# Cartan subalgebra

Let $\U0001d524$ be a Lie algebra^{}. Then a Cartan subalgebra^{} is a maximal subalgebra^{} of $\U0001d524$ which is self-normalizing, that is, if $[g,h]\in \U0001d525$ for all $h\in \U0001d525$, then $g\in \U0001d525$ as well. Any Cartan subalgebra $\U0001d525$ is nilpotent^{}, and if $\U0001d524$ is semi-simple^{}, it is abelian^{}. All Cartan subalgebras of a Lie algebra are conjugate by the adjoint action of any Lie group with algebra $\U0001d524$.

The dimension of $\U0001d525$ is called the rank of $\U0001d524$.

Title | Cartan subalgebra |
---|---|

Canonical name | CartanSubalgebra |

Date of creation | 2013-03-22 13:20:09 |

Last modified on | 2013-03-22 13:20:09 |

Owner | bwebste (988) |

Last modified by | bwebste (988) |

Numerical id | 7 |

Author | bwebste (988) |

Entry type | Definition |

Classification | msc 17B20 |

Defines | rank of a Lie algebra |