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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.

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    $\begingroup$ Just for context, the result (vanishing of $H_i$ for all $i\ge 1$ implies the Lie algebra is reduced to $\{0\}$) holds for finite-dimensional Lie algebras: $\mathfrak{g}$: if $\mathfrak{g}$ is unimodular of dimension $n$ then $H_n(\mathfrak{g})$ is 1-dimensional, and if $\mathfrak{g}$ is not unimodular then $H_1(\mathfrak{g})$ is nonzero. However I even believe that for a nonzero finite-dimensional Lie algebra in characteristic zero, $H_1(\mathfrak{g})\oplus H_3(\mathfrak{g})$ is always nonzero. $\endgroup$
    – YCor
    Dec 8 '21 at 13:56
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    $\begingroup$ The 1983 Commentarii paper by Harpe and McDuff EUDML link yields many acyclic groups, typically large automorphism groups. This suggests, by analogy, that large endomorphism algebra might be acyclic. Thus candidates would be the Lie algebra of all linear endomorphisms of an infinite-dimensional vector space, the Lie algebra of bounded operators of an infinite-dimensional Hilbert space, etc. $\endgroup$
    – YCor
    Dec 8 '21 at 14:09
  • $\begingroup$ @YCor Thank you! And thank you for answering my initial question. Yes, I also thought about some automorphism/endomorphism stuff. But were any computations for endomorphism Lie algebras (or vector fields) done? Or is it just an intuition? $\endgroup$ Dec 8 '21 at 16:08
  • $\begingroup$ It's just an analogy suggesting to look at this case. I'm not confident enough to conjecture it should work. $\endgroup$
    – YCor
    Dec 8 '21 at 16:24
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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 $K$).

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 $H_k(\mathfrak{g})=0$.

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.

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 $k$. $\Box$

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, 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

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