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Associated with any closed quantum subgroup 
$G\,\subset \,U_{N}^{+}$ and any index set 
$I\,\subset \,\{1,\,.\,.\,.\,,\,N\}$ is a certain homogeneous space 
${{X}_{G,I}}\subset S_{\mathbb{C},+}^{N-1},$ called an affine homogeneous space. Using Tannakian duality methods, we discuss the abstract axiomatization of the algebraic manifolds 
$X\subset S_{\mathbb{C},+}^{N-1}$ that can appear in this way.
