Twisted Ehresmann Schauenburg bialgebroids

TL;DR

This paper constructs and analyzes twisted Ehresmann Schauenburg bialgebroids via 2-cocycles, revealing their structure as cocycle twists of untwisted bialgebroids in the context of cleft Hopf Galois extensions.

Contribution

It introduces a method to construct invertible 2-cocycles on Ehresmann Schauenburg bialgebroids under specific conditions, and shows these are isomorphic to untwisted bialgebroids, extending the theory to Galois objects.

Findings

Constructed invertible 2-cocycles on bialgebroids.

Proved isomorphism to untwisted bialgebroids via cocycle twists.

Extended the theory to Galois objects without cocommutativity.

Abstract

We construct an invertible normalised 2 cocycle on the Ehresmann Schauenburg bialgebroid of a cleft Hopf Galois extension under the condition that the corresponding Hopf algebra is cocommutative and the image of the unital cocycle corresponding to this cleft Hopf Galois extension belongs to the centre of the coinvariant subalgebra. Moreover, we show that any Ehresmann Schauenburg bialgebroid of this kind is isomorphic to a 2-cocycle twist of the Ehresmann Schauenburg bialgebroid corresponding to a Hopf Galois extension without cocycle, where comodule algebra is an ordinary smash product of the coinvariant subalgebra and the Hopf algebra (i.e. $\C(B/#_{\sigma}H, H)\simeq \C(B\#H, H)^{\tilde{\sigma}}$). We also study the theory in the case of a Galois object where the base is trivial but without requiring the Hopf algebra to be cocommutative.