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

Geometry of Associated Quantum Vector Bundles and the Quantum Gauge Group