Curved L-infinity algebras and lifts of torsors

Consider an extension of finite dimensional nilpotent Lie algebras $0 \to \mathfrak{h} \to \tilde{\mathfrak{g}} \to \mathfrak{g} \to 0$(overafield$k$ of characteristic zero) corresponding to an extension of unipotent algebraic groups $1\to H\to \tilde{G}\to G\to 1$. For a $G$-torsor $P$ on an algebraic variety $X$ over $k$, we study the problem of lifting $P$ to $\tilde{G}$-torsor $\tilde{P}$. Fixing a trivialization of $P$ on open subsets of an affine cover, we give the Cech complex of $\mathfrak{h}$-valued functions the structure of a curved $L_{\mathrm{\infty }}$-algebra and define a curved version of the Deligne-Getzler groupoid. We show that this groupoid is isomorphic the groupoid of cocycle level $\tilde{G}$-lifts of $P$.