Hopf-Galois extensions and twisted Hopf algebroids

TL;DR

This paper demonstrates that quantum principal bundles and Hopf Galois extensions can be described using left Hopf algebroids with antipodes, and explores their properties under cocycle twists, with applications to quantum groups.

Contribution

It establishes the structure of Ehresmann-Schauenburg bialgebroids as left Hopf algebroids with antipodes and analyzes their behavior under Drinfeld cotwists, especially for associative type extensions.

Findings

Ehresmann-Schauenburg bialgebroid of quantum principal bundles is a left Hopf algebroid.

If the quantum group is coquasitriangular, the associated Hopf algebroid has an antipode.

Associative type extensions correspond to Hopf algebroid cotwists, exemplified by quantum groups.

Abstract

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.