Coisotropic Lie bialgebras and complementary dual Poisson homogeneous spaces

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