Nilradical of a Lie algebra

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

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

[math]\displaystyle{ 0 \to \mathfrak{nil}(\mathfrak g)\to \mathfrak g\to \mathfrak{g}^{\mathrm{red}}\to 0 }[/math]

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

[math]\displaystyle{ 0 \to \mathfrak{rad}(\mathfrak g)\to \mathfrak g\to \mathfrak{g}^{\mathrm{ss}}\to 0 }[/math]

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

See also

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

References

• Fulton, William; Harris, Joe (1991) (in en-gb). Representation theory. A first course. Graduate Texts in Mathematics, Readings in Mathematics. 129. New York: Springer-Verlag. doi:10.1007/978-1-4612-0979-9. ISBN 978-0-387-97495-8. OCLC 246650103. https://link.springer.com/10.1007/978-1-4612-0979-9. 
• 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 .