The "components" command gives a description of the group of connected
components of a real reductive group. This group is of the form (Z/2Z)^n
for n less than or equal to the rank.
If G is semisimple and simply connected then the real points of G is
connected. If G is a torus the component group is (Z/2Z)^n where n is
the number of R^x factors.
Example 1: The number of R^* factors of a torus is the number of "s"
terms if the torus is defined without taking a quotient, but not
otherwise.
The real form S^1xR^* of C^*xC^* has component group Z/2Z:
real: type
Lie type: T1.T1
elements of finite order in the center of the simply connected group:
Q/Z.Q/Z
enter kernel generators, one per line
(ad for adjoint, ? to abort):
enter inner class(es): cs
main: components
there is a unique real form: u(1).gl(1,R)
component group is (Z/2)^1
The cartan command exhibits the structure of this real torus:
real: cartan
Name an output file (return for stdout, ? to abandon):
Cartan #0:
split: 1; compact: 1; complex: 0
[other output deleted]
On the other hand a finite quotient of the same complex torus gives a
different result.
empty: type
Lie type: T1.T1
elements of finite order in the center of the simply connected group:
Q/Z.Q/Z
enter kernel generators, one per line
(ad for adjoint, ? to abort):
1/2,1/2
enter inner class(es): sc
main: components
there is a unique real form: gl(1,R).u(1)
real group is connected
real: cartan
Name an output file (return for stdout, ? to abandon):
Cartan #0:
split: 0; compact: 0; complex: 1
[other output deleted]
Example 2: Components of real forms of A1.A1 in the complex inner class
The simply connected complex group SL(2,C)xSL(2,C) has a real form
SL(2,C), which is connected. The corresponding real form of the
adjoint group PSL(2,C)xPSL(2,C) is PSL(2,C), which is also
connected. However the intermediate real form of
SL(2,C)xSL(2,C)/<(-I,-I)> is isomorphic to SO(3,1) (of type D2=A1.A1),
and is disconnected.
real: type
Lie type: A1.A1 sc C
main: components
there is a unique real form: sl(2,C)
real group is connected
real: type
Lie type: A1.A1 ad C
main: components
there is a unique real form: sl(2,C)
real group is connected
real: type
Lie type: A1.A1
elements of finite order in the center of the simply connected group:
Z/2.Z/2
enter kernel generators, one per line
(ad for adjoint, ? to abort):
1/2,1/2
enter inner class(es): C
main: components
there is a unique real form: sl(2,C)
component group is (Z/2)^1