# 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 $T=\varphi^{-1}(t)$ where $\varphi:K\to K$ is the map $k\mapsto ktk^{-1}$, 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 MaximalTorus 2013-03-22 13:23:52 2013-03-22 13:23:52 bwebste (988) bwebste (988) 5 bwebste (988) Definition msc 22E10 msc 22C05