# Nilpotent Lie algebra

In mathematics, a Lie algebra is nilpotent if its lower central series eventually becomes zero.

## Definition

Let $\mathfrak{g}$ be a Lie algebra. $\mathfrak{g}$ is nilpotent if the lower central series

$\mathfrak{g}_n = 0$ for some $n\in \mathbb{N}$.

We can equivalently define a nilpotentency in terms of the adjoint representation. Let $\mathfrak{g}$ be a Lie algebra. $\mathfrak{g}$ is nilpotent if for some $n$ that depends on $\mathfrak{g}$

$\text{ ad }x_1\text{ ad }x_2\dots\text{ad }x_n(y)=0$ for all $x_i,y\in\mathfrak{g}$.

In particular, $(\text{ad }x)^n=0$ for all $x\in \mathfrak{g}$. We call an element $x\in\mathfrak{g}$ ad-nilpotent if $\text{ad }x$ is a nilpotent endomorphism.

## Properties

• Every nilpotent Lie algebra is solvable. This is useful in proving the solvability of a Lie algebra since, in practice, it is usually easier to prove nilpotency rather than solvability. However, in general, the converse of this property is false.
• If a Lie algebra $\mathfrak{g}$ is nilpotent, then all subalgebras and homomorphic images are nilpotent.
• If the quotient algebra $\mathfrak{g}/Z(\mathfrak{g})$, where is $Z(\mathfrak{g})$ is the center of $\mathfrak{g}$, is nilpotent, then so is $\mathfrak{g}$.
• Engel's theorem: A Lie algebra $\mathfrak{g}$ is nilpotent if and only if all elements of $\mathfrak{g}$ are ad-nilpotent.
• The Killing form of a nilpotent Lie algebra is 0.
• A nilpotent Lie algebra has an outer automorphism.

## 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