# Tangent complex of dgla/ twisted dgla

I am looking for a theorem which says if we twist an $L_\infty$ quasi-isomorphism between dgla's by a Maurer-Cartan element, we'll get a new $L_\infty$ quasi-isomorphism.

Let $(\mathfrak{g}, d, [-,-])$ be a DGLA. A degree one element $\omega \in \mathfrak{g}$ is called a MC element (or a Maurer-Cartan solution) if $$d\omega + \frac{1}{2}[\omega, \omega] = 0$$ For a MC element $\omega$, the corresponding tangent complex/twisted DGLA structure on $\mathfrak{g}$ is defined by $$d_\omega = d + [\omega, - ], \qquad [-,-]_\omega = [-,-].$$

Assume that $\mathcal{F}: \mathfrak{g} \to \mathfrak{g}'$ is an $L_\infty$-morphism between DGLA's and that $\omega$ is a MC element in $\mathfrak{g}$. Let $\omega'$ be the formal element in $\mathfrak{g}'$ given by $$\omega' = \sum_{j=1}^\infty \frac{1}{j!} \mathcal{F}_j(\omega^j).$$ There are convergence issues. But if it is convergent, then $\omega'$ is a Maurer-Cartan solution in $\mathfrak{g}'$, and hence $\mathfrak{g}_\omega := (\mathfrak{g}, d_\omega, [-,-]_\omega)$, $\mathfrak{g}_{\omega'}' := (\mathfrak{g}',d_{\omega'}', [-,-]_{\omega'}')$ are both DGLA's. With these DGLA's, one can consider the twisted $L_\infty$-morphism $\mathcal{F}_\omega: \mathfrak{g}_\omega \to \mathfrak{g}_\omega'$, where $\mathcal{F}_\omega$ is defined by $$\mathcal{F}_{\omega, n}(\gamma) = \sum_{j=0}^\infty \frac{1}{j!} \mathcal{F}_{n+j}(\omega^j \gamma)$$ for $\gamma \in \Lambda^n \mathfrak{g}$. Then $\mathcal{F}_\omega$ defines an $L_\infty$-morphism $\mathfrak{g}_\omega \to \mathfrak{g}_{\omega'}'$ provided the formulas are convergent.

Let $C$ be a commutative differential graded algebra, $\mathfrak{g}$ be the DGLA of polyvector fields $\mathcal{T}_{poly}^\bullet (\mathbb{R}^d)$, and $\mathfrak{g}'$ be the DGLA of polydifferential operators $\mathcal{D}_{poly}^\bullet(\mathbb{R}^d)$. Suppose $\mathcal{F}: \mathcal{T}_{poly}^\bullet(\mathbb{R}^d) \to \mathcal{D}_{poly}^\bullet(\mathbb{R}^d)$ is a $L_\infty$-quasi-isomorphism. So we have an induced $L_\infty$-morphism $\mathcal{F}: C\otimes \mathcal{T}_{poly}^\bullet(\mathbb{R}^d) \to C \otimes \mathcal{D}_{poly}^\bullet(\mathbb{R}^d)$.

Question: Assume $\omega \in C \otimes \mathcal{T}_{poly}^\bullet(\mathbb{R}^d)$ is an MC element. Is the twisted morphism $\mathcal{F}_\omega : (C\otimes \mathcal{T}_{poly}^\bullet(\mathbb{R}^d))_\omega \to (C \otimes \mathcal{D}_{poly}^\bullet(\mathbb{R}^d))_{\omega'}$ an $L_\infty$-quasi-isomorphism? What would be a good reference for this question?

• Hi, sorry to resurrect this, but did you ever track down a reference? I am also working in a similar set up to you, where I work with twisted morphisms provided that they convergence. The sources that I have found linked are usually formal, and work with complete/filtered L infinity algebras. Do you know if this works with that assumption? Provided that all formulas in sight convergence appropriately. Thank you! Nov 4, 2021 at 7:52