= Infinite-dimensional simple module for the two-dimensional affine Lie algebra
{title2=$x=t,\quad yf(t)=f(t-1)$}
The <solvable Lie algebra> $\mathbb Cx\oplus\mathbb Cy$, with $[x,y]=y$, has an infinite-dimensional <Irreducible Lie algebra representation> on $\mathbb C[t]$: $xf(t)=tf(t)$ and $yf(t)=f(t-1)$. Indeed $[x,y]f=yf$. An <invariant subspace> is an ideal $p(t)\mathbb C[t]$ because it is invariant under multiplication by $t$. Shift invariance gives $p(t)\mid p(t-1)$, hence $p(t-1)=p(t)$ and $p$ is constant. This shows why the <Lie theorem> needs finite-dimensional representations, even when the algebra itself is finite-dimensional.
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