# Nilpotent Lie algebra

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

It is a Lie algebra analog of a nilpotent group.

## Definition

Let g be a Lie algebra. Then g is nilpotent if the lower central series terminates, i.e. if gn = 0 for some n ∈ ℕ.

Directly from the definition of the lower central series, it follows that

$[X, [X,[\cdots[X, Y]\cdots] = {\mathrm{ad}_X}^{n-1}Y \in \mathfrak{g}_n = 0 \quad \forall X, Y \in \mathfrak{g}.$

Thus adXn − 1 = 0 for all xg. The above holds, again by definition, for

$[X_1, [X_2,[\cdots[X_{n-1}, Y]\cdots] = \mathrm{ad}_{X_1}\mathrm{ad}_{X_2}\mathrm{ad}_{X_{n-1}}Y \in \mathfrak{g}_n = 0 \quad \forall X_1, X_2,\ldots, X_{n-1}, Y \in \mathfrak{g},$

so that adX1adX2 ⋅⋅⋅ adXn − 1 = 0. From this follows that ad g is nilpotent, since the expansion of a (n − 1)-fold nested bracket will have terms of this form. Since ad is a Lie algebra homomorphism, one may write[1]

$[[\cdots[[X_{n+1},X_n],X_{n-1}],\cdots,X_1] = \mathrm{ad}[\cdots[X_{n+1},X_n], \cdots, X_2](X_1),$

and

\begin{align}\mathrm{ad}[\cdots[X_{n+1},X_n], \cdots, X_2] &= [\mathrm{ad}[\cdots[X_{n+1},X_n],\cdots X_3], \mathrm{ad}_{X_2}]\\ &= \ldots = [\cdots[\mathrm{ad}_{X_{n+1}}, \mathrm{ad}_{X_n}], \cdots \mathrm{ad}_{X_2}].\end{align}

If ad g is nilpotent, the last expression is zero, and accordingly the first. Thus g is nilponent if and only if ad g is nilpotent. One can therefore equivalently define a nilpotentency in terms of the adjoint representation as follows. Let g be a Lie algebra. Then g is nilpotent if, for some n that depends on g,

$\mathrm{ ad }x_1\mathrm{ ad }x_2\dots\mathrm{ad }x_n(y)=0 \quad \forall x_i, y \in \mathfrak{g}.$

The last expression implies the nilpotency of ad g. In particular, adxn = 0 for all xg. We call an element xg ad-nilpotent if ad x is a nilpotent endomorphism. The fact that this last condition implies the nilpotency of g is the content of Engel's theorem.

## 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 g is nilpotent, then all subalgebras and homomorphic images are nilpotent.
• If the quotient algebra g/Z(g), where Z(g) is the center of g, is nilpotent, then so is g.
• Engel's theorem: A Lie algebra g is nilpotent if and only if all elements of g are ad-nilpotent.
• The Killing form of a nilpotent Lie algebra is 0.
• A nilpotent Lie algebra has an outer automorphism.