#
Tannaka--Krein duality for compact quantum homogeneous spaces.
I. General theory

##
Kenny De Commer and Makoto Yamashita

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.

Keywords:
compact quantum groups; $C^*$-algebras; Hilbert modules;
ergodic actions; module categories

2010 MSC:
17B37; 20G42; 46L08

*Theory and Applications of Categories,*
Vol. 28, 2013,
No. 31, pp 1099-1138.

Published 2013-10-27.

http://www.tac.mta.ca/tac/volumes/28/31/28-31.dvi

http://www.tac.mta.ca/tac/volumes/28/31/28-31.ps

http://www.tac.mta.ca/tac/volumes/28/31/28-31.pdf

ftp://ftp.tac.mta.ca/pub/tac/html/volumes/28/31/28-31.dvi

ftp://ftp.tac.mta.ca/pub/tac/html/volumes/28/31/28-31.ps

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