Hopf Galois extensions of Hopf algebroids

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.

arXiv commentThis paper is an early version of work that is now contained and much further developed in Hopf BiGalois Theory for Hopf Algebroids (arXiv:2510.17298) and Cleft Extensions for Hopf Algebroids without Antipodes (arXiv:2608.14064). This version is no longer intended for publication, but we have retained it as a separate arXiv submission for technical reasons