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