Galois-Type Extensions and Equivariant Projectivity

Abstract

The theory of general Galois-type extensions is presented, including the interrelations between coalgebra extensions and algebra (co)extensions, properties of corresponding (co)translation maps, and rudiments of entwinings and factorisations. To achieve broad perspective, this theory is placed in the context of far reaching generalisations of the Galois condition to the setting of corings. At the same time, to bring together K-theory and general Galois theory, the equivariant projectivity of extensions is assumed resulting in the centrepiece concept of a principal extension. Motivated by noncommutative geometry, we employ such extensions as replacements of principal bundles. This brings about the notion of a strong connection and yields finitely generated projective associated modules, which play the role of noncommutative vector bundles. Subsequently, the theory of strong connections is developed. It is purported as a basic ingredient in the construction of the Chern character for Galois-type extensions (called the Chern-Galois character).

This is a pre-release of a chapter of the book "Quantum Symmetry in Noncommutative Geometry". We hope to receive some constructive feedback in order to debug or improve the file prior to its publication by the European Mathematical Society Publishing House

References in corpus (8)

• Quantum groupoids
• Solutions of the quantum dynamical Yang-Baxter equation and dynamical quantum groups
• Strong Connections and Chern-Connes Pairing in the Hopf-Galois Theory
• Differentiable cyclic cohomology and Hopf algebraic structures in transverse geometry
• Noncommutative Instantons on the 4-Sphere from Quantum Groups
• Bijectivity of the canonical map for the noncommutative instanton bundle
• Piecewise principal comodule algebras
• Noncommutative geometry through monoidal categories

Cited by in corpus (7)

• Free actions of compact quantum group on unital C*-algebras
• Noncommutative bundles over the multi-pullback quantum complex projective plane
• The Quantum Flag Manifold $\mathbf{S}{\mathbf{U}}_{\mathbf{q}}(3)/{\mathbb{T}}^2$ as an Example of a Noncommutative Sphere Bundle
• The local-triviality dimension of actions of compact quantum groups
• On the Relationship between Classical and Deformed Hopf Fibrations
• Cyclic Homology and Quantum Orbits
• Inertial Hopf-cyclic homology