This discussion comes from this paper.
We are considering a connected reductive group $G/\mathbb{C}$ with a Borel subgroup $B\subset G$ and corresponding Lie algebras $\mathfrak{g},\mathfrak{b}$. Also we have an algebraic curve $Y/\mathbb{C}$ and a $B$-bundle $\mathcal{F}$ on $Y$.
The authors then introduce $\mathfrak{g}_\mathcal{F}$ as the $\mathcal{F}$-twist of a (Lie) algebra $\mathfrak{g}$. What is the twist of an algebra by a bundle?
When I hear twist I usually think of twisting invertible sheaves by one another, and have not come across it being used for algebras. Additionally I can't seem to find a definition online anywhere.