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

$0\to \mathfrak{𝔫}\mathfrak{𝔦}\mathfrak{𝔩}(\mathfrak{𝔤})\to \mathfrak{𝔤}\to {\mathfrak{𝔤}}^{\mathrm{r}\mathrm{e}\mathrm{d}}\to 0$

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

$0\to \mathfrak{𝔯}\mathfrak{𝔞}\mathfrak{𝔡}(\mathfrak{𝔤})\to \mathfrak{𝔤}\to {\mathfrak{𝔤}}^{\mathrm{s}\mathrm{s}}\to 0$

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

See also

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

References

• 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.