Cartan subgroup
Encyclopedia
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.
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.
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.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a Cartan subgroup of a Lie group
Lie group
In mathematics, a Lie group is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure...
or algebraic group
Algebraic group
In algebraic geometry, an algebraic group is a group that is an algebraic variety, such that the multiplication and inverse are given by regular functions on the variety...
G is one of the subgroups whose Lie algebra
Lie algebra
In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. Lie algebras were introduced to study the concept of infinitesimal transformations. The term "Lie algebra" was introduced by Hermann Weyl in the...
is a Cartan subalgebra. The dimension of a Cartan subgroup, and therefore of a Cartan subalgebra, is the rank of G.
Conventions
The identity componentIdentity component
In mathematics, the identity component of a topological group G is the connected component G0 of G that contains the identity element of the group...
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 groupLie group
In mathematics, a Lie group is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure...
is a maximal connected Abelian subgroup (a maximal torus
Maximal torus
In the mathematical theory of compact Lie groups a special role is played by torus subgroups, in particular by the maximal torus subgroups.A torus in a Lie group G is a compact, connected, abelian Lie subgroup of G . A maximal torus is one which is maximal among such subgroups...
). 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
David Vogan
David Alexander Vogan is a mathematician at M.I.T. who works on unitary representations of simple Lie groups. He is one of the participants in the Atlas of Lie Groups and Representations.-Publications:*Some online by Vogan...
, who defines a Cartan subgroup to be the group of elements that normalize a fixed maximal torus
Maximal torus
In the mathematical theory of compact Lie groups a special role is played by torus subgroups, in particular by the maximal torus subgroups.A torus in a Lie group G is a compact, connected, abelian Lie subgroup of G . A maximal torus is one which is maximal among such subgroups...
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.