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# Contents

## Idea

A Lie algebra is nilpotent if acting on any one of its elements with other elements, via the Lie bracket, repeatedly eventually yields zero.

## Definition

The lower central series or descending central series? of a Lie algebra $\mathfrak{g}$ is a sequence of nested ideals $\mathfrak{g}^{k+1} \trianglelefteq \mathfrak{g}^{k}$ defined inductively by $\mathfrak{g}^1 \coloneqq \mathfrak{g}$, $\mathfrak{g}^{k+1} \coloneqq [\mathfrak{g}, \mathfrak{g}^k]$. The Lie algebra is said to be nilpotent if $\mathfrak{g}^{k} = 0$ for some $k \in \mathbb{N}$.

In other words, a Lie algebra $\mathfrak{g}$ is nilpotent if and only the improper ideal $\mathfrak{g}$ is a nilpotent element in the ideal lattice with respect to the ideal product $[-,-]$.

## Properties

• Every abelian Lie algebra is nilpotent, and every nilpotent Lie algebra is solvable?.

### Relation to Sullivan models

A finite-dimensional Lie algebra $\mathfrak{g}$ is nilpotent precisely if its Chevalley-Eilenberg algebra $CE(\mathfrak{g})$ is a Sullivan algebra (necessarily minimal). See also at rational homotopy theory for more on this.