Nilradical of a Lie algebra

This article includes a list of references, related reading, or external links, but its sources remain unclear because it lacks inline citations. Please help improve this article by introducing more precise citations. (December 2023) (Learn how and when to remove this message)

In algebra, the nilradical of a Lie algebra is a nilpotent ideal, which is as large as possible.

The nilradical ${\displaystyle {\mathfrak {nil}}({\mathfrak {g}})}$ of a finite-dimensional Lie algebra ${\displaystyle {\mathfrak {g}}}$ is its maximal nilpotent ideal, which exists because the sum of any two nilpotent ideals is nilpotent. It is an ideal in the radical ${\displaystyle {\mathfrak {rad}}({\mathfrak {g}})}$ of the Lie algebra ${\displaystyle {\mathfrak {g}}}$. The quotient of a Lie algebra by its nilradical is a reductive Lie algebra ${\displaystyle {\mathfrak {g}}^{\mathrm {red} }}$. However, the corresponding short exact sequence

${\displaystyle 0\to {\mathfrak {nil}}({\mathfrak {g}})\to {\mathfrak {g}}\to {\mathfrak {g}}^{\mathrm {red} }\to 0}$

does not split in general (i.e., there isn't always a subalgebra complementary to ${\displaystyle {\mathfrak {nil}}({\mathfrak {g}})}$ in ${\displaystyle {\mathfrak {g}}}$). This is in contrast to the Levi decomposition: the short exact sequence

${\displaystyle 0\to {\mathfrak {rad}}({\mathfrak {g}})\to {\mathfrak {g}}\to {\mathfrak {g}}^{\mathrm {ss} }\to 0}$

does split (essentially because the quotient ${\displaystyle {\mathfrak {g}}^{\mathrm {ss} }}$ is semisimple).

• Levi decomposition
• Nilradical of a ring, a notion in ring theory.

• Fulton, William; Harris, Joe (1991). Representation theory. A first course. Graduate Texts in Mathematics, Readings in Mathematics. Vol. 129. New York: Springer-Verlag. doi:10.1007/978-1-4612-0979-9. ISBN 978-0-387-97495-8. MR 1153249. OCLC 246650103.
• Onishchik, Arkadi L.; Vinberg, Ėrnest Borisovich (1994), Lie Groups and Lie Algebras III: Structure of Lie Groups and Lie Algebras, Springer, ISBN 978-3-540-54683-2.