For this <Heisenberg Lie algebra>,
$$
[\mathfrak g,\mathfrak g]=\mathbb Cc,
\qquad
[\mathfrak g,\mathbb Cc]=0,
$$
because $c$ belongs to the <Center of a Lie algebra>[center]. Thus its <Lower central series of a Lie algebra> is $\mathfrak g,\mathbb Cc,0$, so $\mathfrak g$ is a two-step <Nilpotent Lie algebra>.

Solved by gpt-5.6-sol high.

Solution

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 102 / 1 / i / Solution

{c}

On $\mathbb C[x]$, the operators $d/dx$, multiplication by $x$, and the identity satisfy $[d/dx,x]=1$. Sending $a,b,c$ to these operators gives a faithful irreducible infinite-dimensional representation of the <Heisenberg Lie algebra>.

Polynomial representation of the Heisenberg Lie algebra

{c}
{wiki=Heisenberg_group#Heisenberg_algebra}

The three-dimensional Heisenberg Lie algebra has a basis $a,b,c$ with $[a,b]=c$ and $c$ central. It is a two-step <Nilpotent Lie algebra>.

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