Coisotropic Lie bialgebras and complementary dual Poisson homogeneous spaces (1909.01000v2)

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.