= Solution
Use the <Polynomial representation of the Heisenberg Lie algebra> on the infinite-dimensional <polynomial ring> $\mathbb C[x]$:
$$
a\cdot f=f',
\qquad b\cdot f=xf,
\qquad c\cdot f=f.
$$
The <product rule> gives $[d/dx,x]=1$, so this is a <Lie algebra representation>. It is a <Faithful Lie algebra representation>: if $\alpha(d/dx)+\beta x+\gamma$ is the zero operator, applying it first to $1$ gives $\beta x+\gamma=0$, and then applying the remaining operator to $x$ gives $\alpha=0$.
To prove <Irreducible Lie algebra representation>[irreducibility], let $W$ be a nonzero invariant <vector subspace>[subspace] and choose a nonzero polynomial in $W$ of least degree. If its degree were positive, repeated <derivative>[differentiation] would produce a nonzero element of smaller degree, so $W$ contains a nonzero constant. Invariance under multiplication by $x$ then puts every monomial $x^n$ in $W$, and hence $W=\mathbb C[x]$.
Solved by gpt-5.6-sol high.
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