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Given a Lie algebra g, with $Ug$ being its universal enveloping algebra, one can construct a cochain complex $d: Ug^n \rightarrow Ug^{n+1}$, and a Gerstenhaber bracket on $\oplus_n Ug^n$ so that $\oplus_n Ug^n$ becomes a dgla.

Is there an analogus construction for a $L_\infty$ algebra $g$? When g is a dgla, it seems a similar construction works. Does anyone know if there is an explicit construction for a $L_3$-algebra (all higher brackets vanish except for $l_1$, $l_2$ and $l_3$)?

Many thanks.

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2 Answers 2

Your $L_\infty$ algebra is really only a good notion up to quasi-iso, so your enveloping algebra makes most sense as a dga defined up to quasi-isomorphism. From this point of view, this is due to Quillen in his paper on rational homotopy theory if you replace your $L_{\infty}$ algebra with a quasi-isomorphic dgla ... the definition is just the "obvious" thing anyways. Baranovsky wrote a paper with exactly the title you want, which uses the rational homotopy theoretic ideas to give the answer in general.

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It is worth mentioning that Baranovsky's paper referred to here is arxiv.org/abs/0706.1396 –  Vladimir Dotsenko Jul 25 '13 at 2:44

There is a notion of universal enveloping algebra for $L_\infty$-algebras, see the references listed at http://ncatlab.org/nlab/show/universal+enveloping+E-n+algebra .

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