Two-cocycles and cleft extensions in left braided categories

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 [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.