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 x_i 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