# nilpotent cone

Let $\U0001d524$ be a finite dimensional semisimple Lie algebra^{}. The
*nilpotent cone* $\mathcal{N}$ of $\U0001d524$ is the set of elements that
act nilpotently in all representations of $\U0001d524$. In other words,

$$\mathcal{N}=\{a\in \U0001d524:\rho (a)\text{is nilpotent for all representations}\rho :\U0001d524\to \mathrm{End}(V)\}$$ |

The nilpotent cone is an irreducible (http://planetmath.org/IrreducibleClosedSet)
subvariety (http://planetmath.org/AffineVariety) of $\U0001d524$ (considered as a
$k$-vector space^{}), and is invariant under the adjoint action of $\U0001d524$
on itself.

Example: if $\U0001d524={\mathrm{sl}}_{2}$, then the nilpotent cone
is the variety^{} of all matrices in $\U0001d524$ 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 |