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6.2 A preliminary example: 𝖘𝖑(2, ℂ)

Before we proceed to develop the general theory of real forms, we shall in this section discuss in detail some properties of the real forms of A1 = 𝔰𝔩(2,ℂ). This is a nice example, which exhibits many properties that turn out not to be specific just to the case at hand, but are, in fact, valid also in the general framework of semi-simple Lie algebras. The main purpose of subsequent sections will then be to show how to extend properties that are immediate in the case of 𝔰𝔩(2,ℂ ), to general semi-simple Lie algebras.

6.2.1 Real forms of 𝖘𝖑(2,ℂ )

The complex Lie algebra 𝔰𝔩(2,ℂ) can be represented as the space of complex linear combinations of the three matrices

( ) ( ) ( ) h = 1 0 , e = 0 1 , f = 0 0 (6.7 ) 0 − 1 0 0 1 0
which satisfy the well known commutation relations
[h, e] = 2 e, [h, f ] = − 2 f, [e, f ] = h. (6.8 )
A crucial property of these commutation relations is that the structure constants defined by the brackets are all real. Thus by restricting the scalars in the linear combinations from the complex to the real numbers, we still obtain closure for the Lie bracket on real combinations of h, e and f, defining thereby a real form of the complex Lie algebra 𝔰𝔩(2,ℂ ): the real Lie algebra 𝔰𝔩(2, ℝ)19. As we have indicated above, this real form of 𝔰𝔩(2,ℂ ) is called the “split real form”.

Another choice of 𝔰𝔩(2,ℂ ) generators that, similarly, leads to a real Lie algebra consists in taking i times the Pauli matrices x σ, y σ, z σ, i.e.,

( ) ( ) ( ) τx = i(e + f ) = 0 i , τ y = (e − f ) = 0 1 , τz = ih = i 0 . (6.9 ) i 0 − 1 0 0 − i
The real linear combinations of these matrices form the familiar 𝔰𝔲(2) Lie algebra (a real Lie algebra, even if some of the matrices using to represent it are complex). This real Lie algebra is non-isomorphic (as a real algebra) to 𝔰𝔩(2,ℝ) as there is no real change of basis that maps {h, e,f} on a basis with the 𝔰𝔲(2) commutation relations. Of course, the two algebras are isomorphic over the complex numbers.

6.2.2 Cartan subalgebras

Let 𝔥 be a subalgebra of 𝔰𝔩(2, ℝ). We say that 𝔥 is a Cartan subalgebra of 𝔰𝔩(2,ℝ) if it is a Cartan subalgebra of 𝔰𝔩(2,ℂ) when the real numbers are replaced by the complex numbers. Two Cartan subalgebras 𝔥1 and 𝔥2 of 𝔰𝔩(2,ℝ ) are said to be equivalent (as Cartan subalgebras of 𝔰𝔩(2,ℝ )) if there is an automorphism a of 𝔰𝔩(2,ℝ ) such that a(𝔥1) = 𝔥2.

The subspace ℝh constitutes clearly a Cartan subalgebra of 𝔰𝔩(2, ℝ). The adjoint action of h is diagonal in the basis {e, f, h } and can be represented by the matrix

( ) 2 0 0 ( 0 − 2 0) . (6.10 ) 0 0 0
Another Cartan subalgebra of 𝔰𝔩(2,ℝ) is given by y ℝ (e − f) ≡ ℝ τ, whose adjoint action with respect to the same basis is represented by the matrix
( ) 0 1 1 ( − 2 0 0) . (6.11 ) − 2 0 0
Contrary to the matrix representing adh, in addition to 0 this matrix has two imaginary eigenvalues: ±2 i. Thus, there can be no automorphism a of 𝔰𝔩(2,ℝ ) such that τy = λa(h), λ ∈ ℝ since ada(h) has the same eigenvalues as adh, implying that the eigenvalues of λ ada (h) are necessarily real (λ ∈ ℝ).

Consequently, even though they are equivalent over the complex numbers since there is an automorphism in SL (2,ℂ) that connects the complex Cartan subalgebras ℂ h and y ℂ τ, we obtain

( [ ]) ( [ ]) τ y = i Ad Exp iπ-(e + f ) h, h = i Ad Exp π-τx τy. (6.12 ) 4 4
The real Cartan subalgebras generated by h and τy are non-isomorphic over the real numbers.

6.2.3 The Killing form

The Killing form of SL (2,ℝ ) reads explicitly

( ) 0 4 0 B = ( 4 0 0) (6.13 ) 0 0 8
in the basis {e, f, h }. The Cartan subalgebra ℝh is spacelike while the Cartan subalgebra y ℝτ is timelike. This is another way to see that these are inequivalent since the automorphisms of 𝔰𝔩(2,ℝ) preserve the Killing form. The group Aut [𝔰𝔩(2,ℝ)] of automorphisms of 𝔰𝔩(2,ℝ ) is SO (2,1), while the subgroup Int[𝔰𝔩(2,ℝ )] ⊂ Aut [𝔰𝔩(2,ℝ )] of inner automorphisms is the connected component SO (2,1)+ of SO (2, 1). All spacelike directions are equivalent, as are all timelike directions, which shows that all the Cartan subalgebras of 𝔰𝔩(2,ℝ ) can be obtained by acting on these two inequivalent particular ones by Int[𝔰𝔩(2,ℝ )], i.e., the adjoint action of the group SL (2,ℝ ). The lightlike directions do not define Cartan subalgebras because the adjoint action of a lighlike vector is non-diagonalizable. In particular ℝe and ℝf are not Cartan subalgebras even though they are Abelian.

By exponentiation of the generators h and τy, we obtain two subgroups, denoted 𝒜 and 𝒦:

{ ( t ) } 𝒜 = Exp [th ] = e −0t |t ∈ ℝ ≃ ℝ, (6.14 ) 0 e { ( cos(t) sin (t)) } 𝒦 = Exp [tτ y] = |t ∈ [0,2π [ ≃ ℝ ∕ℤ. (6.15 ) − sin(t) cos(t)
The subgroup defined by Equation (6.14View Equation) is noncompact, the one defined by Equation (6.15View Equation) is compact; consequently the generator h is also said to be noncompact while τ y is called compact.

6.2.4 The compact real form 𝖘 𝖚(2)

The Killing metric on the group 𝔰𝔲(2) is negative definite. In the basis {τx,τy,τz}, it reads

( ) − 8 0 0 B = ( 0 − 8 0 ) . (6.16 ) 0 0 − 8

The corresponding group obtained by exponentiation is SU (2), which is isomorphic to the 3-sphere and which is accordingly compact. All directions in 𝔰𝔲(2) are equivalent and hence, all Cartan subalgebras are SU (2) conjugate to ℝτ y. Any generator provides by exponentiation a group isomorphic to ℝ ∕ℤ and is thus compact.

Accordingly, while 𝔰𝔩(2,ℝ ) admits both compact and noncompact Cartan subalgebras, the Cartan subalgebras of 𝔰𝔲(2) are all compact. The real algebra 𝔰𝔲(2) is called the compact real form of 𝔰𝔩(2,ℂ ). One often denotes the real forms by their signature. Adopting Cartan’s notation A1 for 𝔰𝔩(2,ℂ ), one has 𝔰𝔩(2,ℝ) ≡ A1 (1) and 𝔰𝔲 (2) ≡ A1(−3). We shall verify before that there are no other real forms of 𝔰𝔩(2,ℂ ).

6.2.5 𝖘𝖚 (2) and 𝖘 𝖑(2, ℝ) compared and contrasted – The Cartan involution

Within 𝔰𝔩(2, ℂ), one may express the basis vectors of one of the real subalgebras 𝔰𝔲 (2 ) or 𝔰𝔩(2,ℝ) in terms of those of the other. We obtain, using the notations t = (e − f) and x = (e + f ):

x x x = − iτ , τ = i x, h = − iτz, τz = i h, (6.17 ) t = τy, τy = t.
Let us remark that, in terms of the generators of 𝔰𝔲(2), the noncompact generators x and h of 𝔰𝔩(2,ℝ ) are purely imaginary but the compact one t is real.

More precisely, if τ denotes the conjugation20 of 𝔰𝔩(2,ℂ ) that fixes x y z {τ , τ , τ }, we obtain:

τ(x) = − x, τ(t) = +t, τ(h) = − h, (6.18 )
or, more generally,
∀X ∈ 𝔰𝔩(2, ℂ) : τ(X ) = − X †. (6.19 )
Conversely, if we denote by σ the conjugation of 𝔰𝔩(2,ℂ ) that fixes the previous 𝔰𝔩(2,ℝ) Cartan subalgebra in 𝔰𝔩(2,ℂ), we obtain the usual complex conjugation of the matrices:
-- σ(X ) = X. (6.20 )

The two conjugations τ and σ of 𝔰𝔩(2, ℂ) associated with the real subalgebras 𝔰𝔲(2) and 𝔰𝔩(2,ℝ ) of 𝔰𝔩(2, ℂ) commute with each other. Each of them, trivially, fixes pointwise the algebra defining it and globally the other algebra, where it constitutes an involutive automorphism (“involution”).

The Killing form is neither positive definite nor negative definite on 𝔰𝔩(2,ℝ ): The symmetric matrices have positive norm squared, while the antisymmetric ones have negative norm squared. Thus, by changing the relative sign of the contributions associated with symmetric and antisymmetric matrices, one can obtain a bilinear form which is definite. Explicitly, the involution θ of 𝔰𝔩(2,ℝ) defined by t θ(X ) = − X has the feature that

B θ(X, Y ) = − B (X, θY ) (6.21 )
is positive definite. An involution of a real Lie algebra with that property is called a “Cartan involution” (see Section 6.4.3 for the general definition).

The Cartan involution θ is just the restriction to 𝔰𝔩(2,ℝ) of the conjugation τ associated with the compact real form 𝔰𝔲(2) since for real matrices † t X = X. One says for that reason that the compact real form 𝔰𝔲(2) and the noncompact real form 𝔰𝔩(2,ℝ ) are “aligned”.

Using the Cartan involution θ, one can split 𝔰𝔩(2,ℝ ) as the direct sum

𝔰𝔩(2,ℝ) = 𝔨 ⊕ 𝔭, (6.22 )
where 𝔨 is the subspace of antisymmetric matrices corresponding to the eigenvalue +1 of the Cartan involution while 𝔭 is the subspace of symmetric matrices corresponding to the eigenvalue − 1. These are also eigenspaces of τ and given explicitly by 𝔨 = ℝt and 𝔭 = ℝx ⊕ ℝh. One has
𝔰𝔲(2) = 𝔨 ⊕ i𝔭, (6.23 )
i.e., the real form 𝔰𝔩(2,ℝ ) is obtained from the compact form 𝔰𝔲(2) by inserting an “i” in front of the generators in 𝔭.

6.2.6 Concluding remarks

Let us close these preliminaries with some remarks.

  1. The conjugation τ allows to define a Hermitian form on 𝔰𝔩(2,ℂ ):
    X ∙ Y = − Tr (Yτ (X )). (6.24)
  2. Any element of the group SL (2,ℝ) can be written as a product of elements belonging to the subgroups 𝒦, 𝒜 and 𝒩 = Exp [ℝe ] (Iwasawa decomposition),
    ( ) eacosθ n eacosθ + e−a sin θ Exp[θ t] Exp [a h] Exp[n e] = − easinθ e−acos θ − nea sin θ . (6.25)
  3. Any element of 𝔭 is conjugated via 𝒦 to a multiple of h ,
    ( ) ( ) cos α2- sin α2- cos α2- − sin α2- ρ(cos αh + sin αx ) = − sin α2- cos α2- ρ h sin α2- cos α2- , (6.26)
    so, denoting by ğ”ž = ℝ h the (maximal) noncompact Cartan subalgebra of 𝔰𝔩(2,ℝ ), we obtain
    𝔭 = Ad(𝒦 )ğ”ž. (6.27)
  4. Any element of SL (2,ℝ ) can be written as the product of an element of 𝒦 and an element of Exp [𝔭 ]. Thus, as a consequence of the previous remark, we have SL (2,ℝ ) = 𝒦 𝒜 𝒦 (Cartan)21.
  5. When the Cartan subalgebra of 𝔰𝔩(2,ℝ ) is chosen to be ℝ h, the root vectors are e and f. We obtain the compact element t, generating a non-equivalent Cartan subalgebra by taking the combination
    t = e + θ(e). (6.28)
    Similarly, the normalized root vectors associated with t are (up to a complex phase) E = 1(h ∓ ix) ±2i 2:
    [t, E2i] = 2iE2i, [t, E− 2i] = − 2i E− 2i, [E2i, E− 2i] = it. (6.29)
    Note that both the real and imaginary components of E ±2i are noncompact. They allow to obtain the noncompact Cartan generators h,x by taking the combinations
    cosα h + sin α x = eiαE2i + e−iαE− 2i. (6.30)

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