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Characterization of principal bundles: the noncommutative algebraic case

William J. Ugalde

Abstract

We review Hopf-Galois extensions, in particular faithfully flat ones, accepted to be the noncommutative algebraic dual of a principal bundle. We also make a short digression into how quantum groups relate to Hopf-Galois extensions. Several examples are given, in order to provide a satisfactory understanding of each topic.

Characterization of principal bundles: the noncommutative algebraic case

Abstract

We review Hopf-Galois extensions, in particular faithfully flat ones, accepted to be the noncommutative algebraic dual of a principal bundle. We also make a short digression into how quantum groups relate to Hopf-Galois extensions. Several examples are given, in order to provide a satisfactory understanding of each topic.
Paper Structure (12 sections, 10 theorems, 161 equations)

This paper contains 12 sections, 10 theorems, 161 equations.

Table of Contents

  1. Introduction
  2. Tracking down the idea of a Hopf--Galois extension
  3. Noncommutative spaces
  4. A possibly non-existing map
  5. Hopf--Galois extensions: preliminaries
  6. Algebras, coalgebras and bialgebras
  7. Actions and coactions
  8. Hopf--Galois extensions and Hopf algebras
  9. Hopf algebras
  10. Quantum groups and Hopf algebras
  11. Quantum principal bundles
  12. Conclusions

Key Result

Lemma 4.4

Given a bialgebra $(H, \mu, \eta, \Delta, \varepsilon)$, if there exists a linear map $S \colon H \to H$ such that $\mu(S \otimes \mathrm{id}_H) \Delta = \eta \varepsilon = \mu(\mathrm{id}_H \otimes S) \Delta$, then the canonical map corresponding to the coaction $\delta = \Delta$ is bijective, mean

Theorems & Definitions (76)

  • Example 2.1
  • Example 2.2
  • Example 2.3
  • Example 2.4
  • Example 2.5
  • Example 2.6
  • Remark
  • Remark
  • Example 3.1
  • Remark
  • ...and 66 more