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For the sake of simplicity, my base field will be the complex numbers.

My question is simple: what are (preferably natural) examples of infinite dimensional simple Lie algebras?

I came up with this question after I've read the PhD thesis of a colleague. He was studying certain representations of the Lie algebras of vector fields on smooth affine varieties. They are simple and infinite dimensional. More precisely, David A. Jordan and Thomas Thiebert have proven the following result:

Theorem. Let $X$ be an affine variety, and $\mathcal{V}_X$ the Lie algebra of vector fields on $X$. Then $X$ is smooth if and only if $\mathcal{V}_X$ is simple.

The other case I know are Cartan infinite dimensional Lie algebras of vector fields on the affine space: $\mathbb{W}$, $\mathbb{S}$, $\mathbb{H}$, $\mathbb{K}$.

And then I realised that these are the only two examples of infinite dimesional simple Lie algebras I know of. Hence, my question.

Improve this question
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    $\begingroup$ I know another type of examples, which may be (possibly) isomorphic to ones of the form $\mathcal{V}_X$. Let k be a field. Take a (not necessarily finitely generated!) additive submonoid $\Gamma \subset k^n$ such that $\Gamma \otimes k = k^n$. Denote this inclusion as $I$. Also fix a basis $e_s$ of $k^n$, and denote corresponding scalar parts of $I$ by $I_s$. Then there's a vector space $E$ of "Eulerian derivations" of a group algebra $k[\Gamma]$: $\gamma \mapsto \sum I_s (\gamma)\gamma$. Then the Lie algebra generated by $E$ and $k[\Gamma]$ in $End_k(k[\Gamma])$ would have simple commutator. $\endgroup$
    – Denis T
    Commented Apr 24 at 0:52

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