We define two-cocycles and cleft extensions in categories that are not necessarily braided, but where specific objects braid from one direction, like for a Hopf algebra [Formula: see text] a Yetter–Drinfeld module braids from the left with [Formula: see text]-modules. We will generalize classical results to this context and give some application for the categories of Yetter–Drinfeld modules and [Formula: see text]-modules. In particular, we will describe liftings of coradically graded Hopf algebras in the category of Yetter–Drinfeld modules with these techniques.

中文翻译：

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左编织类别中的两个cocycles和cleft extensions

我们在不一定编织的类别中定义两个cocycles和裂缝扩展，但特定对象从一个方向编织，例如对于Hopf代数[公式：参见文本]，Yetter-Drinfeld模块从左侧编织[公式：参见文本]-模块。我们将把经典结果推广到这种情况下，并对 Yetter-Drinfeld 模块和 [Formula: see text] 模块的类别给出一些应用。特别是，我们将描述使用这些技术在 Yetter-Drinfeld 模块类别中的 coradically 分级 Hopf 代数的提升。