nilpotent cone

Let 𝔤 be a finite dimensional semisimple Lie algebraMathworldPlanetmath. The nilpotent cone 𝒩 of 𝔤 is the set of elements that act nilpotently in all representations of 𝔤. In other words,

𝒩={a𝔤:ρ(a) is nilpotent for all representations ρ:𝔤End(V)}

The nilpotent cone is an irreducible ( subvariety ( of 𝔤 (considered as a k-vector spaceMathworldPlanetmath), and is invariant under the adjoint action of 𝔤 on itself.

Example: if 𝔤=sl2, then the nilpotent cone is the varietyPlanetmathPlanetmath of all matrices in 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