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

In this new “diagonal” description, the conjugation (see Equation (6.94)) becomes where The Cartan involution has the following realisation: In terms of the four matrices introduced in Equation (6.84), the generators defining this Cartan subalgebra reads Let us emphasize that we have numbered the basis generators of by first choosing those in , then those in .

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

From the relations (6.130) it is easy to obtain the expression of the in terms of the and thus also the expressions for the simple roots , , and , defined by , It is then straightforward to obtain the action of on the roots, which, when expressed in terms of the simple roots themselves, is given by We see that the root is real while , and are complex. As a check of these results, we may, for instance, verify that In fact, this kind of computation provides a simpler way to obtain Equation (6.133).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

(We have put in fourth position, rather than in second, to follow usual conventions.) The action of on this basis reads These new simple roots are now all complex.

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:

which are the positive roots of the (non-reduced) root system. The first four roots are degenerate twice, while the last two roots are nondegenerate. For instance, the two simple roots and project on the same restricted root , while the two simple roots and project on the same restricted root .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

so that a maximally noncompact Cartan subalgebra can be taken to be diagonal and is explicitly given by The corresponding in the algebra is still aligned with the natural matrix representation of . The Cartan involution is given by where . One has where the noncompact part is one-dimensional and spanned by , while the compact part is three-dimensional and spanned by , and .

In terms of the ’s, the standard simple roots now read

The Cartan involution acts as showing that and are imaginary, is real, while is complex.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,

which transform as under the Cartan involution. Note that in this system, the simple roots and are imaginary and hence fixed by the Cartan involution, while the other two simple roots are complex.

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

which are the positive roots of the (non-reduced) root system. The first root is six times degenerate, while the second one is nondegenerate. The simple roots and project on the same restricted root , while the imaginary root and project on zero (as does also the non-simple, positive, imaginary root ).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

where is an involutive permutation of the indices of the elements of . Accordingly, contains cycles of one or two elements. In the Tits–Satake diagram, we connect by a double arrow all pairs of distinct simple roots and in the same two-cycle orbit. For instance, for and , we obtain the diagrams in Figure 36.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
subsystem^{28}
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 ,

Solving these equations for the unknown coefficients is always possible because the Killing metric is nondegenerate on .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

where , called as we have indicated above the real rank of , is given by the number of cycles of the permutation (since two simple white roots joined by a double-arrow project on the same simple restricted root [5, 93]). These two sets of data allow one to determine the dimension of (without missing generators) [5, 93]. Another useful information, which can be directly read off from the Tits–Satake diagrams is the dimension of the noncompact subspace appearing in the splitting . It is given (see Section 6.6.6) by This can be illustrated in the two previous examples. For , one gets , and . It follows that and since has no semi-simple part (no black root), it reduces to and has dimension 2. This yields , and, by substraction, ( is easily verified to be equal to ). Similarly, for , one gets , and . It follows that and since the semi-simple part of is read from the black roots to be , which has rank two, one deduces and . This yields , and, by substraction, ( is easily verified to be equal to in this case).Finally, from the knowledge of , we may obtain the restricted root space by projecting the root space according to

and restricting their action on since and project on the same restricted root [5, 93].

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:

A basis of simple roots is The corresponding Dynkin diagram can be obtained from Figure 37 by ignoring the painting of the vertices. To the real Lie algebra, denoted in [28], is associated the Tits–Satake diagram of the left hand side of Figure 37. We immediately obtain from this diagram the following information: Accordingly, has signature (compact part) + (rank of ) and is denoted . Moreover, solving a system of three equations, we obtain: , i.e., This shows that the projection defining the reduced root system consists of projecting any given root orthogonally onto its component. Thus we obtain , with multiplicity 8 for (resulting from the projection of the eight roots ) and 7 for (resulting from the projection of the seven roots ).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

But this means that for the root , is again a root, which is impossible as we shall see below.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 remind the reader that, in this case, the Cartan involution is such that and . From Equation (6.69) we obtain and using we deduce that Furthermore, Equation (6.32) and the fact that the structure constants are rational yield the following relations: On the other hand, the commutativity of and implies with In particular, if the root belongs to , defined in Equation (6.146), then and thus , i.e.,Let us denote by and the subsets of corresponding to the imaginary noncompact and imaginary compact roots, respectively. We have

Obviously, for , belongs to , while for , belongs to . Moreover, if we find These remarks lead to the following explicit constructions of the complexifications of and : Furthermore, since fixes all the elements of , the subspace belongs to the centralizerUsing, as before, the basis in Equation (6.147) we obtain for the roots belonging to , i.e., for an index :

Thus As , where the coefficients are non-negative integers, the matrix must be a permutation matrix and it follows that where is an involutive permutation of .A fundamental property of is

To show this, note that if , it would imply that belongs to , which is impossible for the root lattice of a semi-simple Lie algebra. If and , then . Thus we obtain using Equation (6.35) and taking into account that is maximal Abelian in , that , i.e., and From this result we deduce i.e., which is incompatible with equation (6.160). Thus, the statement (6.170) follows.http://www.livingreviews.org/lrr-2008-1 |
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