We have a good notion of dgc algebra resolutions of commutative algebras. Is there an explicit construction of a dg Lie algebra resolution of a Lie algebra?
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$\begingroup$ Can't you just copy what Tate did? Pick a presentation; start with the free Lie algebra on the generators, add a generators $y$ of degree $1$ per relation $r$, and define $d(y)=r$. Find generators for the homology of this dg, lift them to cycles, add generators to kill them and so on. $\endgroup$– Mariano Suárez-ÁlvarezCommented May 11, 2013 at 23:42
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$\begingroup$ Sure - is it written somewhere other than what you have? $\endgroup$– Jim StasheffCommented Feb 3, 2017 at 21:48
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1 Answer
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Going back to Quillen in 1969 there is a resolution adjunction between dg-coalgebras and dg-Lie algebras, which restricts to a rectification resolution adjunction between $L_\infty$-algebras and dg-Lie algebras. This is an equivalence of homotopy theories due to theorem 3.2 in
- Vladimir Hinich, DG coalgebras as formal stacks (arXiv:math/9812034)
Here the resolution functor sends an $L_\infty$-algebra (and hence in particular a Lie algebra) to a dg-Lie algebra whose underlying graded Lie algebra is free on the underlying chain complex.
More details and more pointers are at
http://www.ncatlab.org/nlab/show/model+structure+on+dg-Lie+algebras#RectificationResolution .
answered May 12, 2013 at 4:26