At the end of Section 6.5.3, we obtained a matrix representation of a maximally noncompact
Cartan subalgebra of in terms of the natural description of this algebra. To facilitate
the forthcoming discussion, we find it useful to use an equivalent description, in which the
matrices representing this Cartan subalgebra are diagonal, as this subalgebra will now play a
central role. It is obtained by performing a similarity transformation
, where
The standard matrix representation of constitutes a compact real Lie subalgebra of
aligned with the diagonal description of the real form
. Moreover, its Cartan subalgebra
generated by purely imaginary combinations of the four diagonal matrices
is such that its
complexification
contains
. Accordingly, the roots it defines act both on
and
. Note that on
, the roots take only real values.
Our first task is to compute the action of the Cartan involution on the root lattice. With
this aim in view, we introduce two distinct bases on
. The first one is
,
which is dual to the basis
and is adapted to the relation
. The
second one is
, dual to the basis
, which is adapted
to the decomposition
. The Cartan involution acts on these root space bases as
The basis allows to define a different ordering on the root lattice, merely by
considering the corresponding lexicographic order. In terms of this new ordering we obtain for instance
since the first nonzero component of
(in this case
along
) is strictly negative.
Similarly, one finds
,
,
,
,
,
,
,
,
. A basis of simple roots, according to this
ordering, is given by
The restricted roots are obtained by considering only the action of the roots on the noncompact Cartan
generators and
. The two-dimensional vector space spanned by the restricted roots
can be identified with the subspace spanned by
and
; one simply projects out
and
. In the notations
and
, one gets as positive restricted roots:
Counting multiplicities, there are ten restricted roots – the same number as the number of positive roots
of . No root of
projects onto zero. The centralizer of
consists only of
.
Let us now perform the same analysis within the framework of . Starting from the natural
description (6.92
) of
, we first make a similarity transformation using the matrix
In terms of the ’s, the standard simple roots now read
A calculation similar to the one just described above, using as ordering rules the lexicographic ordering
defined by the dual of the basis in Equation (6.139), leads to the new system of simple roots,
The restricted roots are obtained by considering the action of the roots on the single noncompact Cartan
generator . The one-dimensional vector space spanned by the restricted roots can be identified with the
subspace spanned by
; one now simply projects out
,
and
. With the notation
,
we get as positive restricted roots
Let us finally emphasize that the centralizer of in
is now given by
, where
is
the center of
in
(i.e., the subspace generated by the compact generators that commute with
)
and contains more than just the three compact Cartan generators
,
and
. In fact,
involves also the root vectors
whose roots restrict to zero. Explicitly, expressed in the basis of
Equation (6.85
), these roots read
with
and are orthogonal to
.
The algebra
constitutes a rank 3, 9-dimensional Lie algebra, which can be identified with
.
We may associate with each of the constructions of these simple root bases a Tits–Satake diagram as
follows. We start with a Dynkin diagram of the complex Lie algebra and paint in black () the imaginary
simple roots, i.e., the ones fixed by the Cartan involution. The others are represented by a white vertex
(
). Moreover, some double arrows are introduced in the following way. It can be easily proven (see
Section 6.6.4) that for real semi-simple Lie algebras, there always exists a basis of simple roots
that
can be split into two subsets:
whose elements are fixed by
(they correspond to
the black vertices) and
(corresponding to white vertices) such that
Tits–Satake diagrams provide a lot of information about real semi-simple Lie algebras. For instance, we can
read from them the full action of the Cartan involution as we now briefly pass to show. More information
may be found in [5, 93
].
The Cartan involution allows one to define a closed
subsystem28
of
:
To determine completely we just need to know its action on a basis of simple roots. For those
belonging to
it is the identity, while for the other ones we have to compute the coefficients
in
Equation (6.145
). These are obtained by solving the linear system given by the scalar products of these
equations with the elements of
,
The black roots of a Tits–Satake diagram represent and constitute the Dynkin diagram of the
compact part
of the centralizer of
. Because
is compact, it is the direct sum of a semi-simple
compact Lie algebra and one-dimensional, Abelian
summands. The rank of
(defined as the
dimension of its maximal Abelian subalgebra; diagonalizability is automatic here because one is in the
compact case) is equal to the sum of the rank of its semi-simple part and of the number of
terms,
while the dimension of
is equal to the dimension of its semi-simple part and of the number of
terms. The Dynkin diagram of
reduces to the Dynkin diagram of its semi-simple
part.
The rank of the compact subalgebra is given by
Finally, from the knowledge of , we may obtain the restricted root space by projecting the root space
according to
The Lie algebra is a 52-dimensional simple Lie algebra of rank 4. Its root vectors can be expressed in
terms of the elements of an orthonormal basis
of a four-dimensional Euclidean space:
Let us mention that, contrary to the Vogan diagrams, any “formal Tits–Satake diagram” is
not admissible. For instance if we consider the right hand side diagram of Figure 37 we get
Let us recall some crucial aspects of the discussion so far. Let be a real form of the complex
semi-simple Lie algebra
and
be the conjugation it defines. We have seen that there
always exists a compact real Lie algebra
such that the corresponding conjugation
commutes with
. Moreover, we may choose a Cartan subalgebra
of
such that its
complexification
is invariant under
, i.e.,
. Then the real form
is
said to be normally related to
. As previously, we denote by the same letter
the
involution defined by duality on
(and also on the root lattice with respect to
:
) by
.
When and
are normally related, we may decompose the former into compact and noncompact
components
such that
. As mentioned, the starting point consists of choosing a
maximally Abelian noncompact subalgebra
and extending it to a Cartan subalgebra
, where
. This Cartan subalgebra allows one to consider the real Cartan subalgebra
Let us denote by and
the subsets of
corresponding to the imaginary noncompact
and imaginary compact roots, respectively. We have
Using, as before, the basis in Equation (6.147) we obtain for the roots belonging to
, i.e., for an
index
:
A fundamental property of is
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