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 byreal 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
Finally, from the knowledge of , we may obtain the restricted root space by projecting the root space according to[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:, is associated the Tits–Satake diagram of the left hand side of Figure 37. We immediately obtain from this diagram the following information:
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 have29 of and thus is empty if is maximally Abelian in . Taking this remark into account, we immediately obtain the dimension formulas (6.149, 6.150).
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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