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

• 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.
• Anker, Jean-Philippe; Orsted, Bent (2006), Lie Theory: Unitary Representations and Compactifications of Symmetric Spaces, Progress in Mathematics 229, Birkhäuser, p. 166, ISBN 9780817644307.

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