Abstract
We study the Ehresmann–Schauenburg bialgebroid of a noncommutative principal bundle as a quantization of the gauge groupoid of a classical principal bundle. We show that the gauge group of the noncommutative bundle is isomorphic to the group of bisections of the bialgebroid, and we give a crossed module structure for the bisections and the automorphisms of the bialgebroid. Examples include: Galois objects of Taft algebras, a monopole bundle over a quantum sphere and a not faithfully flat Hopf–Galois extension of commutative algebras. For each of the latter two examples, there is in fact a suitable invertible antipode for the bialgebroid making it a Hopf algebroid.
Highlights
The study of groupoids on the one hand and gauge theories on the other hand is important in different areas of mathematics and physics
In the present paper, having in mind applications to noncommutative gauge theories, we consider the Ehresmann–Schauenburg bialgebroid associated with a noncommutative principal bundle as a quantization of the classical gauge groupoid
Bisections of the gauge groupoid are closely related to gauge transformations. In parallel with this result, we show that in a rather general context the gauge group of a noncommutative principal bundle is group isomorphic to the group of bisections of the corresponding Ehresmann–Schauenburg bialgebroid
Summary
The study of groupoids on the one hand and gauge theories on the other hand is important in different areas of mathematics and physics. We work out all the details of the gauge group of the principal bundle and of the bialgebroid with corresponding group of bisections, for the noncommutative U(1) bundle over the quantum standard Podles 2-sphere, and for a commutative not faithfully flat Hopf– Galois extension obtained in [3] from a particular coaction on the algebra O(SL(2)). Part of the paper concerns the Ehresmann–Schauenburg bialgebroid of a Galois object and corresponding groups of bisections, been they algebra maps from the bialgebroid to the ground field (and characters) or more general transformations. For these bialgebroids, some of the results we report could be and have been obtained in an abstract and categorical way. We work out this construction for the Taft algebras; more general results will be reported elsewhere
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
