Nilpotent cone

Template:Inline In mathematics, the nilpotent cone $𝒩$ of a finite-dimensional semisimple Lie algebra $𝔤$ is the set of elements that act 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 subvariety of $𝔤$ (considered as a vector space).

Example

The nilpotent cone of ${\mathrm{sl}}_2$, 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 $1.$

References

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