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
Canonical name	NilpotentCone
Date of creation	2013-03-22 13:58:36
Last modified on	2013-03-22 13:58:36
Owner	rmilson (146)
Last modified by	rmilson (146)
Numerical id	9
Author	rmilson (146)
Entry type	Definition
Classification	msc 17B20
Synonym	nilcone