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6.5 Vogan diagrams

Let 𝔤0 be a real semi-simple Lie algebra, 𝔤 its complexification, θ a Cartan involution leading to the Cartan decomposition
𝔤0 = 𝔨0 ⊕ 𝔭0, (6.81 )
and 𝔥0 a Cartan θ-stable subalgebra of 𝔤0. Using, if necessary, successive Cayley transformations, we may build a maximally compact θ-stable Cartan subalgebra 𝔥0 = 𝔱0 ⊕ ğ”ž0, with complexification 𝔥 = 𝔱 ⊕ ğ”ž. As usual we denote by Δ the set of (nonzero) roots of 𝔤 with respect to 𝔥. This set does not contain any real root, the compact dimension being assumed to be maximal.

From Δ we may define a positive subset + Δ by choosing the first set of indices from a basis of i𝔱0, and then the next set from a basis of ğ”ž0. Since there are no real roots, the roots in Δ+ have at least one non-vanishing component along i𝔱0, and the first non-zero one of these components is strictly positive. Since θ = +1 on 𝔱0, and since there are no real roots: + + θΔ = Δ. Thus θ 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 𝔤.

A Vogan diagram is associated to the triple (𝔤0,𝔥0,Δ+ ) as follows. It corresponds to the standard Dynkin diagram of Δ+, with additional information: the 2-element orbits under θ 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).

6.5.1 Illustration – The 𝖘𝖑(5, ℂ) case

The complex Lie algebra 𝔰𝔩(5,ℂ) can be represented as the algebra of traceless 5 × 5 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 24 × 24 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

B (X, Y) = Tr (adX ad Y ) = 10 Tr (XY ). (6.82 )
The coefficient 10 appearing in this relation is known as the Coxeter index of 𝔰𝔩(5, ℂ).

A Cartan–Weyl basis is obtained by taking the 20 nilpotent generators Kpq (with p ⁄= q) corresponding to matrices, all elements of which are zero except the one located at the intersection of row p and column q, which is equal to 1,

(Kpq )α= δαpδβq (6.83 ) β
and the four diagonal ones,
( ) ( ) 1 0 0 0 0 0 0 0 0 0 | 0 − 1 0 0 0| | 0 1 0 0 0| || || || || H1 = | 0 0 0 0 0| , H2 = | 0 0 − 1 0 0| , ( 0 0 0 0 0) ( 0 0 0 0 0) 0 0 0 0 0 0 0 0 0 0 ( ) ( ) (6.84 ) 0 0 0 0 0 0 0 0 0 0 | 0 0 0 0 0| | 0 0 0 0 0| || || || || H3 = | 0 0 1 0 0| , H4 = | 0 0 0 0 0| , ( 0 0 0 − 1 0) ( 0 0 0 1 0) 0 0 0 0 0 0 0 0 0 − 1
which constitute a Cartan subalgebra 𝔥.

The root space is easily described by introducing the five linear forms εp, acting on diagonal matrices d = diag (d1,..., d5) as follows:

εp(d ) = dp. (6.85 )
In terms of these, the dual space ⋆ 𝔥 of the Cartan subalgebra may be identified with the subspace
{ } ∑ p || ∑ p ε = A εp A = 0 . (6.86 ) p p
The 20 matrices Kp q are root vectors,
[Hk, Kpq ] = (εp[Hk ] − εq[Hk])Kpq, (6.87 )
i.e., p K q is a root vector associated to the root εp − εq.

𝖘𝖑(5,ℝ ) and 𝖘𝖚 (5)

By restricting ourselves to real combinations of these generators we obtain the real Lie algebra 𝔰𝔩(5,ℝ ). The conjugation η that it defines on 𝔰𝔩(5,ℂ) is just the usual complex conjugation. This 𝔰𝔩(5,ℝ ) constitutes the split real form 𝔰 0 of 𝔰𝔩(5, ℂ). Applying the construction given in Equation (6.45View Equation) to the generators of 𝔰𝔩(5,ℝ), we obtain the set of antihermitian matrices

p q p q iHk, K q − K p, i(K q + K p) (p > q ), (6.88 )
defining a basis of the real subalgebra 𝔰𝔲(5). This is the compact real form 𝔠0 of 𝔰𝔩(5,ℂ ). The conjugation associated to this algebra (denoted by τ) is minus the Hermitian conjugation,
† τ (X ) = − X . (6.89 )
Since [η,τ] = 0, τ induces a Cartan involution θ on 𝔰𝔩(5,ℝ ), providing a Euclidean form on the previous 𝔰𝔩(5,ℝ) subalgebra
θ t B (X, Y ) = 10 Tr(XY ), (6.90 )
which can be extended to a Hermitian form on 𝔰𝔩(5, ℂ),
Bθ(X, Y ) = 10 Tr(XY †). (6.91 )
Note that the generators iHk and i(Kpq + Kqp) are real generators (although described by complex matrices) since, e.g., (iHk )† = − iH † k, i.e., τ(iHk ) = iHk.

The other real forms

The real forms of 𝔰𝔩(5,ℂ) that are not isomorphic to 𝔰𝔩(5,ℝ ) or 𝔰𝔲(5) are isomorphic either to 𝔰𝔲 (3,2) or 𝔰𝔲 (4,1). In terms of matrices these algebras can be represented as

( ) A Γ X = Γ † B where A = − A† ∈ ℂp×p, B = − B † ∈ ℂq×q, (6.92 ) Tr A + TrB = 0, Γ ∈ ℂp×q with p = 3 (respectively 4) and q = 2 (respectively 1).
We shall call these ways of describing 𝔰𝔲(p,q) the “natural” descriptions of 𝔰𝔲 (p, q). Introducing the diagonal matrix
( ) I = Idp×p , (6.93 ) p,q − Idq×q
the conjugations defined by these subalgebras are given by:
σ (X ) = − I X †I . (6.94 ) p,q p,q p,q

Vogan diagrams

The Dynkin diagram of 𝔰𝔩(5,ℂ) is of A 4 type (see Figure 26View Image).

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Figure 26: The A 4 Dynkin diagram.

Let us first consider an 𝔰𝔲(3,2) 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 ε1 > ε2 > ε3 > ε4 > ε5, the simple roots α1 = ε1 − ε2, α2 = ε2 − ε3, α4 = ε4 − ε5 are compact but α3 = ε3 − ε4 is noncompact. The corresponding Vogan diagram is displayed in Figure 27View Image.

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Figure 27: A Vogan diagram associated to 𝔰𝔲 (3,2).

However, if instead of the natural order we define positive roots by the rule ε1 > ε2 > ε4 > ε5 > ε3, the simple positive roots are &tidle;α1 = ε1 − ε2 and &tidle;α3 = ε4 − ε5 which are compact, and &tidle;α2 = ε2 − ε4 and &tidle;α4 = ε5 − ε3 which are noncompact. The associated Vogan diagram is shown in Figure 28View Image.

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Figure 28: Another Vogan diagram associated to 𝔰𝔲(3,2).

Alternatively, the choice of order ε1 > ε5 > ε3 > ε4 > ε2 leads to the diagram in Figure 29View Image.

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Figure 29: Yet another Vogan diagram associated to 𝔰𝔲 (3, 2).

There remain seven other possibilities, all describing the same subalgebra 𝔰𝔲(3,2). These are displayed in Figure 30View Image.

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Figure 30: The remaining Vogan diagrams associated to 𝔰𝔲(3,2).

In a similar way, we obtain four different Vogan diagrams for 𝔰𝔲(4,1), displayed in Figure 31View Image.

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Figure 31: The four Vogan diagrams associated to 𝔰𝔲(4,1).

Finally we have two non-isomorphic Vogan diagrams associated with 𝔰𝔲 (5) and 𝔰𝔩(5,ℝ ). These are shown in Figure 32View Image.

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Figure 32: The Vogan diagrams for 𝔰𝔲(5) and 𝔰𝔩(5, ℝ).

6.5.2 The Borel and de Siebenthal theorem

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 [129Jump To The Next Citation Point]. 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 {αp } be the basis of simple roots and {Λq} its dual basis, i.e., (Λq |αp ) = δpq. Then the single painted simple root αp may be chosen so that there is no q with (Λp − Λq|Λq) > 0. 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 E 8 (see Figure 33View Image), it is easy to compute the dual basis and the matrix of scalar products Bp q = (Λp − Λq|Λq ).

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Figure 33: The Dynkin diagram of E8. Seen as a Vogan diagram, it corresponds to the maximally compact form of E8.

We obtain

( − 0 − 7 − 20 − 12 − 6 − 2 − 0 − 3) | | | − 3 − 0 − 10 − 4 − 0 − 2 − 2 − 2| || − 6 − 6 − 0 − 4 − 6 − 6 − 4 − 7|| || − 4 − 2 − 6 − 0 − 3 − 4 − 3 − 4|| (Bp q) = | − 2 − 2 − 12 − 5 − 0 − 2 − 2 − 1| , (6.95 ) || − 0 − 6 − 18 − 10 − 4 − 0 − 1 − 2|| |( |) − 2 − 10 − 24 − 15 − 8 − 3 − 0 − 5 − 1 − 4 − 15 − 8 − 3 − 0 − 1 − 0
from which we see that there exist, besides the compact real form, only two other non-isomorphic real forms of E8, described by the Vogan diagrams in Figure 34View Image26.
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Figure 34: Vogan diagrams of the two different noncompact real forms of E8: E8 (− 24) and E8 (8). The lower one corresponds to the split real form.

6.5.3 Cayley transformations in 𝖘 𝖚(3,2 )

Let us now illustrate the Cayley transformations. For this purpose, consider again 𝔰𝔲(3,2 ) with the imaginary diagonal matrices as Cartan subalgebra and the natural ordering of the εk defining the positive roots. As we have seen, α3 = ε3 − ε4 is an imaginary noncompact root. The associated 𝔰𝔩(2,ℂ ) generators are

E α = K3 , E-α-= σK3 = K4 , i H3. (6.96 ) 3 4 3 4 3
From the action of α3 on the Cartan subalgebra D = span {iHk, k = 1,...,4}, we may check that
ker(α3|D ) = span{i H1, i(2 H2 + H3 ), i(2 H4 + H3 )}, (6.97 )
and that ′ ---- 3 4 H = (E α3 + E α3) = (K4 + K 3) is such that ′ ′ θH = − H and ′ ′ σH = H. Moreover ′ H commutes with ker(α3|D ). Thus
′ C = ker(α3|D ) ⊕ ℝ H (6.98 )
constitutes a θ-stable Cartan subalgebra with one noncompact dimension H ′. Indeed, we have B (H ′,H ′) = 20. If we compute the roots with respect to this new Cartan subalgebra, we obtain twelve complex roots (expressed in terms of their components in the basis dual to the one implicitly defined by Equations (6.97View Equation) and (6.98View Equation),
±(i,− 3i,i,±1 ), ±(0,i,− 3i,±1 ), ± (i,i,− i,±1 ), (6.99 )
six imaginary roots
±i(2,− 2,0,0), ±i(1,− 2,− 2,0), ±i(1,0,2,0), (6.100 )
and a pair of real roots ±(0,0, 0,2).

Let us first consider the Cayley transformation obtained using, for instance, the real root (0,0,0, 2). An associated root vector, belonging to 𝔤0, reads

( 0 0 0 0 0) | | | 0 0 0i 0i 0| E = || 0 0 2 − 2 0|| . (6.101 ) ( 0 0 i2 − i2 0) 0 0 0 0 0
The corresponding compact Cartan generator is
( ) 0 0 0 0 0 | 0 0 0 0 0| || || h = | 0 0 i 0 0| , (6.102 ) ( 0 0 0 − i 0) 0 0 0 0 0
which, together with the three generators in Equation (6.97View Equation), provide a compact Cartan subalgebra of 𝔰𝔲(3,2 ).

If we consider instead the imaginary roots, we find for instance that K52 = − &tidle;θK52 is a noncompact complex root vector corresponding to the root β = i(1,− 2,− 2,0). It provides the noncompact generator 2 5 K 5 + K 2 which, together with

ker(β |C ) = span{i(2 H1 + 2 H2 + H3 ), i(2H1 + H3 + 2 H4 ), K34 + K43 }, (6.103 )
generates a maximally noncompact Cartan subalgebra of 𝔰𝔲(3,2). A similar construction can be done using, for instance, the roots ±i(1,0,2, 0), but not with the roots ±i(2,− 2,0,0) as their corresponding root vectors 1 K 2 and 2 K 1 are fixed by &tidle; θ and thus are compact.

6.5.4 Reconstruction

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 [129Jump To The Next Citation Point] 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 (6.45View Equation).

We know the action of θ on the simple roots. This implies that the set Δ of all roots is invariant under θ. 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 θ of all the positive roots, up to level n are in Δ. If γ is a root of level n + 1, choose a simple root α such that (γ |α ) > 0. Then the Weyl transformed root sαγ = β is also a positive root, but of smaller level. Since θ(α) and θ(β ) are then known to be in Δ, and since the involution acts as an isometry, θ(γ) = sθ(α)(θ(β)) is also in Δ.

One can transfer by duality the action of θ on 𝔥∗ to the Cartan subalgebra 𝔥, and then define its action on the root vectors associated to the simple roots according to the rules

( E if α is unpainted and invariant, { α θE α = ( − E α if α is painted and invariant, (6.104 ) − E θ[α] if α belongs to a 2-cycle.
These rules extend θ to an involution of 𝔤.

This involution is such that θEα = a αE θ[α], with aα = ±127. Furthermore it globally fixes 𝔠 0, θ 𝔠 = 𝔠 0 0. Let 𝔨 and 𝔭 be the +1 or − 1 eigenspaces of θ in 𝔤 = 𝔨 ⊕ 𝔭. Define 𝔨0 = 𝔠0 ∩ 𝔨 and 𝔭0 = i(𝔠0 ∩ 𝔭) so that 𝔠0 = 𝔨0 ⊕ i𝔭0. Set

𝔤0 = 𝔨0 ⊕ 𝔭0. (6.105 )
Using θ𝔠 = 𝔠 0 0, one then verifies that 𝔤 0 constitutes the desired real form of 𝔤 [129Jump To The Next Citation Point].

6.5.5 Illustrations: 𝖘 𝖑(4, ℝ) versus 𝖘𝖑(2, ℍ)

We shall exemplify the reconstruction of real algebras from Vogan diagrams by considering two examples of real forms of 𝔰𝔩(4,ℂ ). The diagrams are shown in Figure 35View Image.

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Figure 35: The Vogan diagrams associated to a 𝔰𝔩(4ℝ ) and 𝔰𝔩(2 ℍ) subalgebra.

The θ involutions they describe are (the upper signs correspond to the left-hand side diagram, the lower signs to the right-hand side diagram):

θHα1 = H α3, H α2 = H α2, θH α3 = H α1 θE = E , θE = ∓E , θE = E . (6.106 ) α1 α3 α2 α2 α3 α1
Using the commutations relations
[E α1, E α2] = E α1+α2, [E α2, E α3] = E α2+α3, (6.107 ) [E α1+ α2, E α3] = E α1+α2+α3 = [Eα1, Eα2+α3]
we obtain
θE = ±E , θE = ±E , θE = ∓E . (6.108 ) α1+α2 α2+ α3 α2+α3 α1+α2 α1+ α2+ α3 α1+α2+α3
Let us consider the left-hand side diagram. The corresponding +1 θ-eigenspace 𝔨 has the following realisation,
𝔨 = span {H α1 + H α3, H α2, Eα1 + Eα3, E− α1 + E −α3, Eα1+α2 + E α2+α3, E −α1−α2 + E −α2−α3}(6,.109 )
and the − 1 θ-eigenspace 𝔭 is given by
𝔭 = span {H α1 − H α3, E ±α2, Eα1 − E α3, E −α1 − E− α3, Eα1+α2 − E α2+α3, E −α1−α2 − E −α2−α3, E ±(α1+α2+α3)}. (6.110 )
The intersection 𝔠0 ∩ 𝔨 then leads to the 𝔰𝔬(4,ℝ ) = 𝔰𝔬(3,ℝ ) ⊕ 𝔰𝔬 (3, ℝ) algebra
𝔨 = span {i(H + H ), (E + E − E − E ), i(E + E + E + E )} 0 { α1 α3 α1 α3 −α1 − α3 α1 α3 −α1 − α3 ⊕ span i(H α1 + 2 H α2 + Hα3), (E α1+α2 + E α2+ α3 − E −(α1+α2) − E −(α2+α3)), i(E + E + E + E )}, (6.111 ) α1+α2 α2+α3 −(α1+ α2) −(α2+ α3)
and the remaining noncompact generator subspace 𝔭0 = i(𝔠0 ∩ 𝔭) becomes
𝔭0 = span {H α1 − H α3, (E α1 − E α3 + E−α1 − E −α3), i(E α1 − E α3 − E− α1 + E −α3), (E α1+α2 − Eα2+ α3 + E −(α1+α2) − E −(α2+α3)), i(E α1+α2 − Eα2+α3 − E −(α1+ α2) + E −(α2+ α3)), E α2 + E− α2, i(Eα2 − E −α2), E α1+α2+α3 + E −(α1+ α2+ α3), i(E α1+α2+α3 − E −(α1+α2+ α3))}. (6.112 )

Doing the same exercise for the second diagram, we obtain the real algebra 𝔰 𝔩(2,ℍ ) with 𝔨0 = 𝔰𝔬(5,ℝ ) = 𝔰𝔭(4,ℝ ), which is a 10-parameter compact subalgebra, and 𝔭0 given by

𝔭0 = span{H α1 − H α3, (E α1 − Eα3 + E −α1 − E−α3), i(E α1 − Eα3 − E −α1 + E− α3), (E + E + E + E ), α1+α2 α2+α3 −(α1+α2) −(α2+α3) i(E α1+α2 + E α2+ α3 − E −(α1+α2) − E −(α2+α3))}. (6.113 )

6.5.6 A pictorial summary – All real simple Lie algebras (Vogan diagrams)

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 [129Jump To The Next Citation Point] for the derivation.


Table 16: Vogan diagrams (An series)
An series, n ≥ 1 Vogan diagram Maximal compact subalgebra
𝔰𝔲(n + 1) PIC
No painted root
𝔰𝔲(n + 1)
𝔰𝔲(p,q) PIC
Only the th p root is painted
𝔰𝔲(p) ⊕ 𝔰𝔲(q) ⊕ u(1)
𝔰𝔩(2n,ℝ ) PIC
Odd number of roots
𝔰𝔬(2n)
𝔰𝔩(2n + 1,ℝ ) PIC
Even number of (unpainted) roots
𝔰𝔬(2n + 1)
𝔰𝔩(n + 1, ℍ) PIC
Odd number of (unpainted) roots
𝔰𝔭(n + 1)


Table 17: Vogan diagrams (Bn series)
Bn series, n ≥ 2 Vogan diagram Maximal compact subalgebra
𝔰𝔬 (2n + 1) PIC
No painted root
𝔰𝔬(2n + 1)
𝔰𝔬(p,q)
1 p≤ n− 2, q = 2n +1 − p
PIC
Each of the roots, once painted, leads to the algebra mentioned under it.
𝔰𝔬(p) ⊕ 𝔰𝔬(q)


Table 18: Vogan diagrams (Cn series)
Cn series, n ≥ 3 Vogan diagram Maximal compact subalgebra
𝔰𝔭(n) PIC
No painted root,
𝔰𝔭 (n )
𝔰𝔭(p, q)
0< p≤ n2, q = n− p
𝔰𝔭(n, ℝ)
PIC
Each of the roots, once painted, corresponds to the algebra mentioned near it.
𝔰𝔭(p) ⊕ 𝔰𝔭(q)

𝔲(n)


Table 19: Vogan diagrams (Dn series)
Dn series, n ≥ 4 Vogan diagram Maximal compact subalgebra
𝔰𝔬(2n ) PIC
No painted root
𝔰𝔬(2n)
𝔰𝔬(2p, 2q)
0 <p ≤ n , q =n − p 2
∗ 𝔰𝔬 (2n )
PIC
Each of the roots, once painted, corresponds to the algebra mentioned near it.
𝔰𝔬(2p) ⊕ 𝔰𝔬(2q)

𝔲(n)
𝔰𝔬(2p + 1,2q + 1)
0 <p ≤ n−1 2,
q =n − p− 1
PIC
Each of the roots, once painted, corresponds to the algebra mentioned near it.
No root painted corresponds to
𝔰𝔬(1,2n− 1).
𝔰𝔬(2p + 1) ⊕ 𝔰𝔬 (2q + 1 )


Table 20: Vogan diagrams (G2 series)
G2 Vogan diagram Maximal compact subalgebra
G2 PIC
No painted root, providing the real compact form
G2
G2(2) PIC 𝔰𝔲(2) ⊕ 𝔰𝔲(2)


Table 21: Vogan diagrams (F4 series)
F4 series Vogan diagram Maximal compact subalgebra
F4 PIC
No painted root, providing the real compact form
F4
F4(4) PIC 𝔰𝔭(3) ⊕ 𝔰𝔲(2)
F4(−20) PIC 𝔰𝔬(9)


Table 22: Vogan diagrams (E6 series)
E6 Vogan diagram Maximal compact subalgebra
E 6 PIC
No painted root, providing the real compact form
E 6
E6(6) PIC 𝔰𝔭(4)
E6(2) PIC 𝔰𝔲 (6) ⊕ 𝔰𝔲(2)
E6(−14) PIC 𝔰𝔲(10 ) ⊕ 𝔲(1)
E6(−26) PIC F4


Table 23: Vogan diagrams (E7 series)
E7 Vogan diagram Maximal compact subalgebra
E7 PIC
No painted root, providing the real compact form
E7
E7(7) PIC 𝔰𝔲(8)
E7 (43) PIC 𝔰𝔬(12) ⊕ 𝔰𝔲(2)
E7(−25) PIC E6 ⊕ 𝔲 (1 )


Table 24: Vogan diagrams (E8 series)
E8 Vogan diagram Maximal compact subalgebra
E8 PIC
No painted root, providing the real compact form
E8
E 8(8) PIC 𝔰𝔬(16)
E 8(−24) PIC E ⊕ 𝔰𝔲(2) 7

Using these diagrams, the matrix Ip,q defined by Equation (6.93View Equation), and the three matrices

( n×n) Jn = 0 Id , (6.114 ) − Idn×n 0 ( Idp×p 0 0 0 ) | q×q | Kp,q = | 0 − Id 0p×p 0 | , (6.115 ) ( 0 0 Id 0 ) 0 0 0 − Idq×q ( p×p ) 0 0 Id 0q×q L = K J = || 0 0 0 − Id || , (6.116 ) p,q p,q p+q ( − Idp×p 0 0 0 ) 0 Idq×q 0 0
we may check that the involutive automorphisms of the classical Lie algebras are all conjugate to one of the types listed in Table 25.


Table 25: List of all involutive automorphisms (up to conjugation) of the classical compact real Lie algebras [93Jump To The Next Citation Point]. The first column gives the complexification 𝔲ℂ 0 of the compact real algebra 𝔲 0, the second 𝔲0, the third the involution τ that 𝔲0 defines in ℂ 𝔲, and the fourth a non-compact real subalgebra 𝔤0 of 𝔲ℂ aligned with the compact one. In the second table, the second column displays the involution that 𝔤0 defines on 𝔲ℂ, the third the involutive automorphism of 𝔲0, i.e, the Cartan conjugation θ = σ τ, and the last column indicates the common compact subalgebra 𝔨0 of 𝔲0 = 𝔨0 ⊕ i𝔭0 and 𝔤0 = 𝔨0 ⊕ 𝔭0.
𝔲ℂ 𝔲 0 τ 𝔤 0
𝔰𝔩(n, ℂ) 𝔰𝔲(n) − X † A I ğ”°ğ”©(n,ℝ )
       
𝔰𝔩(2n,ℂ ) 𝔰𝔲 (2n) † − X A II ∗ 𝔰𝔲 (2n)
       
𝔰𝔩(p + q,ℂ ) 𝔰𝔲(p + q) − X † A III ğ”°ğ”² (p, q)
       
𝔰𝔬(p + q,ℂ) 𝔰𝔬(p + q,ℝ ) --- X B I, D I ğ”°ğ”¬(p,q)
       
𝔰𝔬(2n,ℂ ) 𝔰𝔬(2n, ℝ) --- X D III 𝔰𝔬∗(2n)
       
𝔰𝔭 (n, ℂ) 𝔲 𝔰𝔭(n) --- − JnXJn C I ğ”°ğ”­(n,ℝ )
       
𝔰𝔭(p + q,ℂ) 𝔲𝔰𝔭(p + q) --- − Jp+qXJp+q C III ğ”°ğ”­ (p, q)

𝔲ℂ σ θ 𝔨0
𝔰𝔩(n, ℂ) --- X − Xt 𝔰𝔬(n, ℝ)
       
𝔰𝔩(2n,ℂ ) − J XJ- n n J XtJ n n 𝔲 𝔰𝔭(2n)
       
𝔰𝔩(p + q,ℂ ) − Ip,qX †Ip,q Ip,qXIp,q 𝔰𝔬(n, ℝ)
       
𝔰𝔬(p + q,ℂ) I XI- p,q p,q I XI p,q p,q 𝔰𝔬(p,ℝ ) ⊕ 𝔰𝔬(q,ℝ)
       
𝔰𝔬(2n,ℂ ) --- − JnXJn − JnXJn 𝔰𝔲(n ) ⊕ 𝔲(1)
       
𝔰𝔭 (n, ℂ) --- X − JnXJn 𝔰𝔲(n ) ⊕ 𝔲(1)
       
𝔰𝔭(p + q,ℂ) † − Kp,qX Kp,q t Lp,qX Lp,q 𝔰𝔭(p) ⊕ 𝔰𝔭(q)

For completeness we remind the reader of the definitions of matrix algebras (𝔰𝔲(p, q) has been defined in Equation (6.93View Equation)):

{ | --- } 𝔰𝔲∗(2n) = X |XJn − JnX = 0, TrX = 0, X ∈ ℂ2n×2n { ( A C ) ||A, C ∈ ℂn×n } = -- -- || , (6.117 ) − C A Re [Tr A ] = 0 { | t t (p+q)× (p+q)} 𝔰𝔬(p, q) = X |XIp,q + Ip,qX = 0, X = − X , X ∈ ℝ { ( A C ) ||A = − At ∈ ℝp×p, B = − Bt ∈ ℝq×q,} = t || p×q , (6.118 ) C B C ∈ ℝ ∗ { || t --- t 2n×2n } 𝔰𝔬 (2n) = {X( X Jn +)Jn X = 0, X = − X , X ∈ ℂ } A B | t † n×n = − B- A- |A = − A , B = B ∈ ℂ , (6.119 ) 𝔰𝔭(n, ℝ) = {X ||XtJ + J X = 0, Tr X = 0, X ∈ ℝ2n ×2n} { ( n ) n } A B || t t n×n = C − At A, B = B , C = C ∈ ℝ , (6.120 ) { | } 𝔰𝔭(n, ℂ)) = X |XtJn + JnX = 0, Tr X = 0, X ∈ ℂ2n ×2n { ( ) } = A B ||A, B = Bt, C = Ct ∈ ℂn ×n , (6.121 ) C − At { | --- } 𝔰𝔭(p, q) = X |XtKp,q + Kp,qX = 0, TrX = 0, X ∈ ℂ(p+q)×(p+q) ( ( ) || p×p ) ||{ A† P Qt R |A, Q ∈ ℂ p×q q×p ||} = || P-- B- R-- S|| ||P, R ∈ ℂ , S ∈ ℂ , (6.122 ) || ( − Q R-- A − P) ||A = − A †, B = − B † || ( R † − S − P t B |Q = Qt, S = St ) 𝔲𝔰𝔭 (2p, 2q) = 𝔰𝔲(2p, 2q) ∩ 𝔰𝔭 (2p + 2q ). (6.123 )
Alternative definitions are:
--- 𝔰𝔭(p,q) = {X ∈ 𝔤𝔩(p + q,ℍ )|X Ip,q + Ip,q X = 0 }, t 𝔰𝔭(n,ℝ ) = {X ∈ 𝔤𝔩(2 n,ℝ )|X- Jn + Jn X = 0}, (6.124 ) 𝔰𝔩(n,ℍ ) = {X ∈ 𝔤𝔩(n, ℍ)|X + X = 0}, 𝔰𝔬∗(2n ) = {X ∈ 𝔰𝔲(n,n)|Xt K + K X = 0}. n n

For small dimensions we have the following isomorphisms:

𝔰𝔬(1,2) ≃ 𝔰𝔲(1,1) ≃ 𝔰𝔭(1,ℝ ) ≃ 𝔰𝔩(2,ℝ ), 𝔰𝔩(2,ℂ ) ≃ 𝔰𝔬(1,3), ∗ 𝔰𝔬∗(4) ≃ 𝔰𝔲(2) ⊕ 𝔰𝔲(1,1), 𝔰𝔬 (6) ≃ 𝔰𝔲(3,1), 𝔰𝔭(1,1) ≃ 𝔰𝔬(1,4), (6.125 ) 𝔰 𝔩(2,ℍ ) ≃ 𝔰𝔬(1,5), 𝔰𝔲(2,2) ≃ 𝔰𝔬(2,4), 𝔰𝔩(4,∗ℝ ) ≃ 𝔰𝔬(3,3), 𝔰𝔬 (8) ≃ 𝔰𝔬(2,6).

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