PlanetMath: maximal torus

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maximal torus

(Definition)

Let be a compact group, and let $t\in K$ be an element whose centralizer has minimal dimension (such elements are dense in ). Let be the centralizer of . 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 with the Cartan subgroup of its complexification, and hence a torus, since (and thus ) is compact. We call a maximal torus of .

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

"maximal torus" is owned by bwebste.

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Cross-references: algebraic torus, semisimple group, complex, term, torus, complexification, intersection, abelian, map, closed, subgroup, dense in, dimension, minimal, centralizer, group, compact
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This is version 2 of maximal torus, born on 2003-01-28, modified 2003-08-21.
Object id is 3937, canonical name is MaximalTorus.
Accessed 2560 times total.

Classification:

AMS MSC:	22E10 (Topological groups, Lie groups :: Lie groups :: General properties and structure of complex Lie groups)
	22C05 (Topological groups, Lie groups :: Compact groups)

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