We construct noncommutative principal bundles deforming principal bundles with a Drinfeld twist (2-cocycle). If the twist is associated with the structure group then we have a deformation of the fibers. If the twist is associated with the automorphism group of the principal bundle, then we obtain noncommutative deformations of the base space as well. Combining the two twist deformations we obtain noncommutative principal bundles with both noncommutative fibers and base space. More in general, the natural isomorphisms proving the equivalence of a closed monoidal category of modules and its twist related one are used to obtain new Hopf-Galois extensions as twists of Hopf-Galois extensions. A sheaf approach is also considered, and examples presented.

We construct a quantum version of the SU (2) Hopf bundle S 7 → S 4 . The quantum sphere S 7 q arises from the symplectic group Sp q (2) and a quantum 4sphere S 4 q is obtained via a suitable self-adjoint idempotent p whose entries generate the algebra A(S 4 q ) of polynomial functions over it. This projection determines a deformation of an (anti-)instanton bundle over the classical sphere S 4 . We compute the fundamental K-homology class of S 4 q and pair it with the class of p in the Ktheory getting the value −1 for the topological charge. There is a right coaction of SU q (2) on S 7 q such that the algebra A(S 7 q ) is a non trivial quantum principal bundle over A(S 4 q ) with structure quantum group A(SU q (2)).

We study noncommutative principal bundles (Hopf-Galois extensions) in the context of coquasitriangular Hopf algebras and their monoidal category of comodule algebras. When the total space is quasi-commutative, and thus the base space subalgebra is central, we define the gauge group as the group of vertical automorphisms or equivalently as the group of equivariant algebra maps. We study Drinfeld twist (2-cocycle) deformations of Hopf-Galois extensions and show that the gauge group of the twisted extension is isomorphic to the gauge group of the initial extension. In particular noncommutative principal bundles arising via twist deformation of commutative principal bundles have classical gauge group. We illustrate the theory with a few examples.

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We build an SU (2)-Hopf bundle over a quantum toric four-sphere whose radius is noncentral. The construction is carried out using local methods in terms of sheaves of Hopf-Galois extensions. The associated instanton bundle is presented and endowed with a connection with anti-self-dual curvature.Mathematics Subject Classification. 81R60, 16T05, 81T75.