# Twisting an L-infinity-morphism with “non-associated” Maurer-Cartan elements

## Background

Suppose we are given $L_\infty$-algebras $(g,Q)$ and $(g',Q')$ and an $L_\infty$-morphism $F$ from $(g,Q)$ to $(g',Q')$. Furthermore, we have a Maurer-Cartan element $\pi$ of $(g,Q)$. One can twist $(g,Q)$ with the Maurer-Cartan element $\pi$ and obtains a new $L_\infty$-algebra that we call $(g,Q_\pi)$. Furthermore, we can construct a Maurer-Cartan element $\pi'$ of $(g',Q')$ by the formula

$\pi' = \sum_{n=1}^\infty \frac{1}{n!} F_n(\pi, \ldots , \pi)$,

where $F_n$ is the $\bigwedge^n g \rightarrow g'$-part of $F$. I don't know whether there is a (better) term, so I call $\pi$ and $\pi'$ associated Maurer-Cartan elements

One can twist the morphism $F$ with the Maurer-Cartan elements $\pi$ and $\pi'$ and obtain an $L_\infty$-morphism $F_\pi$ from $(g,Q_\pi)$ to $(g',Q'_{\pi'})$. The references I found are Dolgushev: A Proof of Tsygan's Formality Conjecture for an Arbitrary Smooth Manifold (section 2.4) and Yekutieli: Continuous and Twisted L_infinity Morphisms (section 3).

## Question

Given $L_\infty$-algebras $(g,Q)$ and $(g',Q')$, an $L_\infty$-morphism $F$ from $(g,Q)$ to $(g',Q')$, a Maurer-Cartan elements $\pi$ of $(g,Q)$ and a Maurer-Cartan element $\omega$, $\omega\neq \pi'$, of $(g',Q')$. Can one construct an $L_\infty$-morphisms between $(g,Q)$ twisted with the Maurer-Cartan element $\pi$ and $(g',Q')$ twisted with the Maurer-Cartan element $\omega$, where the Maurer-Cartan elements are not "associated"? I.e. can one construct an $L_\infty$-morphism between $(g,Q_\pi)$ to $(g',Q'_\omega)$?

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If I understand your question correctly, my guess is that the answer is yes. Maurer-Cartan elements are to be thought of as deformations of the "underlying formal manifold". $Q$ is a "vector field" on this formal manifold. "Twisting" on either side should just correspond to deforming the vector field... But I'm not sure. Again this is my guess. –  Kevin H. Lin Apr 26 '10 at 14:33
Unfortunately, I know too little about the formal manifold language of L-infinity algebras, so I did not get any intuition from there. Do you or anyone know a good introductory reference for this point of view? I only know the Kontsevich letter. (Maybe I should make this a separate question?) –  C. Jost Apr 26 '10 at 16:35

First, answer this simplified question: If $\alpha$ is an MC-element of $(h,P)$, is there a morphism $(h,P_\alpha) \to (h,P)$ and a morphism $(h,P)\to (h,P_\alpha)$? (This corresponds to $\pi$ or $\omega$ being zero and $g=g'$, $Q=Q'$ in the original question.) Then compose the morphisms
$(g,Q_\pi) \to (g,Q) \to (g',Q') \to (g',Q'_\omega)$,
where the first arrow comes from $h=g$, $P=Q_\pi$, and $\alpha = -\pi$, the second is the given morphism, and the third comes from $h=g'$, $P = Q'$, and $\alpha = \omega$.