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# Tannakian duality for affine homogeneous spaces

Associated to any closed quantum subgroup $G\subset U_N^+$ and any index set $I\subset\{1,\dots,N\}$ is a certain homogeneous space $X_{G,I}\subset S^{N-1}_{\mathbb C,+}$, called affine homogeneous space. We discuss here the abstract axiomatization of the algebraic manifolds $X\subset S^{N-1}_{\mathbb C,+}$ which can appear in this way, by using Tannakian duality methods.
 MSC Classifications: 46L65 - Quantizations, deformations 46L89 - Other noncommutative'' mathematics based on $C^*$-algebra theory [See also 58B32, 58B34, 58J22]