Hopf Galois extensions of Hopf algebroids

TL;DR

This paper generalizes Hopf Galois theory from Hopf algebras to Hopf algebroids, introducing new structures and proving equivalences among various types of extensions, including twisted and cleft extensions.

Contribution

It extends the theory of Hopf Galois extensions to Hopf algebroids, introducing (skew-)regular comodules, structure theorems, and equivalences among different extension types.

Findings

Generalized structure theorem for relative Hopf modules.

Established equivalence of cleft extensions, twisted crossed products, and Galois extensions.

Analyzed twisted Drinfeld doubles as examples.

Abstract

We study Hopf Galois extensions of Hopf algebroids as a generalization of the theory for Hopf algebras. More precisely, we introduce (skew-)regular comodules and generalize the structure theorem for relative Hopf modules. Also, we show that if $N\subseteq P$ is a left $\mathcal{L}$-Galois extension and $\Gamma$ is a 2-cocycle of $\mathcal{L}$, then for the twisted comodule algebra ${}_{\Gamma}P$, $N\subseteq{}_{\Gamma}P$ is a left Hopf Galois extension of the twisted Hopf algebroid $\mathcal{L}^{\Gamma}$. We study twisted Drinfeld doubles of Hopf algebroids as examples for the Drinfeld twist theory. Finally, we introduce cleft extension and $\sigma$-twisted crossed products of Hopf algebroids. Moreover, we show the equivalence of cleft extensions, $\sigma$-twisted crossed products, and Hopf Galois extensions with normal basis properties, which generalize the theory of cleft extensions of Hopf algebras.