العربية
中国
Français
Deutsch
Ελληνική
हिन्दी
Italiano
日本語
Português
Русский
EspañolCARTAN SUBGROUP
In mathematics, a 'Cartan subgroup' of a Lie group or algebraic group ''G'' is one of the subgroups whose Lie algebra
is a Cartan subalgebra. The dimension of a Cartan subgroup, and therefore of a Cartan subalgebra, is the 'rank' of ''G''.
The identity component of a subgroup has the same Lie algebra. There is no ''standard'' convention for which one of the subgroups with this property is called ''the'' Cartan subgroup, especially in the case of disconnected groups.
A 'Cartan subgroup' of a 'compact connected Lie group' is a maximal connected Abelian subgroup (a maximal torus). Its Lie algebra is a Cartan subalgebra.
For 'disconnected compact Lie groups' there are several inequivalent definitions of a Cartan subgroup. The most common seems to be the one given by David Vogan, who defines a Cartan subgroup to be the group of elements that normalize a fixed maximal torus and fix the fundamental Weyl chamber. This is sometimes called the 'large Cartan subgroup'. There is also a 'small Cartan subgroup', defined to be the centralizer of a maximal torus. These Cartan subgroups need not be abelian in general.
For 'connected algebraic groups' over an algebraically closed field a 'Cartan subgroup' is usually defined as the centralizer of a maximal torus. In this case the Cartan subgroups are connected, nilpotent, and are all conjugate.
★ Cartan subalgebra
★ Carter subgroup
★ Knapp, Vogan, ''Cohomological induction and unitary representations'', ISBN 0-691-03756-6
★ Borel, ''Linear algebraic groups'', ISBN 3-540-97370-2
is a Cartan subalgebra. The dimension of a Cartan subgroup, and therefore of a Cartan subalgebra, is the 'rank' of ''G''.
| Contents |
| Conventions |
| Definitions |
| See also |
| References |
Conventions
The identity component of a subgroup has the same Lie algebra. There is no ''standard'' convention for which one of the subgroups with this property is called ''the'' Cartan subgroup, especially in the case of disconnected groups.
Definitions
A 'Cartan subgroup' of a 'compact connected Lie group' is a maximal connected Abelian subgroup (a maximal torus). Its Lie algebra is a Cartan subalgebra.
For 'disconnected compact Lie groups' there are several inequivalent definitions of a Cartan subgroup. The most common seems to be the one given by David Vogan, who defines a Cartan subgroup to be the group of elements that normalize a fixed maximal torus and fix the fundamental Weyl chamber. This is sometimes called the 'large Cartan subgroup'. There is also a 'small Cartan subgroup', defined to be the centralizer of a maximal torus. These Cartan subgroups need not be abelian in general.
For 'connected algebraic groups' over an algebraically closed field a 'Cartan subgroup' is usually defined as the centralizer of a maximal torus. In this case the Cartan subgroups are connected, nilpotent, and are all conjugate.
See also
★ Cartan subalgebra
★ Carter subgroup
References
★ Knapp, Vogan, ''Cohomological induction and unitary representations'', ISBN 0-691-03756-6
★ Borel, ''Linear algebraic groups'', ISBN 3-540-97370-2
This article provided by Wikipedia. To edit the contents of this article, click here for original source.
psst.. try this: add to faves
