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 $π$ which is compatible with a Poisson-Lie structure $Π$ on $G$. Since the infinitesimal version of $Π$ defines a unique Lie bialgebra structure $δ$ 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 $κ$-deformation, where the cosmological constant $Λ$ 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.