From Schlessinger-Stasheff we know that a deformation problem should come with an associated $L_\infty$-algebra, so that gauge-equivalence classes of solutions to its Maurer-Cartan equation (the "MC moduli space") correspond to deformations. (More accurately the MC moduli space is formulated as a deformation functor but these details are not important for this question.)

Is anything known about the putative $L_\infty$-algebra whose MC moduli space corresponds to deformations of some fixed $L_\infty$-algebra? I am also interested in the corresponding question for $A_\infty$-algebras.

This seems to have been considered in the physics literature, albeit in a context with more structure: Hata-Zwiebach write down an "actional" whose critical points are solutions of the quantum master equation i.e. loop $L_\infty$-algebras. Insofar as their "actional" defines a (loop) $L_\infty$-algebra (which Hata and Zwiebach suggest but is not entirely obvious to me), that (loop) $L_\infty$-algebra would be the object we are looking for, albeit in the case where the original $L_\infty$-algebra in question --- whose deformations we are considering --- has this extra structure making it into a loop $L_\infty$-algebra.

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    $\begingroup$ You're looking for the DGLA of $L_{\infty}$-derivations of the fixed $L_{\infty}$-algebra. $\endgroup$ Sep 28, 2019 at 14:24
  • $\begingroup$ Thanks @JonPridham! What is an $L_\infty$-derivation? $\endgroup$ Sep 28, 2019 at 14:26
  • $\begingroup$ Is it the same as in arxiv.org/pdf/1409.1691.pdf Proposition 4.2? $\endgroup$ Sep 28, 2019 at 14:38
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    $\begingroup$ A $L_\infty$-structure is a coderivation on the cofree cocommutative coaugmented coalgebra on the shift of your vector space which vanishes on degree $0$ elements and squares to zero. All coderivations, not necessarily squaring to $0$, form a dgla, and $L_\infty$-structures are precisely the Maurer-Cartan elements. Thus deformations of a given $L_\infty$-structure are represented by the twist of this dgla by the MC element corresponding to it. For $A_\infty$, just drop cocommutative everywhere; the result is the Hochschild cochain complex. For references, see arxiv.org/abs/1502.03280 $\endgroup$ Sep 28, 2019 at 16:17
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    $\begingroup$ For any operad $O$ and any $O$-algebra $A$, take a cofibrant replacement $B$ of $A$. Then the Lie algebra you're looking for is the Lie algebra $\mathsf{Der}(B)$ of derivations of $B$. For $L_\infty$ or operads with Koszul duality you can instead look at (co)derivations on a rectification. $\endgroup$
    – Pedro
    Sep 28, 2019 at 17:02


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