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For sh Lie algebra cohomology, is there written anywhere a description of H^1(L;L) as sh derivations mod inner ones?

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  • $\begingroup$ What does "sh" stand for here? $\endgroup$ – Victor Protsak Apr 6 '13 at 23:40
  • $\begingroup$ @Victor Protsak: I believe sh comes from 'strongly homotopy'. @jim stasheff: I changed the tag 'liealgebracohomology' to 'lie-algebra-cohomology' as it already existed; also I merged the other tag(s) to what I assume was intended. Generally, the more common tag format is to join separate words by a hyphen. $\endgroup$ – user9072 Apr 7 '13 at 10:13
  • $\begingroup$ @jim stasheff: I doubt a statement like this is true for sh Lie algebras. Thinking about homology as opposed to cohomology, for regular Lie algebras the entire differential on the chain complex is determined by the differential from degree 2 to degree 1, which basically comes from the Lie bracket itself. For an sh Lie algebra the sh Lie operation's effect on the differential occurs arbitrarily high up in the chain complex, so it seems to me that $H^1$ would be insufficient to guarantee being a derivation with respect to the whole sh structure. $\endgroup$ – Jim Conant Apr 7 '13 at 17:51
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    $\begingroup$ @Jim Conant: higher sh Lie operations have arity in arbitrary positive degree, but they also have an inner cohomological degree which is precisely 1-arity. So that their total degree is still 1. $\endgroup$ – DamienC Apr 7 '13 at 19:52

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