Quantum differentials and the \(q\)-monopole revisited (Q1275638)

With the rapid growth of the theory of quantum groups which play a role as quantum symmetries, it becomes natural to develop a notion of quantum group gauge theory, i.e. a quantum formulation of differential calculus on principal fibre bundles. In defining a quantum principal bundle, it is natural to replace the base space of a principal bundle by a quantum algebra and replace the group fibres by quantum groups. In noncommutative differential geometry, the available universal differential calculus for a quantum algebra is usually too large to be practically useful, and so a suitably chosen smaller differential calculus becomes important. In this paper, the authors propose a way to construct a non-universal differential calculus on the quantum total space from a `horizontal form' related to the base space and a bicovariant calculus on the quantum group fibre. Then they apply this construction to some examples, including \(q\)-monopole, finite gauge theory, and certain cross product Hopf algebras.