nLab
nilpotent Lie algebra

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

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

Properties

Relation to Sullivan models

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

References