When talking about the cohomology space of a Lie algebras, it comes naturally to refer to the Chevalley-Eilenberg cohomology, is there other interesting type of cohomology for Lie algebra?
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2$\begingroup$ Just via a few Wikipedia clicks I got to: en.wikipedia.org/wiki/Gelfand%E2%80%93Fuks_cohomology $\endgroup$– Sam HopkinsCommented Sep 26, 2020 at 16:12
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$\begingroup$ If we think of this cohomology as right derived functors of the fixed-vector functor, aren't these "named" cohomologies just computations by differing resolutions, which we know (modulo technicalities) must produce "the same" outcome? A more elementary case is finite-group cohomology, where homology and cohomology are often pseudo-elementarily "defined" in terms of specific resolutions... but can be understood more instrinsically as derived functors. $\endgroup$– paul garrettCommented Sep 26, 2020 at 16:37
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1$\begingroup$ @paulgarrett Maybe the question is about cohomologies just not isomorphic to those derived functors? I am not aware, for example, of the universal determination of the Gelfand-Fuks cohomology via derived functors. Kapranov seems to be relating it to factorization algebra formalism, but I don't know the details or whether this gives some derived functor definition. $\endgroup$– მამუკა ჯიბლაძეCommented Sep 26, 2020 at 17:06
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1$\begingroup$ @მამუკა ჯიბლაძე the work of Kapranov is joint with Ben Hennion. The factorization formalism is, afaik, used to reduce computations to local setting of Vir type algebras in higher dimensions. I don't think it gives a univ prop. In the Clausen Scholze work on condensed math the authors seem to claim to give der functor interps for continuous group cohom, and so I imagine their formalism works for Lie algebras too. $\endgroup$– user108998Commented Sep 26, 2020 at 21:34
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1$\begingroup$ @dylan thanks for the comments, ur remarks about the tgt category construction are well taken! My point was more that at the level of gen of the current question there are ofc lots of ways to produce examples, in ur tgt cat formalism eg by composing with endofunctors of cat of Lie algebras, or by mapping to other cats and using same formalism (HH for Ug, Poisson cohom of Sg, Leibniz cohom as YCor suggests, etc.) certainly VOA cohom of loop lie alg is defined in a manner reminiscent of CE cohom, but I don't think it reduces to it in any obv sense afaik $\endgroup$– user108998Commented Sep 26, 2020 at 23:08
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No, it's not the only one. For instance, Leibniz cohomology is interesting for Lie algebras themselves. See this answer of mine for references.
