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 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 $$1.$$