Translation map in quantum principal bundles
Tomasz Brzezinski

TL;DR
This paper introduces the concept of a translation map in quantum principal bundles and uses it to characterize trivial bundles, cross sections, and gauge transformations in the quantum setting.
Contribution
It defines a translation map in quantum principal bundles and establishes key correspondences for cross sections and automorphisms, advancing the understanding of quantum fiber bundle structures.
Findings
Cross sections correspond to equivariant maps
A quantum principal bundle is trivial if it has an algebra map cross section
Vertical automorphisms correspond to ad-covariant maps
Abstract
The notion of a translation map in a quantum principal bundle is introduced. A translation map is then used to prove that the cross sections of a quantum fibre bundle associated to a quantum principal bundle are in bijective correspondence with equivariant maps , and that a quantum principal bundle is trivial if it admits a cross section which is an algebra map. The vertical automorphisms and gauge transformations of a quantum principal bundle are discussed. In particular it is shown that vertical automorphisms are in bijective correspondence with -covariant maps .
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