nilpotent cone

Let $𝔤$ be a finite dimensional semisimple Lie algebra. Thenilpotent cone $𝒩$ of $𝔤$ is the set of elements thatact nilpotently in all representations of $𝔤$. In other words,

$$𝒩=\{a\in 𝔤:\rho (a)\text{ is nilpotent for all representations }\rho :𝔤\to \mathrm{End}(V)\}$$

The nilpotent cone is an irreducible (http://planetmath.org/IrreducibleClosedSet)subvariety (http://planetmath.org/AffineVariety) of $𝔤$ (considered as a$k$-vector space), and is invariant under the adjoint action of $𝔤$on itself.

Example: if $𝔤={\mathrm{sl}}_2$, then the nilpotent coneis the variety of all matrices in $𝔤$ with rank $1$.