The basic question is:


Does vanishing of homology with trivial coefficients imply triviality of an infinite-dimensional Lie algebra?


My question is motivated by acylic groups in group theory. In particular, there is a an acylic group $\textrm{Aut}\_f(\mathbb{R})$ of autohomeomorphisms of $\mathbb{R}$ with finite support.


In the same vein is it true that Lie algebra of finite vector fields on $\mathbb{R}$ is acyclic? I know that the cohomology of all vector fields is described by Goncharova's theorem, but I don't know the answer for finite vector fields (although I have seen that the answer is mentioned as known somewhere).


This is an extension of a [post on MSE](https://math.stackexchange.com/questions/4325519/finite-dimensional-lie-algebras-with-trivial-homology).



Infinite dimensional Lie algebras with trivial homology
There exist acyclic infinite-dimensional Lie algebras, i.e., for which the trivial homology vanishes in all nonzero degree (recall that $H\_0$ of every Lie algebra is always 1-dimensional over the ground field $K$).



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> **Lemma.** Let $\mathfrak{g}$ be the increasing union of Lie subalgebras $\mathfrak{g}\_n$. Suppose that for fixed $k$, we have $H\_k(\mathfrak{g}\_n)=0$ for all $n$ large enough (say $n\ge N$). Then $H\_k(\mathfrak{g})=0$.
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Proof: this is essentially formal "diagram chasing". If $b\in Z\_k(\mathfrak{g})\subset\bigwedge^k\mathfrak{g}$ (spaces of $k$-cycles and $k$-chains), there exists $n$, which we can choose $\ge N$, such that $b\in \bigwedge^k\mathfrak{g}\_n$. Then $d\_k(b)=0$, so $b\in Z\_k(\mathfrak{g}\_n)$. Since $H\_k(\mathfrak{g}\_n)=0$, there exists $c\in\Lambda^{k-1}\mathfrak{g}\_n$ such that $d\_{k-1}(c)=b$. Hence $b\in B\_k(\mathfrak{g})$ (space of $k$-boundaries). This proves the lemma.



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> **Corollary.** Under the same assumptions, if for some sequence $(N\_k)$, we have $H\_k(\mathfrak{g}\_n)=0$ for all $n\ge N\_k$ and all $k\ge 1$, then $H\_k(\mathfrak{g})=0$ for all $k$. $\Box$
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So it is enough to have such an "increasing sequence of Lie algebras" $(\mathfrak{g}\_n)$ at disposal. We can't choose them finite-dimensional (since $H\_1$ and $H\_3$ [can't be both zero then](https://math.stackexchange.com/a/4327464/35400), at least in char. zero). However there are examples in the literature:


For instance choose $\mathfrak{g}\_n$ as the Lie algebra of formal power series vector fields $$\sum\_{i=1}^nf\_i\frac{\partial}{\partial x\_i},\quad f\_i\in K[\![x\_1,\dots,x\_n]\!].$$


Gelfand and Fuks proved in [1] (see [2], Corollary 1) that $H\_k(\mathfrak{g}\_n)=0$ for all $1\le k\le n$.


References:


[1] I. M. Gel′fand and D. B. Fuks, Cohomologies of the Lie algebra of formal vector fields, Izv. Akad. Nauk SSSR Ser. Mat. 34 (1970), 322–337 (Russian). MR 0266195


[2] Victor Guillemin and Steven Shnider, Some stable results on the cohomology of the classical infinite-dimensional Lie algebras, Trans. Amer. Math. Soc. 179 (1973), 275-280.
[Link at AMS site, unrestricted access](https://www.ams.org/journals/tran/1973-179-00/S0002-9947-1973-0327856-3/)


