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.

References

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

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