ON HOPF-GALOIS EXTENSIONS AND THE GAUGE GROUP OF GALOIS OBJECTS

For starters, we will recall the fundamental concept of a Hopf-Galois extension, and instantiate it through quantum principal SU(2)-bundles with noncommutative seven-spheres as total spaces and noncommutative four-spheres as base spaces. Then we will recall the construction of the Ehresmann-Schauenburg bialgebroid of a Hopf-Galois extension, which is a noncommutative analogue of the Ehresmann groupoid of a classical principal bundle. Next, we will show that, when the base-space subalgebra is in the centre of the total-space algebra of a noncommutative principal bundle, the gauge group of  this bundle is isomorphic to the group of bisections of its Ehresmann-Schauenburg bialgebroid. Then, under the same assumption, we will prove that the group of bisections and the group of automorphisms of the bialgebroid form a crossed module. In particular, we will consider Galois objects (non-trivial noncommutative principal bundles over a point). Then the base-space subalgebra is the ground field and the corresponding Ehresmann-Schauenburg bialgebroid becomes a Hopf algebra. Examples will include Galois objects over group Hopf algebras and Taft algebras.