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6.6 Tits–Satake diagrams

The classification of real forms of a semi-simple Lie algebra, using Vogan diagrams, rests on the construction of a maximally compact Cartan subalgebra. On the other hand, the Iwasawa decomposition emphasizes the role of a maximally noncompact Cartan subalgebra. The consideration of these Cartan subalgebras leads to another way to classify real forms of semi-simple Lie algebras, developed mainly by Araki [5Jump To The Next Citation Point], and based on so-called Tits–Satake diagrams [161155].

6.6.1 Example 1: π–˜π–š (3,2)

Diagonal description

At the end of Section 6.5.3, we obtained a matrix representation of a maximally noncompact Cartan subalgebra of 𝔰𝔲(3,2) 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 X ↦→ ST X S, where

( ) 1 0 0 0 0 | 0 √1- 0 0 √1-| || 2 1 1 2|| S = | 0 0 √2- √2- 0| . (6.126 ) |( 0 0 − 1√-- 1√-- 0|) √1- 2 2 √1- 0 − 2 0 0 2
In this new “diagonal” description, the conjugation σ (see Equation (6.94View Equation)) becomes
† σ(X ) = − &tidle;I3,2X &tidle;I3,2, (6.127 )
where
( 1 0 0 0 0) | | T | 0 0 0 0 1| &tidle;I3,2 = S I3,2S = || 0 0 0 1 0|| . (6.128 ) ( 0 0 1 0 0) 0 1 0 0 0
The Cartan involution has the following realisation:
θ (X ) = I&tidle;3,2X &tidle;I3,2. (6.129 )
In terms of the four matrices introduced in Equation (6.84View Equation), the generators defining this Cartan subalgebra π”₯ reads
h = H , h = H + H + H , 1 3 2 2 3 4 (6.130 ) h3 = i(2 H1 + 2H2 + H3), h4 = i(2 H1 + H2 + H3 + H4 ).
Let us emphasize that we have numbered the basis generators of π”₯ = π”ž ⊕ 𝔱 by first choosing those in π”ž, then those in 𝔱.

Cartan involution and roots

The standard matrix representation of 𝔰𝔲(5) constitutes a compact real Lie subalgebra of 𝔰𝔩(5,β„‚ ) aligned with the diagonal description of the real form 𝔰𝔲(3,2 ). Moreover, its Cartan subalgebra π”₯0 generated by purely imaginary combinations of the four diagonal matrices Hk is such that its complexification π”₯β„‚ contains π”₯. Accordingly, the roots it defines act both on π”₯0 and π”₯. Note that on π”₯ = iπ”₯ ℝ 0, the roots take only real values.

Our first task is to compute the action of the Cartan involution θ on the root lattice. With this aim in view, we introduce two distinct bases on ∗ π”₯ℝ. The first one is 1 2 3 4 {F ,F ,F ,F }, which is dual to the basis {H1, H2, H3, H4} and is adapted to the relation π”₯ℝ = iπ”₯0. The second one is {f 1,f 2,f3,f4}, dual to the basis {h1, h2, − ih3, − ih4}, which is adapted to the decomposition π”₯ = π”ž ⊕ i𝔱 ℝ. The Cartan involution acts on these root space bases as

θ{f1,f2,f 3,f4} = {− f1,− f2,f 3,f 4}. (6.131 )
From the relations (6.130View Equation) it is easy to obtain the expression of the {F k}(k = 1,⋅⋅⋅ ,4) in terms of the {fk} and thus also the expressions for the simple roots α1 = 2F 1 − F 2, α = − F 1 + 2F 2 − F 3 2, α = − F 2 + 2F 3 − F 4 3 and α = − F 3 + 2F 4 4, defined by π”₯ 0,
2 3 4 α1 = − f1+ 2f2 + 33 f ,4 α2 = − f + f + f − f , (6.132 ) α3 = 2f 1, α4 = − f1 + f2 − f3 + f4.
It is then straightforward to obtain the action of θ on the roots, which, when expressed in terms of the π”₯0 simple roots themselves, is given by
θ[α1] = α1 + α2 + α3 + α4, θ[α2] = − α4, θ[α ] = − α , (6.133 ) 3 3 θ[α4] = − α2.
We see that the root α3 is real while α1, α2 and α4 are complex. As a check of these results, we may, for instance, verify that
1 1 θE α1 = &tidle;I3,2K 2I&tidle;3,2 = K 5 = Eα1+α2+α3+ α4. (6.134 )
In fact, this kind of computation provides a simpler way to obtain Equation (6.133View Equation).

The basis 1 2 3 4 {f ,f ,f ,f } 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 α1 < 0 since the first nonzero component of α1 (in this case − 1 along f2) is strictly negative. Similarly, one finds α2 < 0, α3 > 0, α4 < 0, α1 + α2 < 0, α2 + α3 > 0, α3 + α4 > 0, α + α + α > 0 1 2 3, α + α + α > 0 2 3 4, α + α + α + α > 0 1 2 3 4. A basis of simple roots, according to this ordering, is given by

&tidle;α1 = − α4 = f1 − f2 + f3 − f4, &tidle;α2 = α1 + α2 + α3 + α4 = f2 + 2 f3 + 3f4, 2 3 4 (6.135 ) &tidle;α3 = − α1 = f1− 2f2 − 33 f ,4 &tidle;α4 = − α2 = f − f − f + f .
(We have put &tidle;α 4 in fourth position, rather than in second, to follow usual conventions.) The action of θ on this basis reads
θ[&tidle;α ] = −α&tidle; , θ[&tidle;α ] = − &tidle;α , θ [&tidle;α ] = − &tidle;α , θ[α&tidle; ] = − &tidle;α . (6.136 ) 1 4 2 3 3 2 4 1
These new simple roots are now all complex.

Restricted roots

The restricted roots are obtained by considering only the action of the roots on the noncompact Cartan generators h 1 and h 2. The two-dimensional vector space spanned by the restricted roots can be identified with the subspace spanned by f1 and f2; one simply projects out f3 and f4. In the notations β1 = f1 − f2 and β2 = f2, one gets as positive restricted roots:

β1, β2, β1 + β2, β1 + 2β2, 2β2, 2(β1 + β2), (6.137 )
which are the positive roots of the (BC )2 (non-reduced) root system. The first four roots are degenerate twice, while the last two roots are nondegenerate. For instance, the two simple roots &tidle;α1 and &tidle;α4 project on the same restricted root β1, while the two simple roots &tidle;α2 and &tidle;α3 project on the same restricted root β2.

Counting multiplicities, there are ten restricted roots – the same number as the number of positive roots of 𝔰𝔩(5,β„‚ ). No root of 𝔰𝔩(5,β„‚ ) projects onto zero. The centralizer of π”ž consists only of π”ž ⊕ 𝔱.

6.6.2 Example 2: π–˜π–š (4,1)

Diagonal description

Let us now perform the same analysis within the framework of 𝔰𝔲 (4,1). Starting from the natural description (6.92View Equation) of 𝔰𝔲(4,1 ), we first make a similarity transformation using the matrix

( ) 1 0 0 0 0 || 0 1 0 0 0|| S = | 0 0 1 0 0| , (6.138 ) |( 0 0 0 √1- √1|) √21- √21- 0 0 0 − 2 2
so that a maximally noncompact Cartan subalgebra can be taken to be diagonal and is explicitly given by
h1 = H4, h2 = iH1, h3 = iH2, h4 = i(2 H3 + H4 ). (6.139 )
The corresponding 𝔰𝔲(4,1) in the 𝔰𝔩(5,β„‚ ) algebra is still aligned with the natural matrix representation of 𝔰𝔲(5). The Cartan involution is given by X ↦→ &tidle;I4,1X &tidle;I4,1 where T I&tidle;4,1 = S I4,1S. One has π”₯ = π”ž ⊕ 𝔱 where the noncompact part π”ž is one-dimensional and spanned by h1, while the compact part 𝔱 is three-dimensional and spanned by h2, h3 and h4.

Cartan involution and roots

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

2 3 α1 = 2f − f , α2 = − f2 + 2f3 − 2 f4, α3 = − f1 − f2 + 3f 4, (6.140 ) α4 = 2f 1.
The Cartan involution acts as
θ[α1] = α1, θ[α2] = α2, (6.141 ) θ[α3] = α3 + α4, θ[α4] = − α4,
showing that α1 and α2 are imaginary, α4 is real, while α3 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 (6.139View Equation), leads to the new system of simple roots,

&tidle;α1 = − α1 − α2 − α3, &tidle;α2 = α1 + α2, (6.142 ) &tidle;α3 = − α2, &tidle;α4 = α2 + α3 + α4,
which transform as
θ[α&tidle;1 ] = − &tidle;α4 − &tidle;α2 − α&tidle;3, θ[α&tidle;2 ] = α&tidle;2, θ[α&tidle; ] = α&tidle; , (6.143 ) 3 3 θ[α&tidle;4 ] = − &tidle;α1 − &tidle;α2 − α&tidle;3
under the Cartan involution. Note that in this system, the simple roots &tidle;α2 and &tidle;α3 are imaginary and hence fixed by the Cartan involution, while the other two simple roots are complex.

Restricted roots

The restricted roots are obtained by considering the action of the roots on the single noncompact Cartan generator h1. The one-dimensional vector space spanned by the restricted roots can be identified with the subspace spanned by f1; one now simply projects out f2, f3 and f4. With the notation β1 = f1, we get as positive restricted roots

β1, 2β1, (6.144 )
which are the positive roots of the (BC )1 (non-reduced) root system. The first root is six times degenerate, while the second one is nondegenerate. The simple roots &tidle;α1 and α&tidle;4 project on the same restricted root β1, while the imaginary root &tidle;α2 and &tidle;α3 project on zero (as does also the non-simple, positive, imaginary root α&tidle;2 + &tidle;α3).

Let us finally emphasize that the centralizer of π”ž in 𝔰𝔲(4,1) is now given by π”ž ⊕ π”ͺ, where π”ͺ is the center of π”ž in 𝔨 (i.e., the subspace generated by the compact generators that commute with H4) and contains more than just the three compact Cartan generators h2, h3 and h4. In fact, π”ͺ involves also the root vectors E β whose roots restrict to zero. Explicitly, expressed in the basis of Equation (6.85View Equation), these roots read β = εp − εq with p,q = 1,2, or 3 and are orthogonal to α4. The algebra π”ͺ constitutes a rank 3, 9-dimensional Lie algebra, which can be identified with 𝔰𝔲(3) ⊕ 𝔲(1).

6.6.3 Tits–Satake diagrams: Definition

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 (βˆ™) 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 B that can be split into two subsets: B0 = {αr+1,..., αl} whose elements are fixed by θ (they correspond to the black vertices) and B βˆ– B0 = {α1, ...,αr} (corresponding to white vertices) such that

∑l ∀αk ∈ B βˆ– B0 : θ[αk] = − α π(k) + ajk αj, (6.145 ) j=r+1
where π is an involutive permutation of the r indices of the elements of B βˆ– B0. Accordingly, π contains cycles of one or two elements. In the Tits–Satake diagram, we connect by a double arrow all pairs of distinct simple roots αk and α π(k) in the same two-cycle orbit. For instance, for 𝔰𝔲(3,2) and 𝔰𝔲(4,1 ), we obtain the diagrams in Figure 36View Image.
View Image

Figure 36: Tits–Satake diagrams for 𝔰𝔲 (3, 2) and 𝔰𝔲(4,1).

6.6.4 Formal considerations

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 [5Jump To The Next Citation Point93Jump To The Next Citation Point].

The Cartan involution allows one to define a closed subsystem28 Δ0 of Δ:

Δ0 = {α ∈ Δ |θ[α ] = α }, (6.146 )
which is the system of imaginary roots. These project to zero when restricted to the maximally noncompact Cartan subalgebra. As we have seen in the examples, it is useful to use an ordering adapted to the Cartan involution. This can be obtained by considering a basis of π”₯ constituted firstly by elements of π”ž followed by elements of 𝔱. If we use the lexicographic order defined by the dual of this basis, we obtain a root ordering such that if α ⁄∈ Δ0 is positive, θ[α] is negative since the real part comes first and changes sign. Let B be a simple root basis built with respect to this ordering and let B0 = B ∩ Δ0. Then we have
B = { α1,...,αl} and B0 = {αr+1, ...,αl}. (6.147 )
The subset B0 is a basis for Δ0. To see this, let B βˆ– B0 = {α1,...,αr }. If ∑l k β = k=1 b αk is, say, a positive root (i.e., with coefficients bk ≥ 0) belonging to Δ0, then β − θ[β] = 0 is given by a sum of positive roots, weighted by non-negative coefficients, ∑r bk (α − θ[α ]) k=1 k k. As a consequence, the coefficients k b are all zero for k = 1,⋅⋅⋅ ,r and B0 constitutes a basis of Δ0, as claimed.

To determine completely θ we just need to know its action on a basis of simple roots. For those belonging to B 0 it is the identity, while for the other ones we have to compute the coefficients aj k in Equation (6.145View Equation). These are obtained by solving the linear system given by the scalar products of these equations with the elements of B0,

∑l (θ[αk ] + α π(k)|αq ) = aj (αj|αq ). (6.148 ) j=r+1 k
Solving these equations for the unknown coefficients j ak is always possible because the Killing metric is nondegenerate on B0.

The black roots of a Tits–Satake diagram represent B 0 and constitute the Dynkin diagram of the compact part π”ͺ of the centralizer of π”ž. Because π”ͺ is compact, it is the direct sum of a semi-simple compact Lie algebra and one-dimensional, Abelian 𝔲 (1 ) summands. The rank of π”ͺ (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 𝔲 (1) terms, while the dimension of π”ͺ is equal to the dimension of its semi-simple part and of the number of 𝔲(1) terms. The Dynkin diagram of π”ͺ reduces to the Dynkin diagram of its semi-simple part.

The rank of the compact subalgebra π”ͺ is given by

rank π”ͺ = rank 𝔀 − rank𝔭, (6.149 )
where rank𝔭, called as we have indicated above the real rank of 𝔀, is given by the number of cycles of the permutation π (since two simple white roots joined by a double-arrow project on the same simple restricted root [5Jump To The Next Citation Point93Jump To The Next Citation Point]). These two sets of data allow one to determine the dimension of π”ͺ (without missing 𝔲(1) generators) [5Jump To The Next Citation Point93Jump To The Next Citation Point]. Another useful information, which can be directly read off from the Tits–Satake diagrams is the dimension of the noncompact subspace 𝔭 appearing in the splitting 𝔀 = 𝔨 ⊕ 𝔭. It is given (see Section 6.6.6) by
1- dim 𝔭 = 2(dim 𝔀 − dim π”ͺ + rank 𝔭). (6.150 )
This can be illustrated in the two previous examples. For 𝔰𝔲(3,2), one gets dim 𝔀 = 24, rank 𝔀 = 4 and rank 𝔭 = 2. It follows that rank π”ͺ = 2 and since π”ͺ has no semi-simple part (no black root), it reduces to π”ͺ = 𝔲 (1) ⊕ 𝔲(1) and has dimension 2. This yields dim 𝔭 = 12, and, by substraction, dim 𝔨 = 12 (𝔨 is easily verified to be equal to 𝔰𝔲(3) ⊕ 𝔰𝔲(2) ⊕ 𝔲(1)). Similarly, for 𝔰𝔲(4,1), one gets dim 𝔀 = 24, rank 𝔀 = 4 and rank 𝔭 = 1. It follows that rank π”ͺ = 3 and since the semi-simple part of π”ͺ is read from the black roots to be 𝔰𝔲(3), which has rank two, one deduces π”ͺ = 𝔰𝔲(3) ⊕ 𝔲(1) and dim π”ͺ = 9. This yields dim 𝔭 = 8, and, by substraction, dim 𝔨 = 16 (𝔨 is easily verified to be equal to 𝔰𝔲(4) ⊕ 𝔲(1) in this case).

Finally, from the knowledge of θ, we may obtain the restricted root space by projecting the root space according to

-- 1- Δ → Δ : α ↦→ ¯α = 2(α − θ[α ]) (6.151 )
and restricting their action on π”ž since α and − θ(α) project on the same restricted root [5Jump To The Next Citation Point93Jump To The Next Citation Point].

6.6.5 Illustration: F4

The Lie algebra F4 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 {e |k = 1,...,4} k of a four-dimensional Euclidean space:

{ } ΔF = ±ei ± ej|i < j } ∪ {±ei } ∪ {1(±e1 ± e2 ± e3 ± e4) . (6.152 ) 4 2
A basis of simple roots is
1 α1 = e2 − e3, α2 = e3 − e4, α3 = e4, α4 = -(e1 − e2 − e3 − e4). (6.153 ) 2
The corresponding Dynkin diagram can be obtained from Figure 37View Image by ignoring the painting of the vertices. To the real Lie algebra, denoted F II in [28], is associated the Tits–Satake diagram of the left hand side of Figure 37View Image. We immediately obtain from this diagram the following information:
1- rank 𝔭 = 1, rank π”ͺ = 3, π”ͺ = 𝔰𝔬(7), dim 𝔭 = 2(52 − 21 + 1) = 16. (6.154 )
Accordingly, F II has signature − 21 (compact part) + 1 (rank of 𝔭) = − 20 and is denoted F4 (−20). Moreover, solving a system of three equations, we obtain: θ[α4] = − α4 − α1 − 2 α2 − 3α3, i.e.,
θ[e1] = − e1 and θ[ek] = ek if k = 2,3,4. (6.155 )
This shows that the projection defining the reduced root system Σ consists of projecting any given root orthogonally onto its e1 component. Thus we obtain 1 Σ = {± 2e1, ±e1 }, with multiplicity 8 for 1 2e1 (resulting from the projection of the eight roots { 12(e1 ± e2 ± e3 ± e4)}) and 7 for e1 (resulting from the projection of the seven roots {e1 ± ek|k = 2,3,4} ∪ {e1}).
View Image

Figure 37: On the left, the Tits–Satake diagram of the real form F 4(−20). On the right, a non-admissible Tits–Satake diagram.

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 37View Image we get

θ[e ] = − e , θ[e ] = − e , and θ[e ] = e if k = 3 or 4. (6.156 ) 1 2 2 1 k k
But this means that for the root α = e1, α + θ∗[α ] = e1 − e2 is again a root, which is impossible as we shall see below.

6.6.6 Some more formal considerations

Let us recall some crucial aspects of the discussion so far. Let 𝔀σ be a real form of the complex semi-simple Lie algebra 𝔀ℂ and σ be the conjugation it defines. We have seen that there always exists a compact real Lie algebra 𝔲τ such that the corresponding conjugation τ commutes with σ. Moreover, we may choose a Cartan subalgebra π”₯ of 𝔲 τ such that its complexification π”₯β„‚ is invariant under σ, i.e., σ(π”₯β„‚) = π”₯β„‚. Then the real form 𝔀σ is said to be normally related to (𝔲 θ, π”₯). As previously, we denote by the same letter θ the involution defined by duality on (π”₯β„‚ )∗ (and also on the root lattice with respect to π”₯β„‚: Δ) by θ = τ σ.

When 𝔀σ and 𝔲τ are normally related, we may decompose the former into compact and noncompact components 𝔀σ = 𝔨 ⊕ 𝔭 such that 𝔲 τ = 𝔨 ⊕ i𝔭. As mentioned, the starting point consists of choosing a maximally Abelian noncompact subalgebra π”ž ⊂ 𝔭 and extending it to a Cartan subalgebra π”₯ = 𝔱 ⊕ π”ž, where 𝔱 ⊂ 𝔨. This Cartan subalgebra allows one to consider the real Cartan subalgebra

π”₯ℝ = i𝔱 ⊕ 𝔭 = ∑ ℝ H . (6.157 ) α α∈Δ
Let us remind the reader that, in this case, the Cartan involution θ = σ τ = τ σ is such that θ|𝔨 = +1 and θ|𝔭 = − 1. From Equation (6.69View Equation) we obtain
θ(E ) = ρ E , (6.158 ) α α θ[α]
and using θ2 = 1 we deduce that
ρα ρθ[α] = 1. (6.159 )
Furthermore, Equation (6.32View Equation) and the fact that the structure constants are rational yield the following relations:
ρα ρβN θ[α],θ[β] = ρα+ βNα,β, θ(H α) = H θ[α], (6.160 ) ρα ρ−α = 1.
On the other hand, the commutativity of τ and σ implies
σ(H α) = − Hσ[α], σ(E α) = κα Eσ[α], (6.161 )
with
κα = − ¯ρα, σ[α] = − θ[α ]. (6.162 )
In particular, if the root α belongs to Δ0, defined in Equation (6.146View Equation), then θ[α] = α and thus ρ2α = 1, i.e.,
ρ = − κ = ±1. (6.163 ) α α

Let us denote by Δ0,− and Δ0,+ the subsets of Δ0 corresponding to the imaginary noncompact and imaginary compact roots, respectively. We have

Δ0,− = {α ∈ Δ0|ρα = − 1} and Δ0,+ = {α ∈ Δ0 |ρα = +1 }. (6.164 )
Obviously, for α ∈ Δ0,−, E α belongs to β„‚ 𝔭, while for α ∈ Δ0,+, E α belongs to β„‚ 𝔨. Moreover, if α ∈ Δ βˆ– Δ0 we find
β„‚ β„‚ E α + θ(Eα ) ∈ 𝔨 and E α − θ (E α) ∈ 𝔭 . (6.165 )
These remarks lead to the following explicit constructions of the complexifications of 𝔨 and 𝔭:
⊕ ⊕ 𝔨ℂ = 𝔱ℂ ⊕ β„‚ E ⊕ β„‚ (E + θ(E )), α α α α∈⊕Δ0,+ α∈Δ⊕βˆ–Δ0 (6.166 ) 𝔭ℂ = π”ž β„‚ ⊕ β„‚ E α ⊕ β„‚ (E α − θ(Eα )). α∈Δ 0,− α∈Δ βˆ–Δ0
Furthermore, since θ fixes all the elements of Δ0, the subspace ⊕ α∈Δ0,− β„‚ Eα belongs to the centralizer29 of π”ž and thus is empty if π”ž is maximally Abelian in 𝔭. Taking this remark into account, we immediately obtain the dimension formulas (6.149View Equation, 6.150View Equation).

Using, as before, the basis in Equation (6.147View Equation) we obtain for the roots belonging to B βˆ– B0, i.e., for an index i ≤ r:

∑ j ∑ j j j − θ[αi ] = piαj + qiαj with pi, qi ∈ β„•. (6.167 ) j=1,...,r j=r+1,...,l
Thus
2 ∑ j k ∑ j k ∑ j αi = (− θ) [αi] = pipjαk + piqjαk − qiαj. (6.168 ) j = 1,...,r j = 1,...,r j=r+1,...,l k =1,...,r k =r+ 1,...,l
As ∑ j k j j=1,...,r pipj = δi, where the coefficients j pi are non-negative integers, the matrix j (pi) must be a permutation matrix and it follows that
θ[α ] = − α (mod Δ ), (6.169 ) i π(i) 0
where π is an involutive permutation of {1, ...,r}.

A fundamental property of Δ is

∀α ∈ Δ : θ[α] + α ⁄∈ Δ. (6.170 )
To show this, note that if α ∈ Δ0, it would imply that 2 α belongs to Δ, which is impossible for the root lattice of a semi-simple Lie algebra. If α ∈ Δ βˆ– Δ0 and θ[α] + α ∈ Δ, then θ[α] + α ∈ Δ0. Thus we obtain using Equation (6.35View Equation) and taking into account that π”ž is maximal Abelian in 𝔭, that ρα = +1, i.e.,
σ (E σ[α]−α) = +E θ[α]− α (6.171 )
and
[E α, σ (E−α )] = ρ−α N α,− σ[α]E α−σ[α], ---- [σ(E α), E −α] = ρ−α N α,− σ[α]E σ[α]−α (6.172 ) = ρ−α N σ[α],−α E σ[α]−α.
From this result we deduce
---- ραN σ[α],− α = ρ−α N α,−σ[α] = − ρα N α,−σ[α], (6.173 )
i.e., ---- ρα = − ρ−α which is incompatible with equation (6.160View Equation). Thus, the statement (6.170View Equation) follows.
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