Differential structures on quantum principal bundles

A fully constructive approach to differential calculus on quantum principal bundles is presented. The calculus on the bundle is built in an intrinsic way, starting from given graded (differential) ∗-algebras representing horizontal forms on the bundle and differential forms on the quantum base space, together with a family of antiderivations acting on horizontal forms, playing the role of covariant derivatives of certain special connections. These connections are used as global counterparts of local trivializations. In this conceptual framework, a natural differential calculus on the structure quantum group is described. Higher-order calculi on the structure quantum group coming from both universal envelopes and braided exterior algebras are considered.