Example: strictly upper triangular matrices - Imported Knowlpedia Content

Let $\mathfrak n_n\subset \mathfrak{gl}_n(\mathbb R)$ be the vector space of strictly upper triangular matrices (zeros on and below the diagonal), with Lie bracket the commutator $[A,B]=AB-BA$ .

This is a standard example of a nilpotent Lie algebra.

Concrete bracket computation in the matrix-unit basis

For $i<j$ , let $E_{ij}$ be the matrix unit. A direct multiplication gives

E_{ij}E_{kl}=\delta_{jk}E_{il},

[E_{ij},E_{kl}] = E_{ij}E_{kl}-E_{kl}E_{ij} = \delta_{jk}E_{il}-\delta_{li}E_{kj}.

In particular, if $i<j$ and $k<l$ , then $[E_{ij},E_{kl}]$ is either $0$ or another strictly upper triangular matrix unit.

Lower central series (explicit nilpotency mechanism)

Define the “height” of $E_{ij}$ as $j-i\ge 1$ . From the formula above, any nonzero commutator of strictly upper triangular matrices increases height: the product $E_{ij}E_{jk}=E_{ik}$ has height $(k-i)=(j-i)+(k-j)$ .

Consequently, iterated commutators eventually vanish: the lower central series

\gamma_1=\mathfrak n_n,\qquad \gamma_{r+1}=[\mathfrak n_n,\gamma_r]

satisfies $\gamma_{n}=0$ . Thus $\mathfrak n_n$ is nilpotent of class at most $n-1$ , hence solvable (compare nilpotent implies solvable and derived series).

Small case

n=3

(fully explicit)

A general element is

\begin{pmatrix} 0 & a & b\\ 0 & 0 & c\\ 0 & 0 & 0 \end{pmatrix} = aE_{12}+bE_{13}+cE_{23}.

Using $[E_{12},E_{23}]=E_{13}$ and other brackets $0$ , we get

[\mathfrak n_3,\mathfrak n_3]=\mathrm{span}\{E_{13}\}, \qquad [\mathfrak n_3,[\mathfrak n_3,\mathfrak n_3]]=0,

which matches the Heisenberg algebra pattern.