Nilpotent cone

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

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

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

Example

The nilpotent cone of $\operatorname{sl}_2$ , the Lie algebra of 2×2 matrices with vanishing trace, is the variety of all 2×2 traceless matrices with rank less than or equal to $1.$

This article incorporates material from Nilpotent cone on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.