From we may define a positive subset
by choosing the first set of indices from a basis of
, and then the next set from a basis of
. Since there are no real roots, the roots in
have at least one non-vanishing component along
, and the first non-zero one of
these components is strictly positive. Since
on
, and since there are no real roots:
. Thus
permutes the simple roots, fixes the imaginary roots and exchanges in
2-tuples the complex roots: it constitutes an involutive automorphism of the Dynkin diagram of
.
A Vogan diagram is associated to the triple as follows. It corresponds to the standard
Dynkin diagram of
, with additional information: the 2-element orbits under
are exhibited by
joining the correponding simple roots by a double arrow and the 1-element orbit is painted in black
(respectively, not painted), if the corresponding imaginary simple root is noncompact (respectively,
compact).
The complex Lie algebra can be represented as the algebra of traceless
complex matrices,
the Lie bracket being the usual commutator. It has dimension 24. In principle, in order to compute the
Killing form, one needs to handle the
matrices of the adjoint representation. Fortunately, the
uniqueness (up to an overall factor) of the bi-invariant quadratic form on a simple Lie algebra leads to the
useful relation
A Cartan–Weyl basis is obtained by taking the 20 nilpotent generators (with
)
corresponding to matrices, all elements of which are zero except the one located at the intersection of row
and column
, which is equal to 1,
The root space is easily described by introducing the five linear forms , acting on diagonal matrices
as follows:
By restricting ourselves to real combinations of these generators we obtain the real Lie algebra .
The conjugation
that it defines on
is just the usual complex conjugation. This
constitutes the split real form
of
. Applying the construction given in
Equation (6.45
) to the generators of
, we obtain the set of antihermitian matrices
The real forms of that are not isomorphic to
or
are isomorphic
either to
or
. In terms of matrices these algebras can be represented as
The Dynkin diagram of is of
type (see Figure 26
).
Let us first consider an subalgebra. Diagonal matrices define a Cartan subalgebra whose all
elements are compact. Accordingly all associated roots are imaginary. If we define the positive roots using
the natural ordering
, the simple roots
,
,
are compact but
is noncompact. The corresponding Vogan diagram is
displayed in Figure 27
.
However, if instead of the natural order we define positive roots by the rule , the
simple positive roots are
and
which are compact, and
and
which are noncompact. The associated Vogan diagram is shown in Figure 28
.
Alternatively, the choice of order leads to the diagram in Figure 29
.
There remain seven other possibilities, all describing the same subalgebra . These are displayed
in Figure 30
.
In a similar way, we obtain four different Vogan diagrams for , displayed in Figure 31
.
Finally we have two non-isomorphic Vogan diagrams associated with and
. These are
shown in Figure 32
.
As we just saw, the same real Lie algebra may yield different Vogan diagrams only by changing the
definition of positive roots. But fortunately, a theorem of Borel and de Siebenthal tells us that we may
always choose the simple roots such that at most one of them is noncompact [129]. In other words, we may
always assume that a Vogan diagram possesses at most one black dot.
Furthermore, assume that the automorphism associated with the Vogan diagram is the identity (no
complex roots). Let be the basis of simple roots and
its dual basis, i.e.,
.
Then the single painted simple root
may be chosen so that there is no
with
.
This remark, which limits the possible simple root that can be painted, is particularly helpful when
analyzing the real forms of the exceptional groups. For instance, from the Dynkin diagram of
(see Figure 33
), it is easy to compute the dual basis and the matrix of scalar products
.
We obtain
from which we see that there exist, besides the compact real form, only two other non-isomorphic real forms ofLet us now illustrate the Cayley transformations. For this purpose, consider again with the
imaginary diagonal matrices as Cartan subalgebra and the natural ordering of the
defining the positive
roots. As we have seen,
is an imaginary noncompact root. The associated
generators are
Let us first consider the Cayley transformation obtained using, for instance, the real root .
An associated root vector, belonging to
, reads
If we consider instead the imaginary roots, we find for instance that is a noncompact
complex root vector corresponding to the root
. It provides the noncompact generator
which, together with
We have seen that every real Lie algebra leads to a Vogan diagram. Conversely, every Vogan diagram
defines a real Lie algebra. We shall sketch the reconstruction of the real Lie algebras from the Vogan
diagrams here, referring the reader to [129] for more detailed information.
Given a Vogan diagram, the reconstruction of the associated real Lie algebra proceeds as
follows. From the diagram, which is a Dynkin diagram with extra information, we may first
construct the associated complex Lie algebra, select one of its Cartan subalgebras and build
the corresponding root system. Then we may define a compact real subalgebra according to
Equation (6.45).
We know the action of on the simple roots. This implies that the set
of all roots is invariant
under
. This is proven inductively on the level of the roots, starting from the simple roots (level 1).
Suppose we have proven that the image under
of all the positive roots, up to level
are in
. If
is a root of level
, choose a simple root
such that
. Then the Weyl transformed
root
is also a positive root, but of smaller level. Since
and
are then
known to be in
, and since the involution acts as an isometry,
is also in
.
One can transfer by duality the action of on
to the Cartan subalgebra
, and
then define its action on the root vectors associated to the simple roots according to the rules
This involution is such that , with
27.
Furthermore it globally fixes
,
. Let
and
be the
or
eigenspaces
of
in
. Define
and
so that
. Set
We shall exemplify the reconstruction of real algebras from Vogan diagrams by considering two examples of
real forms of . The diagrams are shown in Figure 35
.
The involutions they describe are (the upper signs correspond to the left-hand side diagram, the
lower signs to the right-hand side diagram):
Doing the same exercise for the second diagram, we obtain the real algebra with
, which is a 10-parameter compact subalgebra, and
given by
The following tables provide all real simple Lie algebras and the corresponding Vogan diagrams. The
restrictions imposed on some of the Lie algebra parameters eliminate the consideration of isomorphic
algebras. See [129] for the derivation.
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Using these diagrams, the matrix defined by Equation (6.93
), and the three matrices
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For completeness we remind the reader of the definitions of matrix algebras ( has been defined
in Equation (6.93
)):
For small dimensions we have the following isomorphisms:
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