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Hopf-Galois objects over bicrossed product Hopf algebras and twisting maps

Julien Bichon, Agustín García

Abstract

We describe Hopf-Galois objects over bicrossed product Hopf algebras. More precisely, we show that any right Hopf-Galois object over a bicrossed product of Hopf algebras is obtained from Hopf-Galois objects over the two factors and a certain twisting map, while the unique Hopf algebra making it into a bi-Galois object is again a bicrossed product.

Hopf-Galois objects over bicrossed product Hopf algebras and twisting maps

Abstract

We describe Hopf-Galois objects over bicrossed product Hopf algebras. More precisely, we show that any right Hopf-Galois object over a bicrossed product of Hopf algebras is obtained from Hopf-Galois objects over the two factors and a certain twisting map, while the unique Hopf algebra making it into a bi-Galois object is again a bicrossed product.

Paper Structure

This paper contains 26 sections, 23 theorems, 140 equations.

Table of Contents

  1. Introduction
  2. Hopf-Galois objects and pairings
  3. Hopf-Galois objects
  4. Hopf $2$-cocycles
  5. Cogroupoids
  6. Pairings
  7. Smash products
  8. Hopf-Galois structure on smash products arising from pairings
  9. Application to differential operators on affine group schemes
  10. Twisting maps and bicrossed products of Hopf algebras
  11. Twisting maps
  12. Bicrossed products of Hopf algebras
  13. Skew-pairings over bicrossed products
  14. Hopf-Galois objects for bicrossed product Hopf algebras
  15. General setting
  16. ...and 11 more sections

Key Result

Lemma 2.1

Let $\mathcal{C}$ be a connected cogroupoid, $X\in\mathop{\mathrm{ob}}\nolimits \mathcal{C}$ and let $H\subseteq \mathcal{C}(X,X)$ be a Hopf subalgebra. Consider, for $Y,Z \in \mathop{\mathrm{ob}}\nolimits \mathcal{C}$, the algebra This defines a connected subcogroupoid $\mathcal{C}_H$ of $\mathcal{C}$ (with $\mathop{\mathrm{ob}}\nolimits\mathcal{C}_H=\mathop{\mathrm{ob}}\nolimits\mathcal{C}$ and

Theorems & Definitions (59)

  • Lemma 2.1
  • Remark 2.2
  • Proposition 2.3
  • Proposition 2.4
  • Proposition 2.5
  • Example 3.1
  • Example 3.2
  • Lemma 3.3
  • proof
  • Definition 3.4
  • ...and 49 more