Past exam of the mathematics course of the University of Cambridge / 2007 / iii / Paper 4 / 1

All brackets belong to the line $\mathbb{C}c$, and $[p,q]=c$ shows this line is exactly the first derived algebra. Since $c$ is central, the derived series of a Lie algebra isThe lower central series of a Lie algebra is likewiseFor completeness, the center of a Lie algebra is $\mathbb{C}c$: an element $ap+bq+dc$ commutes with $p$ only if $b=0$, and with $q$ only if $a=0$. The upper central series of a Lie algebra is therefore $0\subset \mathbb{C}c\subset \mathfrak{g}$, because the quotient by $\mathbb{C}c$ is abelian. Thus the Heisenberg Lie algebra is nilpotent of class two and solvable of derived length two. These properties are not mutually exclusive.

Let $V$ be a nonzero finite-dimensional Irreducible Lie algebra representation. Since $c$ is central, its action is a scalar $\lambda I$ by the Schur lemma. Taking the trace of $[\rho (p),\rho (q)]=\rho (c)$ gives $0=\lambda \dim V$, so $\lambda =0$.

The actions of $p,q$ now commute. Choose an eigenvalue $a$ of $\rho (p)$. Its nonzero eigenspace is preserved by $\rho (q)$, which has an eigenvector there, with some eigenvalue $b$. The span of this common eigenvector is invariant under all of $\mathfrak{g}$. Irreducibility forces it to be the whole representation. ConsequentlyConversely each such assignment is a one-dimensional Lie algebra representation and hence irreducible. Distinct pairs give nonisomorphic representations. This classifies all finite-dimensional irreducible representations; none is faithful because all annihilate $c$.

Use the Polynomial representation of the Heisenberg Lie algebra on $V=\mathbb{C}[x]$:The product rule gives $[d/dx,x]f=f$, verifying the required bracket, and the identity commutes with both operators.

For irreducibility, let $W$ be a nonzero invariant subspace and choose a nonzero polynomial of degree $d$ in it. Its $d$th derivative is a nonzero constant, so $1\in W$. Repeated multiplication by $x$ then puts every monomial in $W$, giving $W=\mathbb{C}[x]$.

For faithfulness, suppose $a\rho (p)+b\rho (q)+dI=0$. Applying this operator to $1$ gives $bx+d=0$, so $b=d=0$. Applying what remains to $x$ gives $a=0$. The representation is therefore faithful and irreducible. It is necessarily infinite-dimensional, by the finite-dimensional classification just proved.

On ${\mathbb{C}}^3$ take the matrix unitsTheir products give $[E_{12},E_{23}]=E_{13}$, while $E_{13}$ commutes with each of the other two. Thus this is a Lie algebra representation. The three images are linearly independent, so its kernel is zero: it is faithful and finite-dimensional. It is reducible, as expected from the classification in part ii; for example, the first coordinate line is invariant.