Coreductive Lie bialgebras and dual homogeneous spaces

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 𝔤=Lie(G), in this new paper (arXiv:1909.01000) we exploit the idea of Lie bialgebra duality in order to introduce the notion of dual homogeneous space of a given homogeneous space M=G/H with respect to the Lie bialgebra δ. Then, by considering the natural notions of reductive and symmetric homogeneous spaces, we extend these concepts to the dual space thus showing that an even richer duality framework arises. In order to analyse the physical implications of this new duality, the case of M being a Minkowski or (Anti-) de Sitter Poisson homogeneous spacetime is fully studied, and the corresponding dual reductive and symmetric spaces 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 the dual space is reductive 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 we show that, despite the dual spaces are not endowed in general with an invariant metric, their geometry can be described by making use of K-structures.