Connections on locally trivial quantum principal fibre bundles
A theory of connections in a locally trivial quantum principal bundle is developed as a part of the general theory of noncommutative differential geometry. A locally trivial quantum principal bundle is a noncommutative algebraic version of a principal bundle of the classical Differential Geometry. It is defined in terms of a Hopf algebra \(H\) (which is considered as the ``structure group'' of the bundle), an algebra \(B\) (``the base space'') and a right \(H\) comodule \(P\) with coaction \(\Delta_P\) and an injective homomorphism \(B \to P\) of modules, together with a system of ideals of \(B\) and some additional conditions. The authors define the notion of covariant differential algebras and connections on a quantum principal bundle. They introduce the notion of covariant derivatives, connection forms, curvatures and curvature forms and establish some relations between these notions. They consider as an example a \(U(1)\) quantum principal bundle over a glued quantum sphere and construct a connection in this bundle. The connection may be considered as a \(q\)-version of a Dirac monopole.
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