| Maximal Torus |
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| CATEGORIES ABOUT MAXIMAL TORUS | |
| lie groups | |
| representation theory of lie groups | |
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For example, the Lie group SO(3) of rotations in three dimensions has as maximal torus ''T'' a Circle Group (a 1-torus, that is). It can be taken to be the group of rotations about the x-axis, parametrised by angle. According to general theory, all the maximal tori form a single Conjugacy Class of subgroups. The related group SU(2) also has rank 1, with a rotation group as maximal torus. The conjugacy of maximal tori implies that all the maximal tori SO(3) are the rotations about some fixed axis — so that we have surveyed them all. In general SO(2''n'') and SO(2''n''+1) have rank ''n''. In those cases one can easily find explicit parameter angles for the maximal torus: that is, commuting one-parameter families of rotations exhibiting the torus as a product of circle groups. The Weyl Group ''W'' of ''G'' is the Normalizer of ''T'' in ''G'' modulo the Centralizer ; or in other words the group of transformations of ''T'' into itself carried out by conjugation in ''G''. The Representation Theory of ''G'', when it is a connected group at least, is essentially determined by ''T'' and W''. |
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