Coisotropic Lie bialgebras and complementary dual Poisson homogeneous spaces

Understand

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.