Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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.

share|improve this question
add comment

2 Answers

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.

share|improve this answer
    
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
add comment

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 .

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.