Nilpotent cone

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

• Aoki, T.; Majima, H.; Takei, Y.; Tose, N. (2009), Algebraic Analysis of Differential Equations: from Microlocal Analysis to Exponential Asymptotics, Springer, p. 173, ISBN 9784431732402, https://books.google.com/books?id=oqx-p6N2XSEC&pg=PA173 .
• Anker, Jean-Philippe; Orsted, Bent (2006), Lie Theory: Unitary Representations and Compactifications of Symmetric Spaces, Progress in Mathematics, 229, Birkhäuser, p. 166, ISBN 9780817644307, https://books.google.com/books?id=nc__yHsnHFQC&pg=PA166 .

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