Example: the Heisenberg Lie algebra

A 3D nilpotent Lie algebra with basis $X,Y,Z$ and bracket $[X,Y]=Z$.

Example: the Heisenberg Lie algebra

Let $\mathfrak h_3$ be the 3-dimensional real Lie algebra with basis $\{X,Y,Z\}$ and brackets

[X,Y]=Z,\qquad [X,Z]=[Y,Z]=0.

This is the simplest non-abelian nilpotent example (see nilpotent Lie algebra ).

Concrete matrix model

Inside $\mathfrak{gl}_3(\mathbb R)$ , set

X = E_{12},\quad Y = E_{23},\quad Z = E_{13},

where $E_{ij}$ has a $1$ in the $(i,j)$ entry and $0$ otherwise. Using the commutator bracket,

[E_{12},E_{23}] = E_{12}E_{23}-E_{23}E_{12} = E_{13}-0 = E_{13},

and one checks $[E_{12},E_{13}]=[E_{23},E_{13}]=0$ . Hence this realizes $\mathfrak h_3$ as strictly upper triangular $3\times 3$ matrices (compare strictly upper triangular examples ).

Derived subalgebra and center

The derived subalgebra is

[\mathfrak h_3,\mathfrak h_3] = \mathrm{span}\{Z\}.

Moreover $Z$ commutes with everything, so the center is

Z(\mathfrak h_3)=\mathrm{span}\{Z\}.

Thus $\mathfrak h_3/[\mathfrak h_3,\mathfrak h_3]\cong \mathbb R^2$ is abelian.

Lower central and derived series (explicit)

The lower central series is

\mathfrak h_3=\gamma_1 \supset \gamma_2=[\gamma_1,\gamma_1]=\mathrm{span}\{Z\} \supset \gamma_3=[\mathfrak h_3,\gamma_2]=0,

so $\mathfrak h_3$ is nilpotent of class $2$ .

The derived series is

\mathfrak h_3^{(1)}=[\mathfrak h_3,\mathfrak h_3]=\mathrm{span}\{Z\},\qquad \mathfrak h_3^{(2)}=[\mathrm{span}\{Z\},\mathrm{span}\{Z\}]=0,

so $\mathfrak h_3$ is solvable (compare solvable Lie algebra ).

Context. Exponentiating $\mathfrak h_3$ yields the Heisenberg group , a central extension of $\mathbb R^2$ by $\mathbb R$ that is fundamental in geometry and representation theory.