  We associate Popa systems (= standard invariants of subfactors) to the finite
dimensional representations of compact quantum groups. We characterise the
systems arising in this way: these are the ones which can be ``represented'' on
finite dimensional Hilbert spaces. This is proved by an universal construction.
We explicitely compute (in terms of some free products) the operation of going
from representations of compact quantum groups to Popa systems and then back
via the universal construction. We prove a Kesten type result for the
co-amenability of compact quantum groups, which allows us to compare it with
the amenability of subfactors.

Quantum Algebra;Operator Algebras