Two-cocycles and cleft extensions in left braided categories
Istv\'an Heckenberger, Kevin Wolf

TL;DR
This paper generalizes the concepts of two-cocycles and cleft extensions to categories with partial braiding, enabling new applications in the theory of Hopf algebras and Yetter-Drinfeld modules.
Contribution
It introduces a framework for two-cocycles and cleft extensions in non-braided categories with specific braiding objects, extending classical results.
Findings
Describes liftings of coradically graded Hopf algebras in Yetter-Drinfeld categories.
Provides a generalized approach to two-cocycles and cleft extensions.
Applications to categories of Yetter-Drinfeld modules and H-modules.
Abstract
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 a Yetter-Drinfeld module braids from the left with -modules. We will generalize classical results to this context and give some application for the categories of Yetter-Drinfeld modules and -modules. In particular we will describe liftings of coradically graded Hopf algebras in the category of Yetter-Drinfeld modules with these techniques.
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