Geometry of Associated Quantum Vector Bundles and the Quantum Gauge Group

Abstract: It is well--known that if one is given a principal $G$--bundle with a principal connection, then for every unitary finite--dimensional linear representation of $G$ one can induce a linear connection and a Hermitian structure on the 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 on the associated vector bundles. This paper aims to present the {\it non--commutative geometrical} counterpart of all of these {\it classical} facts in the theory of quantum bundles and quantum connections.