Nilpotent Lie algebra

Lie groups and Lie algebras
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Scientists Sophus Lie Henri Poincaré Wilhelm Killing Élie Cartan Hermann Weyl Claude Chevalley Harish-Chandra Armand Borel
Glossary Table of Lie groups
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In mathematics, a Lie algebra ${\displaystyle {\mathfrak {g}}}$ is nilpotent if its lower central series terminates in the zero subalgebra. The lower central series is the sequence of subalgebras

${\displaystyle {\mathfrak {g}}\geq [{\mathfrak {g}},{\mathfrak {g}}]\geq [{\mathfrak {g}},[{\mathfrak {g}},{\mathfrak {g}}]]\geq [{\mathfrak {g}},[{\mathfrak {g}},[{\mathfrak {g}},{\mathfrak {g}}]]]\geq ...}$

We write ${\displaystyle {\mathfrak {g}}_{0}={\mathfrak {g}}}$, and ${\displaystyle {\mathfrak {g}}_{n}=[{\mathfrak {g}},{\mathfrak {g}}_{n-1}]}$ for all ${\displaystyle n>0}$. If the lower central series eventually arrives at the zero subalgebra, then the Lie algebra is called nilpotent. The lower central series for Lie algebras is analogous to the lower central series in group theory, and nilpotent Lie algebras are analogs of nilpotent groups.

The nilpotent Lie algebras are precisely those that can be obtained from abelian Lie algebras, by successive central extensions.

Note that the definition means that, viewed as a non-associative non-unital algebra, a Lie algebra ${\displaystyle {\mathfrak {g}}}$ is nilpotent if it is nilpotent as an ideal.