= Witt algebra
{c}
{title2=$[\ell_m,\ell_n]=(m-n)\ell_{m+n}$}
{wiki}
Over $\mathbb C$, the Witt algebra is the <Lie algebra> of <algebraic derivations> of the <Laurent polynomial ring> $\mathbb C[z,z^{-1}]$. It has <basis> $\ell_n=-z^{n+1}\,d/dz$ for $n\in\mathbb Z$. The <commutator> of these <differential operators> is
$$
[\ell_m,\ell_n]=(m-n)\ell_{m+n}.
$$
Indeed, the second-derivative terms cancel in the <commutator>, leaving the derivative of the coefficient of each operator. The span of $\ell_{-1},\ell_0,\ell_1$ is a copy of the <sl2 Lie algebra>: take $e=\ell_{-1}$, $h=2\ell_0$, $f=-\ell_1$.
The <Witt algebra> models finite <Fourier series> of complexified <vector fields> on the <circle>. Its nontrivial one-dimensional <central extension of a Lie algebra> is the <Virasoro algebra>, whose central term changes the <representation theory> relevant to <conformal field theory>.
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