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nilpotent Lie algebra

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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 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

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

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

Last revised on November 22, 2017 at 00:55:46. See the history of this page for a list of all contributions to it.