Cleft Extensions for Hopf Algebroids without Antipodes

Abstract

We introduce cleft extensions for Hopf algebroids. We prove the equivalence between cleft extensions, $σ$-twisted crossed products, and Hopf-Galois extensions with the normal basis property, thereby generalizing the theory of cleft extensions for Hopf algebroids developed by Böhm and Brzeziński, and fitting in with the general theory of Galois and biGalois extensions over Hopf algebroids developed by the authors. We investigate the Ehresmann Hopf algebroid associated with a cleft extension and show that it is isomorphic to a generalized version of the Connes-Moscovici Hopf algebroid. A special case of the Connes-Moscovici Hopf algebroid, namely the case where the coinvariants of the cleft extension coincide with the base of the Hopf algebroid, is a Drinfeld twist of a Hopf algebroid by a two-cocycle, generalizing work of Böhm, Han and Majid.