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.

单词	MaximalTorus
释义	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.
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