Real Forms of a Complex Group

We can also find out what are all the real forms of a Lie group.

{COMMENT: I THINK AN EXPLANATION IS NEEDED FOR A WIDER AUDIENCE AS TO WHY WE TALK ABOUT REAL FORMS OF A NON-COMPLEX GROUP AND WHY THE REAL FORMS OF Sp(4,C) ARE DIFFERENT. I THINK THIS WAS NOT COVERED LATER IN THE VIDEOS}:

atlas> set G=Sp(4,R)
Identifier G: RealForm
atlas> set rf=real_forms(G)
Identifier rf: [RealForm]
atlas>
atlas> #rf
Value: 3
atlas> for H in rf do prints(H) od
compact connected real group with Lie algebra 'sp(2)'
connected real group with Lie algebra 'sp(1,1)'
connected split real group with Lie algebra 'sp(4,R)'
Value: [(),(),()]
atlas>

Note the for loop that uses the command prints discussed earlier in the section of basic atlas commands. It tells the software to go through the string of real forms of \(G\) and print the list.

So, for \(G=Sp(4,\mathbb R)\), we have a list of three real forms: the compact real form \(Sp(2)=Sp(2,0)\), an intermediate real form \(Sp(1,1)\) and the split real form \(Sp(4,\mathbb R)\). One corresponding to each Lie algebra above.

Cartan Classes Accross Real Forms

Now something interesting about real forms and Cartan classes is that The Cartan classes are the same across all real forms of the same complex group. In fact even though within the compact real form there is only one conjugacy class of Cartans, we will have the same Cartan classes for \(G=Sp(2)\) than for \(G=Sp(4,\mathbb R)\).:

atlas> rf[0]
Value: compact connected real group with Lie algebra 'sp(2)'
atlas> nr_of_Cartan_classes (rf[0])
Value: 4
atlas>set Gc=rf[0]
Identifier Gc: RealForm
atlas> Gc
Value: compact connected real group with Lie algebra 'sp(2)'
atlas> Cartan_classes (Gc)[0]
Value: Cartan class #0, occurring for 3 real forms and for 1 dual real form
atlas> Cartan_classes (Gc)[1]
Value: Cartan class #1, occurring for 2 real forms and for 1 dual real form
atlas> Cartan_classes (Gc)[3]
Value: Cartan class #3, occurring for 1 real form and for 3 dual real forms
atlas>

Inner Classes and Outer Involutions

Now let’s look at the real forms of \(G=SL(5,\mathbb R)\). It turns out there is only one: \(G\) itself:

atlas> set G=SL(5,R)
Identifier G: RealForm (hiding previous one of type RealForm)
atlas> set rf =real_forms(G)
Identifier rf: [RealForm] (hiding previous one of type [RealForm])
atlas>
atlas> #rf
Value: 1
atlas> rf[0]
Value: connected split real group with Lie algebra 'sl(5,R)'
atlas>

Note that the complexified Lie algebra of \(G\), \(\mathfrak{sl} (5,\mathbb C)\) has other real forms. However, associated to \(G\), there is only one. This is because atlas thinks always in terms of inner classes. That is, a real form is a real form in a given inner class. And there is a distinquished involution in \(G\) which is non trivial and which atlas uses to determine the inner class of the group \(G\). It turns out there is only one real form in this inner class.:

atlas> distinguished_involution(G)
Value:
| 1, 0, 0, 0 |
| 1, 0, 0, -1 |
| 1, 0, -1, 0 |
| 1, -1, 0, 0 |
atlas>

Let’s look at another real form of type A4, namely \(SU(3,2)\). THis gives us another inner class. This inner class has the rest of the real forms of \(\mathfrak{sl}(5,\mathbb C)\):

atlas> H:=SU(3,2)
Value: connected quasisplit real group with Lie algebra 'su(3,2)'
atlas> set rfH=real_forms (H)
Identifier rfH: [RealForm]
atlas> #rfH
Value: 3
atlas> for a in rfH do prints(a) od
compact connected real group with Lie algebra 'su(5)'
connected real group with Lie algebra 'su(4,1)'
connected quasisplit real group with Lie algebra 'su(3,2)'
Value: [(),(),()]
atlas>

So, these three real groups are grouped together because they are all in the same inner class. Note that they all have a compact Cartan, whereas \(SL(5,\mathbb R)\) does not. Also, the same result is achieved if we choose \(H\) to be either of the other two groups in the above list. The distinguished involution for \(H=SU(3,2)\) is the identity:

atlas> distinguished_involution (H)
Value:
| 1, 0, 0, 0 |
| 0, 1, 0, 0 |
| 0, 0, 1, 0 |
| 0, 0, 0, 1 |
atlas>

Inner classes are associated with outer involutions; that is, with a diagram automorphism. So, for \(SU(p,q)\) the inner forms are associated with the trivial automorphism of the Dynkin diagram of A{p+q-1}; and for \(SL(p+q,\mathbb R)\), with the non trivial diagram automorphism.

So what will happen for \(G=SL(6,\mathbb R)\)? We can see in the following example that there is another real form in the same inner class as \(SL(6,\mathbb R)\), namely \(SL(3,\mathbb H)\). This will be true whenever p+q=2n.:

atlas> set G=SL(6,R)
Identifier G: RealForm (hiding previous one of type RealForm)
atlas> set rf=real_forms (G)
Identifier rf: [RealForm] (hiding previous one of type [RealForm])
atlas> #rf
Value: 2
atlas>
atlas> for a in rf do prints(a) od
connected real group with Lie algebra 'sl(3,H)'
connected split real group with Lie algebra 'sl(6,R)'
Value: [(),()]
atlas>

More generally, for \(SL(2n,\mathbb R)\) has another real form in its inner class, the group SL(n,H).

In fact the distinguished involution is the one attached to the non trivial diagram automorphism. And in this case it flips all the roots except the central root:

atlas> set delta=distinguished_involution (G)
Identifier delta: mat
atlas> delta
Value:
| 1, 0, 0, 0, 0 |
| 1, 0, 0, 0, -1 |
| 1, 0, 0, -1, 0 |
| 1, 0, -1, 0, 0 |
| 1, -1, 0, 0, 0 |
atlas>
atlas> simple_roots (G)
Value:
| 1, 0, 0, 0, 1 |
| -1, 1, 0, 0, 1 |
| 0, -1, 1, 0, 1 |
| 0, 0, -1, 1, 1 |
| 0, 0, 0, -1, 2 |
atlas> delta*simple_roots (G)
Value:
| 1, 0, 0, 0, 1 |
| 1, 0, 0, 1, -1 |
| 1, 0, 1, -1, 0 |
| 1, 1, -1, 0, 0 |
| 2, -1, 0, 0, 0 |
atlas>

In other words, delta is the outer automorphism that exchanges alpha[0] and alpha[4]; alpha[1] and alpha[3] and fixes alpha[2].

This gives another way of thinking about coordinates. That is, trying to understand the automorphism delta by looking at what it does to the simple roots.