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Nilradical of a Lie algebra

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

The nilradical n i l ( g ) {\displaystyle {\mathfrak {nil}}({\mathfrak {g}})} of a finite-dimensional Lie algebra g {\displaystyle {\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 r a d ( g ) {\displaystyle {\mathfrak {rad}}({\mathfrak {g}})} of the Lie algebra g {\displaystyle {\mathfrak {g}}} . The quotient of a Lie algebra by its nilradical is a reductive Lie algebra g r e d {\displaystyle {\mathfrak {g}}^{\mathrm {red} }} . However, the corresponding short exact sequence

0 → n i l ( g ) → g → g r e d → 0 {\displaystyle 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 n i l ( g ) {\displaystyle {\mathfrak {nil}}({\mathfrak {g}})} in g {\displaystyle {\mathfrak {g}}} ). This is in contrast to the Levi decomposition: the short exact sequence

0 → r a d ( g ) → g → g s s → 0 {\displaystyle 0\to {\mathfrak {rad}}({\mathfrak {g}})\to {\mathfrak {g}}\to {\mathfrak {g}}^{\mathrm {ss} }\to 0}

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

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