# maximal torus

Let $K$ be a compact group, and let $t\in K$ be an element whose centralizer^{} has minimal dimension^{} (such elements are dense in $K$). Let $T$ be the centralizer of $t$. This subgroup^{} is closed since $T={\phi}^{-1}(t)$ where $\phi :K\to K$ is the map $k\mapsto kt{k}^{-1}$, and abelian^{} since it is the intersection of $K$ with the Cartan subgroup of its complexification, and hence a torus, since $K$ (and thus $T$) is compact. We call $T$ a maximal torus of $K$.

This term is also applied to the corresponding maximal abelian subgroup of a complex semisimple group, which is an algebraic torus.

Title | maximal torus |
---|---|

Canonical name | MaximalTorus |

Date of creation | 2013-03-22 13:23:52 |

Last modified on | 2013-03-22 13:23:52 |

Owner | bwebste (988) |

Last modified by | bwebste (988) |

Numerical id | 5 |

Author | bwebste (988) |

Entry type | Definition |

Classification | msc 22E10 |

Classification | msc 22C05 |