Translation map in quantum principal bundles

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 $E(B,V,A)$ associated to a quantum principal bundle $P(B,A)$ are in bijective correspondence with equivariant maps $V\to P$, 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 $\ad$-covariant maps $A\to P$.