# nilpotent cone

Let $\mathfrak{g}$ be a finite dimensional semisimple Lie algebra. The nilpotent cone $\mathcal{N}$ of $\mathfrak{g}$ is the set of elements that act nilpotently in all representations of $\mathfrak{g}$. In other words,

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

The nilpotent cone is an irreducible (http://planetmath.org/IrreducibleClosedSet) subvariety (http://planetmath.org/AffineVariety) of $\mathfrak{g}$ (considered as a $k$-vector space), and is invariant under the adjoint action of $\mathfrak{g}$ on itself.

Example: if $\mathfrak{g}=\operatorname{sl}_{2}$, then the nilpotent cone is the variety of all matrices in $\mathfrak{g}$ with rank $1$.

Title nilpotent cone NilpotentCone 2013-03-22 13:58:36 2013-03-22 13:58:36 rmilson (146) rmilson (146) 9 rmilson (146) Definition msc 17B20 nilcone