Computing something I have come across a Lie algebra $\def\L{\mathfrak L}\def\CC{\mathbb C}\L_N$ that I would like to identify.
Fix an integer $N$ such that $N\geq2$, let $\L_N$ be the free complex vector space over the set $\{e_{l}:l\geq-N+1\}$, and turn $\L_N$ into a Lie algebra setting $$ [e_l, e_m] = \begin{cases} 0 & \text{if $i+u>N$} \\ & \text{or $l+m<-N+1$;} \\ \dfrac{l(v+1)-m(j+1)}{N-1}\cdot e_{l+m} & \text{othewise.} \end{cases} $$ where $j$ and $v$ are the quotients of dividing $l$ and $m$ by $N-1$, respectively.
Question. What is this algebra?
When $N=2$ this is isomorphic to the algebra $W_1=\operatorname{Der}(\mathbb C[t])$ of vector fields on the line, with $e_l$ corresponding to $-t^{l+1}\dfrac{\mathrm d}{\mathrm d t}$ and, in fact, for all $N$ the subspace $\L_N^0$ spanned by $\{e_{j(N-1)}:j\geq-1\}$ is a subalgebra isomorphic to $W_1$.
In that spirit, one can look at the Lie algebra (I'll hack the notation for Lie algebras of vector fields…) $$ W_{1/(N-1)} = \left\langle -t^{\frac{l}{N-1}+1}\dfrac{\mathrm d}{\mathrm d t}:l\geq-N+1\right\rangle $$ which has brackets $$ \left[ -t^{\frac{l}{N-1}+1}\dfrac{\mathrm d}{\mathrm d t} , -t^{\frac{m}{N-1}+1}\dfrac{\mathrm d}{\mathrm d t} \right] = \frac{l-m}{N-1} \left(-t^{\frac{l+m}{N-1}+1}\dfrac{\mathrm d}{\mathrm d t}\right). $$ This has an air de famille with $\L_N$, of course, but it is not quite it. In any case, this gives a reason not to renormalize the denominator away in the formula for the bracket of $\L_N$.
The difference between $\L_N$ and $W_{1/(N-1)}$ is in the behaviour $\mod N-1$ of the indices of course. One can try loopifying $W_1$ into $W_1\otimes\CC[t]/(t^N)$, but that's something very different, or even something like a twisted loop algebra $W_1\ltimes\CC[t]/(t^N)$, with $W_1$ acting on the second factor by derivations, but this does not give anything extra because $W_1$ is simple and cannot act in many ways. It could be that one needs to further deform the $\CC[t]/(t^N)$-loop algebra along a cocycle.
The algebra $\L_N$ has a unique maximal nilpotent ideal $\def\N{\mathfrak{N}}\N$ , spanned by the $e_l$ with $l$ not divisible by $N-1$, which has index exactly $N-1$. The quotient $\L_N/\N$ is a copy of $W_1$, the extension is in fact split and the action of $W_1$ on $\N$ is semisimple — a direct sum of modules in the so called «intermediate series» for $W_1$, which look like $\CC[t]t^\alpha(dt)^\beta$ (these are «$\beta$-density bundles» that come with a flat connection, so an action of the first Weyl algebra and, by restriction, one of $W_1$) for $N-2$ pairs $(\alpha,\beta)$ that one can compute.
