nLab nilpotent Lie algebra

Idea

A Lie algebra is nilpotent if there exists a uniform $n$ such that acting via the Lie bracket on any one of its elements with other elements $n$ times yields zero. In other words, $\exists n\in\mathbf{N}$ such that for any $x_i$, $i = 1,\ldots, n$, the composition $ad x_1 ad x_2 \ldots ad x_n = 0$.

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 $[-,-]$.

A Lie algebra is locally nilpotent if any finite-dimensional subalgebra is nilpotent.

A Lie algebra is ad-nilpotent if for any $x$, its endomorphism $ad x$ is nilpotent. In general, this is a weaker property.

Engel’s theorem

Every finite-dimensional matrix Lie algebra $L\subset gl(m)$ over a field is nilpotent iff it is ad-nilpotent.

Examples

• Trivially: Every abelian Lie algebra is nilpotent.

• The Lie algebra of even-degree elements of the tensor product of a super translation Lie algebra (super Minkowski super Lie algebra) with a superpoint $\mathbb{R}^{0\vert q}$ is nilpotent.

Properties

• 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.

Related concepts

• nilpotent element

• nilpotent group

• nilpotent topological space

• nilpotent L-infinity algebra

References

• Jean-Pierre Serre: Nilpotent and Solvable Lie Algebras, Chapter V in: Lie Algebras and Lie Groups – 1964 Lectures given at Harvard University, Lecture Notes in Mathematics 1500, Springer (1992) [doi:10.1007/978-3-540-70634-2]

• Arthur A. Sagle, Ralph E. Walde: Nilpotent Lie Groups and Algebras, Chapter 11 in: Introduction to Lie Groups and Lie Algebras, Pure and Applied Mathematics 51, Elsevier (1973) 215-227 [doi:10.1016/S0079-8169(08)61671-2]

• James Milne, pp. 260 in: Algebraic Groups – The theory of group schemes of finite type over a field, Cambridge University Press (2017) [doi:10.1017/9781316711736, webpage, pdf]

• Laurence Corwin, Frederick P. Greenleaf: Section 1.2 of: Representations of Nilpotent Lie Groups and their Applications – Volume 1 Part 1. Basic Theory and Examples, Cambridge University Press (2004) [ISBN:9780521604956]

• Victor Kac (notes by Marc Doss), Nilpotent and Solvable Lie algebras (2010) [pdf, pdf]

• Daniel Huybrechts: Nilpotent Lie Algebras, Section 1 of Stability Structures on Lie Algebras, after Kontsevich and Soibelman (2011) [pdf, pdf]

• Veronique Fischer, Michael Ruzhansky, Section 1.6 in: Quantization on Nilpotent Lie Groups, Birkhäuser (2016) [doi:10.1007/978-3-319-29558-9]

• Nilpotent Lie algebras [pdf]

See also:

• Wikipedia, Nilpotent Lie algebra

• Wikipedia: Engel’s theorem

• eom: Lie algebra, nilpotent

• GroupProps: Lie correspondence between nilpotent Lie algebras and unipotent algebraic groups

On the Sullivan models which are Chevalley-Eilenberg algebra of nilpotent Lie algebras:

• Manuel Amann, pp. 4 of: On quasi-isometric nilpotent Lie groups [arXiv:1710.04542]

On nilpotent super Lie algebras:

• Manfred Scheunert, §III.2 in: The theory of Lie superalgebras. An introduction, Lect. Notes Math. 716 (1979) [doi:10.1007/BFb0070929]

• Luc Frappat, Antonio Sciarrino, Paul Sorba: §26 in: Dictionary on Lie Superalgebras, Academic Press (2000) [arXiv:hep-th/9607161, ISBN:978-0122653407]