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 ${\mathfrak {nil}}({\mathfrak g})$ 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 ${\mathfrak {rad}}({\mathfrak {g}})$ of the Lie algebra ${\mathfrak {g}}$. The quotient of a Lie algebra by its nilradical is a reductive Lie algebra ${\mathfrak {g}}^{{{\mathrm {red}}}}$. However, the corresponding short exact sequence

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

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

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

does split (essentially because the quotient ${\mathfrak {g}}^{{{\mathrm {ss}}}}$ 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. 129. New York: Springer-Verlag. ISBN 978-0-387-97495-8. MR 1153249, ISBN 978-0-387-97527-6.
• 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.