  In Differential Geometry, it is well--known that given a principal
$G$--bundle with a principal connection, for every unitary finite--dimensional
linear representation of $G$, one can induce a linear connection and a
hermitian structure on associated vector bundles, which are compatible.
Furthermore, the gauge group acts on the space of principal connections and on
the space of linear connections defined in associated vector bundles. This
paper aims to present the non--commutative counterpart of all these classical
facts using the theory of quantum bundles and quantum connections.
