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6.4 Classical decompositions

6.4.1 Real forms and conjugations

The compact and split real Lie algebras constitute the two ends of a string of real forms that can be inferred from a given complex Lie algebra. As announced, this section is devoted to the systematic classification of these various real forms.

If 𝔤0 is a real form of 𝔤, it defines a conjugation on 𝔤. Indeed we may express any Z ∈ 𝔤 as Z = X0 + i Y0 with X0 ∈ 𝔤0 and iY0 ∈ i𝔤0, and the conjugation of 𝔤 with respect to 𝔤0 is given by

-- Z ↦→ Z = X0 − iY0. (6.47 )
Using Equation (6.3View Equation), it is immediate to verify that this involutive map is antilinear: ---- ---- λ Z = λ Z, where -- λ is the complex conjugate of the complex number λ.

Conversely, if σ is a conjugation on 𝔤, the set 𝔤σ of elements of 𝔤 fixed by σ provides a real form of 𝔤. Then σ constitutes the conjugation of 𝔤 with respect to 𝔤σ. Thus, on 𝔤, real forms and conjugations are in one-to-one correspondence. The strategy used to classify and describe the real forms of a given complex simple algebra consists of obtaining all the nonequivalent possible conjugations it admits.

6.4.2 The compact real form aligned with a given real form

Let 𝔤0 be a real form of the complex semi-simple Lie algebra 𝔤 ℂ = 𝔤0 ⊗ ℝ ℂ. Consider a compact real form 𝔠 0 of 𝔤ℂ and the respective conjugations τ and σ associated with 𝔠 0 and 𝔤0. It may or it may not be that τ and σ commute. When they do, τ leaves 𝔤0 invariant,

τ(𝔤0) ⊂ 𝔤0

and, similarly, σ leaves 𝔠 0 invariant,

σ (𝔠0) ⊂ 𝔠0.
In that case, one says that the real form 𝔤0 and the compact real form 𝔠0 are “aligned”.

Alignment is not automatic. For instance, one can always de-align a compact real form by applying an automorphism to it while keeping 𝔤 0 unchanged. However, there is a theorem that states that given a real form 𝔤0 of the complex Lie algebra ℂ 𝔤, there is always a compact real form 𝔠0 associated with it [93Jump To The Next Citation Point129Jump To The Next Citation Point]. As this result is central to the classification of real forms, we provide a proof in Appendix B, where we also prove the uniqueness of the Cartan involution.

We shall from now on always consider the compact real form aligned with the real form under study.

6.4.3 Cartan involution and Cartan decomposition

A Cartan involution θ of a real Lie algebra 𝔤0 is an involutive automorphism such that the symmetric, bilinear form θ B defined by

B θ(X, Y ) = − B (X, θY ) (6.48 )
is positive definite. If the algebra 𝔤0 is compact, a Cartan involution is trivially given by the identity.

A Cartan involution θ of the real semi-simple Lie algebra 𝔤 0 yields the direct sum decomposition (called Cartan decomposition)

𝔤0 = 𝔨0 ⊕ 𝔭0, (6.49 )
where 𝔨0 and 𝔭0 are the θ-eigenspaces of eigenvalues +1 and − 1. Explicitly, the decomposition of a Lie algebra element is given by
1- 1- X = 2(X + θ[X ]) + 2(X − θ[X ]). (6.50 )
The eigenspaces obey the commutation relations
[𝔨0, 𝔨0] ⊂ 𝔨0, [𝔨0, 𝔭0] ⊂ 𝔭0, [𝔭0, 𝔭0] ⊂ 𝔨0, (6.51 )
from which we deduce that B (𝔨0,𝔭0) = 0 because the mappings ad[𝔨0] ad[𝔭0] map 𝔭0 on 𝔨0 and 𝔨0 on 𝔭0. Moreover θ[𝔨0] = + 𝔨0 and θ[𝔭0] = − 𝔭0, and hence θ B (𝔨0, 𝔭0) = 0. In addition, since θ B is positive definite, the Killing form B is negative definite on 𝔨0 (which is thus a compact subalgebra) but is positive definite on 𝔭0 (which is not a subalgebra).

Define in 𝔤ℂ the algebra 𝔠0 by

𝔠 = 𝔨 ⊕ i𝔭 . (6.52 ) 0 0 0
It is clear that 𝔠0 is also a real form of 𝔤ℂ and is furthermore compact since the Killing form restricted to it is negative definite. The conjugation τ that fixes 𝔠0 is such that τ(X ) = X (X ∈ 𝔨0), τ(iY ) = iY (Y ∈ 𝔭 0) and hence τ(Y ) = − Y (Y ∈ 𝔭 0). It leaves 𝔤 0 invariant, which shows that 𝔠0 is aligned with 𝔤0. One has
𝔠0 = 𝔨0 ⊕ i𝔭0, 𝔨0 = 𝔤0 ∩ 𝔠0, 𝔭0 = 𝔤0 ∩ i𝔠0. (6.53 )

Conversely, let 𝔠0 be a compact real form aligned with 𝔤0 and τ the corresponding conjugation. The restriction θ of τ to 𝔤 0 is a Cartan involution. Indeed, one can decompose 𝔤0 as in Equation (6.49View Equation), with Equation (6.51View Equation) holding since θ is an involution of 𝔤0. Furthermore, one has also Equation (6.53View Equation), which shows that 𝔨0 is compact and that B θ is positive definite.

This shows, in view of the result invoked above that an aligned compact real form always exists, that any real form possesses a Cartan involution and a Cartan decomposition. If there are two Cartan involutions, θ and ′ θ, defined on a real semi-simple Lie algebra, one can show that they are conjugated by an internal automorphism [93Jump To The Next Citation Point129Jump To The Next Citation Point]. It follows that any real semi-simple Lie algebra possesses a “unique” Cartan involution.

On the matrix algebra ad [𝔤 ] 0, the Cartan involution is nothing else than minus the transposition with respect to the scalar product θ B,

T ad θX = − (ad X ) . (6.54 )
Indeed, remembering that the transpose of a linear operator with respect to B θ is defined by θ θ T B (X, AY ) = B (A X, Y), one gets
θ B (ad θX (Y ), Z) = − B ([θX, Y ], θZ ) = B(Y, [θX, θZ ]) = B (Y, θ[X, Z ]) = − B θ(Y, adX (Z )) = − B θ((ad X )T(Y), Z). (6.55 )
Since B θ is positive definite, this implies, in particular, that the operator adY, with Y ∈ 𝔭0, is diagonalizable over the real numbers since it is symmetric, ad Y = (adY )T.

An important consequence of this [93Jump To The Next Citation Point129Jump To The Next Citation Point] is that any real semi-simple Lie algebra can be realized as a real matrix Lie algebra, closed under transposition. One can also show [93Jump To The Next Citation Point129Jump To The Next Citation Point] that the Cartan decomposition of the Lie algebra of a semi-simple group can be lifted to the group via a diffeomorphism between 𝔨 × ğ”­ ↦→ 𝒢 = 𝒦 exp[𝔭 ] 0 0 0, where 𝒦 is a closed subgroup with 𝔨 0 as Lie algebra. It is this subgroup that contains all the topology of 𝒢.

6.4.4 Restricted roots

Let 𝔤0 be a real semi-simple Lie algebra. It admits a Cartan involution θ that allows to split it into eigenspaces 𝔨 0 of eigenvalue +1 and 𝔭 0 of eigenvalue − 1. We may choose in 𝔭 0 a maximal Abelian subalgebra ğ”ž0 (because the dimension of 𝔭0 is finite). The set {ad H |H ∈ ğ”ž0} is a set of symmetric transformations that can be simultaneously diagonalized on ℝ. Accordingly we may decompose 𝔤0 into a direct sum of eigenspaces labelled by elements of the dual space ğ”ž∗0:

⊕ 𝔤0 = gλ, gλ = {X ∈ 𝔤0|∀H ∈ ğ”ž0 : ad H (X ) = λ(H )X }. (6.56 ) λ

One, obviously non-vanishing, subspace is g 0, which contains ğ”ž 0. The other nontrivial subspaces define the restricted root spaces of 𝔤0 with respect to ğ”ž0, of the pair (𝔤0, ğ”ž0). The λ that label these subspaces g λ are the restricted roots and their elements are called restricted root vectors. The set of all λ is called the restricted root system. By construction the different gλ are mutually B θ-orthogonal. The Jacobi identity implies that [gλ, g μ] ⊂ gλ+ μ, while ğ”ž0 ⊂ 𝔭0 implies that θg λ = g−λ. Thus if λ is a restricted root, so is − λ.

Let 𝔪 be the centralizer of ğ”ž0 in 𝔨0. The space g0 is given by

g = ğ”ž ⊕ 𝔪. (6.57 ) 0 0
If 𝔱0 is a maximal Abelian subalgebra of 𝔪, the subalgebra 𝔥0 = ğ”ž0 ⊕ 𝔱0 is a Cartan subalgebra of the real algebra 𝔤0 in the sense that its complexification 𝔥ℂ is a Cartan subalgebra of 𝔤ℂ. Accordingly we may consider the set of nonzero roots Δ of 𝔤ℂ with respect to 𝔥ℂ and write
⊕ 𝔤ℂ = 𝔥ℂ (gα)ℂ. (6.58 ) α ∈Δ

The restricted root space gλ is given by

⊕ ℂ gλ = 𝔤0 ∩ (gα) (6.59 ) α ∈Δ α|ğ”ž0 =λ
and similarly
ℂ ℂ ⊕ ℂ 𝔪 = 𝔱 (gα) . (6.60 ) α ∈Δ α|ğ”ž0 =0

Note that the multiplicities of the restricted roots λ might be nontrivial even though the roots α are nondegenerate, because distinct roots α might yield the same restricted root when restricted to ğ”ž0.

Let us denote by Σ the subset of nonzero restricted roots and by V Σ the subspace of ğ”ž∗ 0 that they span. One can show [93Jump To The Next Citation Point129Jump To The Next Citation Point] that Σ is a root system as defined in Section 4. This root system need not be reduced. As for all root systems, one can choose a way to split the roots into positive and negative ones. Let Σ+ be the set of positive roots and

⊕ 𝔫 = gλ. (6.61 ) λ∈Σ+
As Σ+ is finite, 𝔫 is a nilpotent subalgebra of 𝔤0 and ğ”ž0 ⊕ 𝔫 is a solvable subalgebra.

6.4.5 Iwasawa and 𝓚 𝓐 𝓚 decompositions

The Iwasawa decomposition provides a global factorization of any semi-simple Lie group in terms of closed subgroups. It can be viewed as the generalization of the Gram–Schmidt orthogonalization process.

At the level of the Lie algebra, the Iwasawa decomposition theorem states that

𝔤0 = 𝔨0 ⊕ ğ”ž0 ⊕ 𝔫. (6.62 )
Indeed any element X of 𝔤0 can be decomposed as
∑ ∑ ∑ X = X0 + X λ = X0 + (X −λ + θX −λ) + (X λ − θX −λ). (6.63 ) λ λ∈ Σ+ λ∈Σ+
The first term X 0 belongs to g = ğ”ž ⊕ 𝔪 ⊂ ğ”ž ⊕ 𝔨 0 0 0 0, while the second term belongs to 𝔨 0, the eigenspace subspace of θ-eigenvalue +1. The third term belongs to 𝔫 since θX −λ ∈ gλ. The sum is furthermore direct. This is because one has obviously 𝔨0 ∩ ğ”ž0 = 0 as well as ğ”ž0 ∩ 𝔫 = 0. Moreover, 𝔨0 ∩ 𝔫 also vanishes because θ𝔫 ∩ 𝔫 = 0 as a consequence of ⊕ θ𝔫 = λ∈Σ+ g−λ.

The Iwasawa decomposition of the Lie algebra differs from the Cartan decomposition and is tilted with respect to it, in the sense that 𝔫 is neither in 𝔨 0 nor in 𝔭 0. One of its virtues is that it can be elevated from the Lie algebra 𝔤0 to the semi-simple Lie group 𝒢. Indeed, it can be shown [93Jump To The Next Citation Point129Jump To The Next Citation Point] that the map

(k,a, n) ∈ 𝒦 × ğ’œ × ğ’© ↦→ k an ∈ 𝒢 (6.64 )
is a global diffeomorphism. Here, the subgroups 𝒦, 𝒜 and 𝒩 have respective Lie algebras 𝔨0, ğ”ž0, 𝔫. This decomposition is “unique”.

There is another useful decomposition of 𝒢 in terms of a product of subgroups. Any two generators of 𝔭0 are conjugate via internal automorphisms of the compact subgroup 𝒦. As a consequence writing an element g ∈ 𝒢 as a product g = k Exp [𝔭0], we may write 𝒢 = 𝒦 𝒜 𝒦, which constitutes the so-called 𝒦 𝒜 𝒦 decomposition of the group (also sometimes called the Cartan decomposition of the group although it is not the exponention of the Cartan decomposition of the algebra). Here, however, the writing of an element of 𝒢 as product of elements of 𝒦 and 𝒜 is, in general, not unique.

6.4.6 θ-stable Cartan subalgebras

As in the previous sections, 𝔤0 is a real form of the complex semi-simple algebra 𝔤, σ denotes the conjugation it defines, τ the conjugation that commutes with σ, 𝔠0 the associated compact aligned real form of 𝔤 and θ the Cartan involution. It is also useful to introduce the involution of 𝔤 given by the product σ τ of the commuting conjugations. We denote it also by θ since it reduces to the Cartan involution when restricted to 𝔤0. Contrary to the conjugations σ and τ, θ is linear over the complex numbers. Accordingly, if we complexify the Cartan decomposition 𝔤0 = 𝔨0 ⊕ 𝔭0, to

𝔤 = 𝔨 ⊕ 𝔭 (6.65 )
with 𝔨 = 𝔨0 ⊗ℝ ℂ = 𝔨0 ⊕ i𝔨0 and 𝔭 = 𝔭0 ⊗ ℝ ℂ = 𝔭0 ⊕ i𝔭0, the involution θ fixes 𝔨 pointwise while θ(X ) = − X for X ∈ 𝔭.

Let 𝔥0 be a θ-stable Cartan subalgebra of 𝔤0, i.e., a subalgebra such that (i) θ(𝔥0) ⊂ 𝔥0, and (ii) 𝔥 ≡ 𝔥 ℂ 0 is a Cartan subalgebra of the complex algebra 𝔤. One can decompose 𝔥0 into compact and noncompact parts,

𝔥0 = 𝔱0 ⊕ ğ”ž0, 𝔱0 = 𝔥0 ∩ 𝔨0, ğ”ž0 = 𝔥0 ∩ 𝔭0. (6.66 )

We have seen that for real Lie algebras, the Cartan subalgebras are not all conjugate to each other; in particular, even though the dimensions of the Cartan subalgebras are all equal to the rank of 𝔤, the dimensions of the compact and noncompact subalgebras depend on the choice of 𝔥0. For example, for 𝔰𝔩(2,ℝ ), one may take 𝔥 = ℝt 0, in which case 𝔱 = 0 0, ğ”ž = 𝔥 0 0. Or one may take 𝔥 = ℝ τy 0, in which case 𝔱0 = 𝔥0, ğ”ž0 = 0.

One says that the θ-stable Cartan subalgebra 𝔥0 is maximally compact if the dimension of its compact part 𝔱0 is as large as possible; and that it is maximally noncompact if the dimension of its noncompact part ğ”ž0 is as large as possible. The θ-stable Cartan subalgebra 𝔥0 = 𝔱0 ⊕ ğ”ž0 used above to introduce restricted roots, where ğ”ž 0 is a maximal Abelian subspace of 𝔭 0 and 𝔱 0 a maximal Abelian subspace of its centralizer 𝔪, is maximally noncompact. If 𝔪 = 0, the Lie algebra 𝔤0 constitutes a split real form of ℂ 𝔤. The real rank of 𝔤0 is the dimension of its maximally noncompact Cartan subalgebras (which can be shown to be conjugate, as are the maximally compact ones [129Jump To The Next Citation Point]).

6.4.7 Real roots – Compact and non-compact imaginary roots

Consider a general θ-stable Cartan subalgebra 𝔥0 = 𝔱0 ⊕ ğ”ž0, which need not be maximally compact or maximally non compact. A consequence of Equation (6.54View Equation) is that the matrices of the real linear transformations ad H are real symmetric for H ∈ ğ”ž 0 and real antisymmetric for H ∈ 𝔱 0. Accordingly, the eigenvalues of ad H are real (and ad H can be diagonalized over the real numbers) when H ∈ ğ”ž0, while the eigenvalues of ad H are imaginary (and adH cannot be diagonalized over the real numbers although it can be diagonalized over the complex numbers) when H ∈ 𝔱0.

Let α be a root of 𝔤, i.e., a non-zero eigenvalue of ad 𝔥 where 𝔥 is the complexification of the θ-stable Cartan subalgebra 𝔥 = 𝔥0 ⊗ ℝ ℂ = 𝔥0 ⊕ i𝔥0. As the values of the roots acting on a given H are the eigenvalues of ad H, we find that the roots are real on ğ”ž0 and imaginary on 𝔱0. One says that a root is real if it takes real values on 𝔥0 = 𝔱0 ⊕ ğ”ž0, i.e., if it vanishes on 𝔱0. It is imaginary if it takes imaginary values on 𝔥0, i.e., if it vanishes on ğ”ž0, and complex otherwise. These notions of “real” and “imaginary” roots should not be confused with the concepts with similar terminology introduced in Section 4 in the context of non-finite-dimensional Kac–Moody algebras.

If 𝔥0 is a θ-stable Cartan subalgebra, its complexification 𝔥 = 𝔥0 ⊗ ℝ ℂ = 𝔥0 ⊕ i𝔥0 is stable under the involutive authormorphism θ = τ σ. One can extend the action of θ from 𝔥 to 𝔥∗ by duality. Denoting this transformation by the same symbol θ, one has

∀H ∈ 𝔥 and ∀α ∈ 𝔥∗, θ(α)(H ) = α(θ−1(H )), (6.67 )
or, since θ2 = 1,
θ(α)(H ) = α (θH ). (6.68 )

Let Eα be a nonzero root vector associated with the root α and consider the vector θE α. One has

[H, θE ] = θ[θH, E ] = α(θH ) θE = θ(α)(H )θE , (6.69 ) α α α α
i.e., θ (g α) = gθ(α) because the roots are nondegenerate, i.e., all root spaces are one-dimensional.

Consider now an imaginary root α. Then for all h ∈ 𝔥0 and a ∈ ğ”ž0 we have α(h + a ) = α (h), while θ(α) (h + a) = α(θ(h + a)) = α(h − a ) = α (h); accordingly α = θ(α). Moreover, as the roots are nondegenerate, one has θE α = ±E α. Writing E α as

E α = X α + iYα with X α, Y α ∈ 𝔤0, (6.70 )
it is easy to check that θE α = +E α implies that X α and Y α belong to 𝔨0, while both are in 𝔭 0 if θE = − E α α. Accordingly, g α is completely contained either in 𝔨 = 𝔨 ⊕ i𝔨 0 0 or in 𝔭 = 𝔭0 ⊕ i𝔭0. If gα ⊂ 𝔨, the imaginary root is said to be compact, and if gα ⊂ 𝔭 it is said to be noncompact.

6.4.8 Jumps between Cartan subalgebras – Cayley transformations

Suppose that β is an imaginary noncompact root. Consider a β-root vector Eβ ∈ gβ ⊂ 𝔭. If this root is expressed according to Equation (6.70View Equation), then its conjugate, with respect to (the conjugation σ defined by) 𝔤0, is

σE β = X β − iYβ with X β, Y β ∈ 𝔭0. (6.71 )
It belongs to g −β because (using ∀H ∈ 𝔥 : σH = H 0)
------- [H, σE β ] = σ [σH, E β] = β(σH )σE β = − β (H )σE β. (6.72 )
Hereafter, we shall denote σE β by -- Eβ. The commutator
[ -- ] -- E β,E β = B (Eβ, E β)H β (6.73 )
belongs to i𝔨0 since -- -- -- σ ([E β,E β]) = [E β,Eβ] = − [Eβ,E β] and can be written, after a renormalization of the generators E β, as
-- [E β,E β] = --2---H β = H ′ ∈ i𝔨0. (6.74 ) (β |β ) β
Indeed as E β ∈ 𝔭, we have -- E β ∈ 𝔭 and thus -- -- θE β = − E β. This implies
-- -- θ -- B (E β ,Eβ) = − B (Eβ, θE β) = B (E β,E β) > 0.

The three generators -- {H β′, E β,E β} therefore define an 𝔰𝔩(2,ℂ ) subalgebra:

[ -- ] [ -- ] -- E β,E β = H β′, [Hβ′, E β] = 2E β, H β′,E β = − 2E β. (6.75 )
We may change the basis and take
-- i -- i -- h = E β + E β, e = -(E β − E β − H β′), f = -(E β − Eβ + H β′), (6.76 ) 2 2
whose elements belong to 𝔤0 (since they are fixed by σ) and satisfy the commutation relations (6.8View Equation)
[e, f] = h, [h, e] = 2e, [h, f] = − 2f. (6.77 )
The subspace
𝔥′ = ker(β |𝔥0) ⊕ ℝh (6.78 ) 0
constitutes a new real Cartan subalgebra whose intersection with 𝔭0 has one more dimension.

Conversely, if β is a real root then θ(β) = − β. Let E β be a root vector. Then -- Eβ is also in gβ and hence proportional to Eβ. By adjusting the phase of E β, we may assume that E β belongs to 𝔤 0. At the same time, θE β, also in 𝔤 0, is an element of g −β. Evidently, θ B (E β, θE β) = − B (E β, E β) is negative. Introducing Hβ′ = 2∕(β|β)H β (which is in 𝔭0), we obtain the 𝔰𝔩(2,ℝ ) commutation relations

[E β, θEβ] = − H β′ ∈ 𝔭0, [Hβ′, E β] = 2E β, [H β′, θE β] = − 2 θE β. (6.79 )
Defining the compact generator Eβ + θE β, which obviously belongs to 𝔤0, we may build a new Cartan subalgebra of 𝔤0:
′ 𝔥0 = ker(β|𝔥0) ⊕ ℝ (E β + θEβ ), (6.80 )
whose noncompact subspace is now one dimension less than previously.

These two kinds of transformations – called Cayley transformations – allow, starting from a θ-stable Cartan subalgebra, to transform it into new ones with an increasing number of noncompact dimensions, as long as noncompact imaginary roots remain; or with an increasing number of compact dimensions, as long as real roots remain. Exploring the algebra in this way, we obtain all the Cartan subalgebras up to conjugacy. One can prove that the endpoints are maximally noncompact and maximally compact, respectively.

Theorem: Let 𝔥0 be a θ stable Cartan subalgebra of 𝔤0. Then there are no noncompact imaginary roots if and only if 𝔥0 is maximally noncompact, and there are no real roots if and only if 𝔥0 is maximally compact [129Jump To The Next Citation Point].

For a proof of this, note that we have already proven that if there are imaginary noncompact (respectively, real) roots, then 𝔥0 is not maximally noncompact (respectively, compact). The converse is demonstrated in [129Jump To The Next Citation Point].

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