Nilpotent cone

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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), and is invariant under the adjoint action of $\mathfrak{g}$ on itself.

[edit] 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 matrices with rank less than or equal to $1.$