# Tannakian duality for affine homogeneous spaces

## TL;DR

This paper develops a Tannakian duality framework to characterize affine homogeneous spaces associated with quantum subgroups of the free unitary quantum group, providing an axiomatic understanding of their algebraic structure.

## Contribution

It introduces an axiomatic approach to classify algebraic manifolds that can be realized as affine homogeneous spaces via Tannakian duality methods.

## Key findings

- Characterization of affine homogeneous spaces using Tannakian duality.
- Axiomatization of algebraic manifolds related to quantum groups.
- Framework for understanding homogeneous spaces in noncommutative geometry.

## Abstract

Associated to any closed quantum subgroup $G\subset U_N^+$ and any index set $I\subset\{1,\ldots,N\}$ 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.