Nilradical of a Lie algebra explained

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

The nilradical

ak{nil}( ak g) of a finite-dimensional Lie algebra ak{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 ak{rad}( ak{g}) of the Lie algebra ak{g} . The quotient of a Lie algebra by its nilradical is a reductive Lie algebra ak{g} ^{red} . However, the corresponding short exact sequence0 \to ak{nil}( ak g)\to ak g\to ak{g} ^{red} \to 0 does not split in general (i.e., there isn't always a subalgebra complementary to ak{nil}( ak g) in ak{g} ). This is in contrast to the Levi decomposition: the short exact sequence0 \to ak{rad}( ak g)\to ak g\to ak{g} ^{ss} \to 0 does split (essentially because the quotient ak{g} ^{ss} is semisimple).

See also

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

References

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