Locally trivial quantum vector bundles and associated vector bundles

We define locally trivial quantum vector bundles (QVB) and construct such QVB associated to locally trivial quantum principal fibre bundles. The construction is quite analogous to the classical construction of associated bundles. A covering of such bundles is induced from the covering of the subalgebra of coinvariant elements of the principal bundle. There exists a differential structure on the associated vector bundle coming from the differential structure on the principal bundle, which allows to define connections on the associated vector bundle associated to connections on the principal bundle. This is the third in a series of papers devoted to locally trivial quantum bundles, following [3] and [4]. The main aim of the present paper is to define associated bundles and connections in the scheme of locally trivial quantum principal bundles of [2] and [4]. Again we follow the idea of gluing all objects from locally given objects. We note that associated bundles have been defined in [6], [7] and [8] as associated bimodules of colinear maps (intertwiners) and in [1], [2], [9] as cotensor products. Both definitions are equivalent (by some duality argument). We use the second definition. We start with some general considerations about covering und gluing of modules. Then we define QVB over algebras with a complete covering. They have as typical fibre a usual vector space, and possess local trivializations with suitable properties. Any LC differential algebra on the basis gives rise to a certain “differential structure” on the QVB, which is an analogue of a module of differential form valued sections in the classical case. Given such a structure, we define the notions of connection (as covariant derivative) and curvature on a QVB. Given a locally trivial quantum principal bundle and a left coaction of its structure group (Hopf algebra) on some vector space, we define an associated QVB as a cotensor product. This is indeed a locally trivial QVB in the sense of our definition, whose transition functions come from the transition functions of the principal bundle in the usual way. As is known from [4], there is a maximal embeddable LC differential algebra related to the differential structure of a locally trivial quantum principal bundle. The differential structure on the QVB defined by this LC differential algebra is isomorphic to the module of horizontal forms “of type ρ” on the principal bundle, where ρ is the left coaction defining the associated bundle. Finally, we show that to every connection on the principal bundle there can be associated a supported by Deutsche Forschungsgemeinschaft, e-mail [email protected] supported by Sächsisches Staatsministerium für Wissenschaft und Kunst, e-mail [email protected] or [email protected]