2
$\begingroup$

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.

Improve this question
$\endgroup$
6
  • 4
    $\begingroup$ You're looking for the DGLA of $L_{\infty}$-derivations of the fixed $L_{\infty}$-algebra. $\endgroup$
    – Jon Pridham
    Sep 28, 2019 at 14:24
  • $\begingroup$ Thanks @JonPridham! What is an $L_\infty$-derivation? $\endgroup$
    – AlexArvanitakis
    Sep 28, 2019 at 14:26
  • $\begingroup$ Is it the same as in arxiv.org/pdf/1409.1691.pdf Proposition 4.2? $\endgroup$
    – AlexArvanitakis
    Sep 28, 2019 at 14:38
  • 4
    $\begingroup$ 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 $\endgroup$
    – Bertram Arnold
    Sep 28, 2019 at 16:17
  • 4
    $\begingroup$ 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

0

Reset to default

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge that you have read and understand our privacy policy and code of conduct.