Hopf-Galois extensions and twisted Hopf algebroids

We show that the Ehresmann-Schauenburg bialgebroid of a quantum principal bundle $P$ or Hopf Galois extension with structure quantum group $H$ is in fact a left Hopf algebroid $L(P,H)$. We show further that if $H$ is coquasitriangular then $L(P,H)$ has an antipode map $S$ obeying certain minimal axioms. Trivial quantum principal bundles or cleft Hopf Galois extensions with base $B$ are known to be cocycle cross products $B\#_\sigma H$ for a cocycle-action pair ($\vartriangleright$,$\sigma$) and we look at these of a certain `associative type' where $\vartriangleright$ is an actual action. In this case also, we show that the associated left Hopf algebroid has an antipode obeying our minimal axioms. We show that if $L$ is any left Hopf algebroid then so is its cotwist $L^\varsigma$ as an extension of the previous bialgebroid Drinfeld cotwist theory. We show that in the case of associative type, $L(B\#_\sigma H,H)=L(B\# H)^{\tilde\sigma}$ for a Hopf algebroid cotwist $\varsigma=\tilde\sigma$. Thus, switching on $\sigma$ of associative type appears at the Hopf algebroid level as a Drinfeld cotwist. We view the affine quantum group $\hat{U_q(sl_2)}$ and the quantum Weyl group of $u_q(sl_2)$ as examples of associative type.