Wavelet transform

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