AI 中文总结
本文引入无对径Hopf代数胚的裂扩张，证明其与σ-扭曲交叉积、带正规基的Hopf-Galois扩张等价，研究相关Ehresmann Hopf代数胚的同构性，推广了相关理论。
AI 中文摘要
我们引入了Hopf代数胚的裂扩张概念，证明了裂扩张、σ-扭曲交叉积及具有正规基性质的Hopf-Galois扩张三者等价，推广了Böhm与Brzeziński发展的Hopf代数胚裂扩张理论，并契合本文作者发展的Hopf代数胚上Galois与双Galois扩张的一般理论。我们研究了与裂扩张相关的Ehresmann Hopf代数胚，证明其同构于Connes-Moscovici Hopf代数胚的推广形式。当裂扩张的余不变量与Hopf代数胚的基重合时，该Connes-Moscovici Hopf代数胚特例是Hopf代数胚被2-上循环的Drinfeld扭，推广了Böhm、Han与Majid的工作。
英文摘要
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 B{ö}hm and Brzezi{ń}ski, 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 B{ö}hm, Han and Majid.
