Translation map in quantum principal bundles

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 $A\to P$.

24 pages. Substantial changes in presentation. New section added with an explicitly computed example. To appear in J. Geom. Phys

References in corpus (7)

• Quantum group gauge theory on quantum spaces
• Projective quantum spaces
• Quantum Groups on Fibre Bundles
• Cross Product Quantisation, Nonabelian Cohomology And Twisting Of Hopf Algebras
• Quantum Principal Bundles
• q-deformed Dirac Monopole With Arbitrary Charge
• Quantum Principal Fiber Bundles: Topological Aspects

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• Extended noncommutative Minkowski spacetimes and hybrid gauge symmetries
• Noncommutative differential geometry on crossed product algebras
• A Note on First Order Differential Calculus on Quantum Principal Bundles
• Coalgebra Extensions and Algebra Coextensions of Galois Type
• On the Relationship between Classical and Deformed Hopf Fibrations
• Classical gauge theory on quantum principal bundles
• Quantum and braided group Riemannian geometry
• On the gauge group of Galois objects
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• Hopf algebroids and twists for quantum projective spaces
• Quantum Homogeneous Spaces and Coalgebra Bundles
• Some Remarks on Quantum and Braided Group Gauge Theory
• On the Durdevic approach to quantum principal bundles
• Gauge groups and bialgebroids
• On modules associated to coalgebra Galois extensions
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• Inertial Hopf-cyclic homology