existence of physically meaningful uncertainty relations between the noncommutative spacetime coordinates. Finally, despite these dual spaces $M^{\perp }$ are not endowed in general with a $G^{\ast }$-invariant metric, we show that their geometry can be described by making use of $K$-structures.

arXiv comment27 pages

References in corpus (38)

• Quantum Field Theory on Noncommutative Spaces
• The quantum structure of spacetime at the Planck scale and quantum fields
• Quantum gravity and minimum length
• Bicrossproduct structure of $\kappa$-Poincare group and non-commutative geometry
• On a Lorentz-Invariant Interpretation of Noncommutative Space-Time and Its Implications on Noncommutative QFT
• The algebraic structure of the generalized uncertainty principle
• Symmetries of quantum spaces. Subgroups and quotient spaces of quantum $SU(2)$ and $SO(3)$ groups
• Quantum symmetry, the cosmological constant and Planck scale phenomenology
• Poisson structures on the Poincare group
• kappa-Minkowski spacetime as the result of Jordanian twist deformation
• Real Forms of Complex Quantum Anti de Sitter Algebra $U_q(Sp(4,C))$ and their Contraction Schemes
• A New Null-Plane Quantum Poincare Algebra
• Lie bialgebra contractions and quantum deformations of quasi-orthogonal algebras
• A non-commutative Minkowskian spacetime from a quantum AdS algebra
• Localization and Reference Frames in $\kappa$-Minkowski Spacetime
• The $\kappa$-(A)dS noncommutative spacetime
• The kappa-(A)dS quantum algebra in (3+1) dimensions
• Deformed Covariant Quantum Phase Spaces as Hopf Algebroids
• Observables and Dispersion Relations in k-Minkowski Spacetime
• Free q-Schrodinger Equation from Homogeneous Spaces of the 2-dim Euclidean Quantum Group
• Curved momentum spaces from quantum (Anti-)de Sitter groups in (3+1) dimensions
• Quantum deformations of D=4 Euclidean, Lorentz, Kleinian and quaternionic o^*(4) symmetries in unified o(4;C) setting
• The $\kappa$-Newtonian and $\kappa$-Carrollian algebras and their noncommutative spacetimes
• Non-commutative relativistic spacetimes and worldlines from 2+1 quantum (anti-)de Sitter groups
• On quantum deformations of (anti-)de Sitter algebras in (2+1) dimensions
• Quantum Homogeneous Spaces as Quantum Quotient Spaces
• Noncommutative spaces of worldlines
• $\kappa$-Deformations and Extended $\kappa$-Minkowski Spacetimes
• Curved momentum spaces from quantum groups with cosmological constant
• Physical Constraints on Quantum Deformations of Spacetime Symmetries
• Classical dynamical r-matrices and homogeneous Poisson structures on $G/H$ and $K/T$
• Quantum deformations of $D=4$ Euclidean, Lorentz, Kleinian and quaternionic ${\mathfrak{o}}^{\star }(4)$ symmetries in unified $\mathfrak{o}(4;\mathbb{C})$ setting -- Addendum
• AdS Poisson homogeneous spaces and Drinfel'd doubles
• Quantum planes and quantum cylinders from Poisson homogeneous spaces
• Quantum algebras as quantizations of dual Poisson-Lie groups
• Gauge fixing in (2+1)-gravity: Dirac bracket and spacetime geometry
• Extended noncommutative Minkowski spacetimes and hybrid gauge symmetries
• Gauge fixing and classical dynamical r-matrices in ISO(2,1)-Chern-Simons theory

Cited by in corpus (8)

• Interplay between spacetime curvature, speed of light and quantum deformations of relativistic symmetries
• Cayley-Klein Lie bialgebras: Noncommutative spaces, Drinfel'd doubles and kinematical applications
• All noncommutative spaces of $\kappa$-Poincaré geodesics
• The noncommutative space of light-like worldlines
• The Weyl-Mellin quantization map for $\kappa$-Minkowski Noncommutative Spacetime
• A general approach to noncommutative spaces from Poisson homogeneous spaces: Applications to (A)dS and Poincaré
• Noncommutative (A)dS and Minkowski spacetimes from quantum Lorentz subgroups
• Noncommutative Lightcones from Quantum SO(2,1) Conformal Groups