Nilpotent Lie algebra

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

Definition

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

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 $ad X n - 1 = 0$ for all $x \in g$ . 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 $ad X 1 ad X 2 \cdot\cdot\cdot ad X n - 1 = 0$ . 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, $ad x n = 0$ for all $x \in g$ . We call an element $x \in g$ 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.

Examples

If $gl (k, ℝ)$ is the set of $k \times k$ matrices with entries in $ℝ$ , then the subalgebra consisting of strictly upper triangular matrices is a nilpotent Lie algebra.
If a Lie algebra $g$ has an automorphism of prime period with no fixed points except at $0$ , then $g$ is nilpotent.
A Heisenberg algebra is nilpotent.
A Cartan subalgebra of a Lie algebra is nilpotent and self-normalizing.

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.

Notes

↑ Knapp 2002 Proposition 1.32.

References