Lower central series

Let $\mathfrak g$ be a Lie algebra .

Definition

The lower central series of $\mathfrak g$ is the descending sequence of ideals

Each $\gamma_k(\mathfrak g)$ is an ideal because it is generated by brackets with $\mathfrak g$.

Nilpotency

The Lie algebra $\mathfrak g$ is nilpotent if $\gamma_{s+1}(\mathfrak g)=0$ for some $s$; the smallest such $s$ is called the nilpotency class (or step).

This filtration is the Lie-algebra analogue of the lower central series of a group and is one of the standard equivalent characterizations recorded in the TFAE criterion for nilpotency .

Example (Heisenberg)

In the 3-dimensional Heisenberg Lie algebra (see the Heisenberg algebra example ) with basis $X,Y,Z$ and bracket $[X,Y]=Z$ (all other brackets between basis elements zero), one computes:

• $\gamma_1(\mathfrak g)=\langle X,Y,Z\rangle$,
• $\gamma_2(\mathfrak g)=[\mathfrak g,\mathfrak g]=\langle Z\rangle$,
• $\gamma_3(\mathfrak g)=[\mathfrak g,\langle Z\rangle]=0$ (since $Z$ is central).

So the Heisenberg algebra is nilpotent of class $2$.