maximal torus

Let K be a compact group, and let tK be an element whose centralizerMathworldPlanetmathPlanetmathPlanetmath has minimal dimensionPlanetmathPlanetmath (such elements are dense in K). Let T be the centralizer of t. This subgroupMathworldPlanetmathPlanetmath is closed since T=φ-1(t) where φ:KK is the map kktk-1, and abelianMathworldPlanetmath 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