Cartan Subgroups
=================

When we give atlas a specific Lie group :math:`G`, the software knows
several characteristics of the group. And depending on what you ask
about :math:`G`, ``atlas`` will provide more information::

     atlas> G:=SL(2,R)
     Value: connected split real group with Lie algebra 'sl(2,R)'
     atlas>  H:=PSL(2,R)
     Value: disconnected split real group with Lie algebra 'sl(2,R)'
     atlas>

Recall that :math:`H\cong SO(2,1)`. 

Now, one of the main structural facts about a group is what its Cartan subgroups are. For that we can type the following sequence of commands::

     atlas> nr_of_Cartan_classes (G)
     Value: 2
     atlas> set cartans =Cartan_classes (G)
     Identifier cartans: [CartanClass] 
     atlas> #cartans
     Value: 2
     atlas> set T=cartans[0]
     Identifier T: CartanClass
     atlas>
     atlas> T
     Value: Cartan class #0, occurring for 2 real forms and for 1 dual real form
     atlas>
     atlas> print_Cartan_info (T)
     compact: 1, complex: 0, split: 0
     canonical twisted involution: e
     twisted involution orbit size: 1; fiber size: 2; strong inv: 2
     imaginary root system: A1
     real root system: empty
     complex factor: empty
     atlas>

Wich gives us the number of conjugacy classes of Cartan subgroups of :math:`G`
and choosing one of those subgroups atlas gives more information about
the Cartan subgroup. The function ``print_Cartan_info`` takes a Cartan class in
:math:`G` and provides the basic structural data of any representative
in the Cartan class.

So this says that the first Cartan subgroup is a real cartan subgroup, meaning
a connected complex torus defined over :math:`\mathbb R`. So, its real
points form a real torus which can be written as a product of
:math:`(S^1)^a`, :math:`({\mathbb C}^{\times })^b` and
:math:`({\mathbb R}^{\times })^c` factors. So, atlas gives the numbers
``(a,b,c)``. In this case the first Cartan subgroup has just ``a=1`` circle
factor, ``b=0`` complex factors and ``c=0`` real factors.

It also tells us the type of roots it has: imaginary, complex or
real. Since the Cartan subgroup is compact, we only have imaginary roots. And
these roots form a system of type ``A1``.

Note that we also see that this Cartan subgroup occurs in different real forms
of complex groups of type ``A1``. The information about twisted involutions will be discussed later.

Now for information about the second Cartan subgroup::

    atlas> set A=cartans[1]
    Identifier A: CartanClass
    atlas> A
    Value: Cartan class #1, occurring for 1 real form and for 2 dual real forms
    atlas> print_Cartan_info (A)
    compact: 0, complex: 0, split: 1
    canonical twisted involution: 1
    twisted involution orbit size: 1; fiber size: 1; strong inv: 1
    imaginary root system: empty
    real root system: A1
    complex factor: empty
    atlas>

Here, the Cartan subgroup has just 1 copy of :math:`{\mathbb R}^{\times }` and
no complex or circle factors. And it just has real roots of type
``A1``. We will discuss later the information about involutions.

In contrast, the group :math:`SL(2,\mathbb C)` has only one conjugacy
class of cartans::

   atlas> set G2=SL(2,C)
   Identifier G2: RealForm (hiding previous one of type string)
   atlas> nr_of_Cartan_classes (G2)
   Value: 1
   atlas> set cartans =Cartan_classes (G2)
   Identifier cartans: [CartanClass] (hiding previous one of type [CartanClass])
   atlas> cartans[0]
   Value: Cartan class #0, occurring for 1 real form and for 1 dual real form
   atlas>
   atlas> print_Cartan_info (cartans[0])
   compact: 0, complex: 1, split: 0
   canonical twisted involution: e
   twisted involution orbit size: 2; fiber size: 1; strong inv: 2
   imaginary root system: empty
   real root system: empty
   complex factor: A1

Now for a larger group like :math:`Sp(4,\mathbb R)`, for example, we
will have a compact cartan which is a product of two circles and all
its roots are imaginary; a split cartan, that is, a product of
:math:`({\mathbb R}^{\times })×({\mathbb R}^{\times })` with all roots
real; and two intermediate cartans; one complex isomorphic to
:math:`{\mathbb C}^{\times }`. This is sometimes called the short root
Cartan. This is the one associated to a Levi factor :math:`Gl(2)`.
Finally, the other Cartan subgroup is isomorphic to
:math:`S^1×{\mathbb R}^{\times }`. The distinction between these two
Cartan subgroups is subtle. Locally they are both isomorphic rank one
Cartan subgroups and look like :math:`S^1×{\mathbb R}^{\times }`. But, one is
:math:`{\mathbb C}^{\times }` and ``atlas`` can distinguish the two.

The root systems of these intermediate Cartan subgroups also transform accordingly. 
For the Compact Cartan subgroup we have an imaginary root system of type ``C2``::

    atlas> set G1=Sp(4,R)
    Identifier G1: RealForm
    atlas> G1
    Value: connected split real group with Lie algebra 'sp(4,R)'
    atlas> nr_of_Cartan_classes (G1)
    Value: 4
    atlas> set cartans =Cartan_classes (G1)
    Identifier cartans: [CartanClass] (hiding previous one of type [CartanClass])
    atlas>
    atlas> print_Cartan_info (cartans[0])
    compact: 2, complex: 0, split: 0
    canonical twisted involution: e
    twisted involution orbit size: 1; fiber size: 4; strong inv: 4
    imaginary root system: C2
    real root system: empty
    complex factor: empty
    
Now for the most split Cartan subgroup, the last one, all of the roots are real::

    atlas> print_Cartan_info (cartans[3])
    compact: 0, complex: 0, split: 2
    canonical twisted involution: 2,1,2,1
    twisted involution orbit size: 1; fiber size: 1; strong inv: 1
    imaginary root system: empty
    real root system: C2
    complex factor: empty
    atlas>

For the complex intermidiate Cartan subgroup, we have an imaginary root system and a real root system, both of type ``A1``::

    atlas> cartans[1]
    Value: Cartan class #1, occurring for 2 real forms and for 1 dual real form
    atlas> print_Cartan_info (cartans[1])
    compact: 0, complex: 1, split: 0
    canonical twisted involution: 2,1,2
    twisted involution orbit size: 2; fiber size: 1; strong inv: 2
    imaginary root system: A1
    real root system: A1
    complex factor: empty
    atlas>

Lastly, the other intermidiate Cartan subgroup has also an imaginary and a real root system of type ``A1``::

    atlas> cartans[2]
    Value: Cartan class #2, occurring for 1 real form and for 2 dual real forms
    atlas> print_Cartan_info (cartans[2])
    compact: 1, complex: 0, split: 1
    canonical twisted involution: 1,2,1
    twisted involution orbit size: 2; fiber size: 2; strong inv: 4
    imaginary root system: A1
    real root system: A1
    complex factor: empty
    atlas>

So the distinction between these last two is burried in the extra information. More about this later.