Nilradical of a Lie algebra

Template:No footnotes In algebra, the nilradical of a Lie algebra is a nilpotent ideal, which is as large as possible.

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

$0\to 𝔫𝔦𝔩(𝔤)\to 𝔤\to {𝔤}^{\mathrm{r}\mathrm{e}\mathrm{d}}\to 0$

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

$0\to 𝔯𝔞𝔡(𝔤)\to 𝔤\to {𝔤}^{\mathrm{s}\mathrm{s}}\to 0$

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

See also

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

References

• Template:Fulton-Harris
• Template:Citation.