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Geometry of Quantum Principal Bundles II-Extended Version
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A general noncommutative-geometric theory of principal bundles is presented. Quantum groups play the role of structure groups. General quantum spaces play the role of base manifolds. A differential calculus on quantum principal bundles is studied. In particular, algebras of horizontal and verticalized differential forms on the bundle are introduced and investigated. The formalism of connections is developed. Operators of horizontal projection, covariant derivative and curvature are constructed and analyzed. A quantum generalization of classical Weil's theory of characteristic classes is sketched. Quantum analogs of infinitesimal gauge transformations are studied. Illustrative examples and constructions are presented.
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Differential calculus on Hopf--Galois extension via the Durdevi\'c braiding
Constructs σ-generated differential calculi on principal comodule algebras via Durdević braiding, proves existence for arbitrary cases, shows natural descent of connections under compatibility conditions, and develops...
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