Geodesic
Tannaka--Krein duality for compact quantum homogeneous spaces. I. General theory
Theory and applications of categories, Tome 28 (2013), pp. 1099-1138.

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An ergodic action of a compact quantum group $G$ on an operator algebra $A$ can be interpreted as a quantum homogeneous space for $G$. Such an action gives rise to the category of finite equivariant Hilbert modules over $A$, which has a module structure over the tensor category $Rep(G)$ of finite-dimensional representations of $G$. We show that there is a one-to-one correspondence between the quantum $G$-homogeneous spaces up to equivariant Morita equivalence, and indecomposable module $C^*$-categories over $Rep(G)$ up to natural equivalence. This gives a global approach to the duality theory for ergodic actions as developed by C. Pinzari and J. Roberts.
Publié le :
Classification : 17B37, 20G42, 46L08
Keywords: compact quantum groups, $C^*$-algebras, Hilbert modules, ergodic actions, module categories 
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     author = {Kenny De Commer and Makoto Yamashita},
     title = {Tannaka--Krein duality for compact quantum homogeneous {spaces.
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Kenny De Commer; Makoto Yamashita. Tannaka--Krein duality for compact quantum homogeneous spaces.
I. General theory. Theory and applications of categories, Tome 28 (2013), pp. 1099-1138. http://geodesic.mathdoc.fr/item/TAC_2013_28_a30/