Connections on locally trivial quantum principal fibre bundles

Abstract

Budzyński and Kondracki [Rep. Math. Phys. 37 (1996) 365] have introduced a notion of locally trivial quantum principal fibre bundle making use of an algebraic notion of covering, which allows a reconstruction of the bundle from local pieces. Following this approach, we construct covariant differential algebras and connections on locally trivial quantum principal fibre bundles by gluing together such locally given geometric objects. We also consider covariant derivatives, connection forms, curvatures and curvature forms and explore the relations between these notions. As an example, a U(1) quantum principal bundle over a glued quantum sphere as well as a connection in this bundle is constructed. The connection may be considered as a q-version of a Dirac monopole.