Nilpotent cone

In mathematics, the nilpotent cone ${\displaystyle {\mathcal {N}}}$ of a finite-dimensional semisimple Lie algebra ${\displaystyle {\mathfrak {g}}}$ is the set of elements that act nilpotently in all representations of ${\displaystyle {\mathfrak {g}}.}$ In other words,

${\displaystyle {\mathcal {N}}=\{a\in {\mathfrak {g}}:\rho (a){\mbox{ is nilpotent for all representations }}\rho$ :{\mathfrak {g}}\to \operatorname {End} (V)\}.}

The nilpotent cone is an irreducible subvariety of ${\displaystyle {\mathfrak {g}}}$ (considered as a vector space).