Nilpotent cone

Template:Inline In mathematics, the nilpotent cone ${𝒩}$ 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,

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

The nilpotent cone is an irreducible subvariety of $\mathfrak{𝔤}$ (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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This article incorporates material from Nilpotent cone on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.

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