There are many variants of a generalization of the concept of homogeneous space to noncommutative geometry. Some of them are sometimes referred to as quantum homogeneous spaces. Variants of quantum flag manifolds, are here the main examples in several different frameworks.

Given a Hopf algebra $H$ and a right $H$-comodule $(E,\rho)$ (where $\rho:E\to E\otimes H$ is a right coaction), a subspace $A\subset E$ is coinvariant subspace for coaction $\rho$ if $\rho(A)\subset A\otimes H$. If $(E,\rho)$ is a $H$-comodule algebra, viewed as a (formal dual to) noncommutative $Spec H$-space, one can consider its coinvariant subalgebras. If $H = E$ and $\rho = \Delta$ then the coinvariant subalgebras represent (the formal duals of) quantum homogenous spaces of the (formal dual of) $H$.

For example, a quotient Hopf algebra$B$ of $H$, with quotient map $\pi: H\to B$, can be considered as representing a quantum subgroup. Then $B$ coacts on $H$ by map $(id\circ\pi)\circ\Delta: H\to H\otimes B$. The space of $B$-coinvariants gives an example of a coinvariant subalgebra in $H$.

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