# Nilpotent Lie algebra

﻿
Nilpotent Lie algebra

In mathematics, a Lie algebra $\mathfrak{g}$ is nilpotent if the lower central series

$\mathfrak{g} > [\mathfrak{g},\mathfrak{g}] > [[\mathfrak{g},\mathfrak{g}],\mathfrak{g}] > [[[\mathfrak{g},\mathfrak{g}],\mathfrak{g}],\mathfrak{g}] > \cdots$

becomes zero eventually. Equivalently, $\mathfrak{g}$ is nilpotent if

$\operatorname{ad}(x_1) \operatorname{ad}(x_2) \operatorname{ad}(x_3) ... \operatorname{ad}(x_r) = 0$

for any sequence xi of elements of $\mathfrak{g}$ of sufficiently large length. (Here, $\operatorname{ad}(x)$ is given by $\operatorname{ad}(x)y = [x, y]$.) Consequences are that $\operatorname{ad}(x)$ is nilpotent (as a linear map), and that the Killing form of a nilpotent Lie algebra is identically zero. (In comparison, a Lie algebra is semisimple if and only if its Killing form is nondegenerate.)

Every nilpotent Lie algebra is solvable; this fact gives one of the powerful ways to prove the solvability of a Lie algebra since, in practice, it is usually easier to prove the nilpotency than the solvability. The converse is not true in general. A Lie algebra $\mathfrak{g}$ is nilpotent if and only if its quotient over an ideal containing the center of $\mathfrak{g}$ is nilpotent.

Most of classic classification results on nilpotency are concerned with finite-dimensional Lie algebras over a field of characteristic 0. Let $\mathfrak{g}$ be a finite-dimensional Lie algebra. $\mathfrak{g}$ is nilpotent if and only if $\operatorname{ad}(\mathfrak{g})$ is nilpotent. Engel's theorem states that $\mathfrak{g}$ is nilpotent if and only if $\operatorname{ad}(x)$ is nilpotent for every $x \in \mathfrak{g}$. $\mathfrak{g}$ is solvable if and only if $[\mathfrak{g}, \mathfrak{g}]$ is nilpotent.

## Examples

• Every subalgebra and quotient of a nilpotent Lie algebra is nilpotent.
• If $\mathfrak{gl}_k$ is the set of $k\times k$ matrices, then the subalgebra consisting of strictly upper triangular matrices, denoted by $\mathfrak{n}_k$, is a nilpotent Lie algebra.
• A Heisenberg algebra is nilpotent.
• A Cartan subalgebra of a Lie algebra is nilpotent and self-normalizing.

## References

• Humphreys, James E. Introduction to Lie Algebras and Representation Theory. Graduate Texts in Mathematics, 9. Springer-Verlag, New York, 1972. ISBN 0-387-90053-5

Wikimedia Foundation. 2010.

### Look at other dictionaries:

• Lie algebra — In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. Lie algebras were introduced to study the concept of infinitesimal transformations. The term… …   Wikipedia

• Lie-Algebra — Eine Lie Algebra, benannt nach Sophus Lie, ist eine algebraische Struktur, die hauptsächlich zum Studium geometrischer Objekte wie Lie Gruppen und differenzierbarer Mannigfaltigkeiten eingesetzt wird. Inhaltsverzeichnis 1 Definition 2 Beispiele 2 …   Deutsch Wikipedia

• Solvable Lie algebra — In mathematics, a Lie algebra g is solvable if its derived series terminates in the zero subalgebra. That is, writing for the derived Lie algebra of g, generated by the set of values [x,y] for x and y in g, the derived series …   Wikipedia

• Malcev Lie algebra — In mathematics, a Malcev Lie algebra, or Mal tsev Lie algebra, is a generalization of a rational nilpotent Lie algebra, and Malcev groups are similar. Both were introduced by Quillen (1969, Appendix A3), based on the work of (Mal cev 1949).… …   Wikipedia

• Abelsche Lie-Algebra — Lie Algebra berührt die Spezialgebiete Mathematik Lineare Algebra Lie Gruppen Physik Eichtheorie ist Spezialfall von Vektorraum …   Deutsch Wikipedia

• Auflösbare Lie-Algebra — Lie Algebra berührt die Spezialgebiete Mathematik Lineare Algebra Lie Gruppen Physik Eichtheorie ist Spezialfall von Vektorraum …   Deutsch Wikipedia

• Einfache Lie-Algebra — Lie Algebra berührt die Spezialgebiete Mathematik Lineare Algebra Lie Gruppen Physik Eichtheorie ist Spezialfall von Vektorraum …   Deutsch Wikipedia

• Ideal (Lie-Algebra) — Lie Algebra berührt die Spezialgebiete Mathematik Lineare Algebra Lie Gruppen Physik Eichtheorie ist Spezialfall von Vektorraum …   Deutsch Wikipedia

• Nilpotente Lie-Algebra — Lie Algebra berührt die Spezialgebiete Mathematik Lineare Algebra Lie Gruppen Physik Eichtheorie ist Spezialfall von Vektorraum …   Deutsch Wikipedia

• Satz von Weyl (Lie-Algebra) — Lie Algebra berührt die Spezialgebiete Mathematik Lineare Algebra Lie Gruppen Physik Eichtheorie ist Spezialfall von Vektorraum …   Deutsch Wikipedia