I have just began to study infinite-dimensional Lie algebras and I am curious whether the Lie algebra $L$ spanned by the vector fields $z^n \partial/\partial z$, $n=0,1,2,3,\dots$ admits any representations (not necessarily irreducible) by finite-dimensional matrices. In particular, the $sl(2)$ subalgebra of $L$ spanned by $z^n \partial/\partial z$, $n=0,1,2$ is well-known to admit infinitely many such representations, and I wonder whether any of those can be somehow extended to the representations of the whole $L$. Many thanks in advance!
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