Two-cocycles and cleft extensions in left braided categories

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 $H$ a Yetter-Drinfeld module braids from the left with $H$-modules. We will generalize classical results to this context and give some application for the categories of Yetter-Drinfeld modules and $H$-modules. In particular we will describe liftings of coradically graded Hopf algebras in the category of Yetter-Drinfeld modules with these techniques.