If (say) $L$ is an $L_\infty$-algebra, then it is known that under certain conditions there exists a quasi-isomorphic $L_\infty$-algebra $L'$ which is a differential graded Lie algebra: all ternary and higher brackets of $L'$ vanish. This is sometimes called rectification.

(I am a physicist, so I assume $L$ is over a field of characteristic $0$; quasi-isomorphisms are then morphisms whose linear parts induce isomorphisms on cohomology. I wouldn't mind hearing about $L$ over formal power series rings, however.)

There exist lots of operad-powered general proofs, such as in this paper by Berger and Moerdijk. Unfortunately, those are not accessible to me, so I am looking for operad-free explicit constructions/algorithms producing $L'$ and the associated quasi-isomorphism. I am also interested in the case where $L$ is endowed with a cyclic inner product, should that make a difference.

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    $\begingroup$ The "bar-cobar resolution" does precisely what you want, do you know of it? $\endgroup$ Feb 16, 2021 at 7:11
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    $\begingroup$ A reference (though arguably as operadic as they come) is Theorem 11.3.6 of irma.math.unistra.fr/~loday/PAPERS/LodayVallette.pdf - unwrapping the definitions, the canonical dgla resolution of a $L_\infty$ algebra $A$ is the free (graded) Lie algebra on $(\bigoplus_{k>0} \Lambda^k A)[k-1]$, where the differential on generators is given by the sum of the Chevalley-Eilenberg differential (which lands in generators) and the reduced coproduct (which lands in brackets of generators). The map down to $A$ is induced by the projection to the summand labeled by $k = 1$. $\endgroup$ Feb 16, 2021 at 7:37
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    $\begingroup$ @AlexArvanitakis in turn, I'd love to hear about the interest on Lie and L-infinity algebras over formal power series rings in physics. They need not have minimal models in the classical sense and the whole theory is a little bit more complicated, but a student of mine worked on this for his thesis and learning about potential new applications would be nice :) $\endgroup$ Feb 16, 2021 at 11:57
  • $\begingroup$ @DanPetersen: I've heard of "bar construction" in this context, given in an off hand comment arxiv.org/abs/0705.3719... $\endgroup$ Feb 16, 2021 at 13:15
  • $\begingroup$ ... which I guess alludes to the proof in @BertramArnold 's link. Also thanks Bertram, I can parse that definition. $\endgroup$ Feb 16, 2021 at 13:19


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