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6.3 The compact and split real forms of a semi-simple Lie algebra

We shall consider here only semi-simple Lie algebras. Over the complex numbers, Cartan subalgebras are “unique”22. These subalgebras may be defined as maximal Abelian subalgebras π”₯ such that the transformations in ad [π”₯] are simultaneously diagonalizable (over β„‚). Diagonalizability is an essential ingredient in the definition. There might indeed exist Abelian subalgebras of dimension higher than the rank (= dimension of Cartan subalgebras), but these would involve non-diagonalizable elements and would not be Cartan subalgebras23.

We denote the set of nonzero roots as Δ. One may complete the Chevalley generators into a full basis, the so-called Cartan–Weyl basis, such that the following commutation relations hold:

[H, E α] = α((H )E α, (6.31 ) { N α,βE α+β if α + β ∈ Δ, [E α,E β] = ( H α if α + β = 0, (6.32 ) 0 if α + β ⁄∈ Δ,
where H α is defined by duality thanks to the Killing form B (X, Y) = Tr(ad X ad Y), which is non-singular on semi-simple Lie algebras:
∀H ∈ π”₯ : α (H ) = B (H ,H ), (6.33 ) α
and the generators are normalized according to (see Equation (6.43View Equation))
B (Eα, E β) = δα+β,0. (6.34 )
The generators Eα associated with the roots α (where α need not be a simple root) may be chosen such that the structure constants N α,β satisfy the relations
N α,β = − N β,α = − N −α,−β = N β,− α−β, (6.35 ) 2 1 N α,β = -q(p + 1)(α|α), p, q ∈ β„•0, (6.36 ) 2
where the scalar product between roots is defined as
(α |β ) = B(H ,H ). (6.37 ) α β
The non-negative integers p and q are such that the string of all vectors β + nα belongs to Δ for − p ≤ n ≤ q; they also satisfy the equation p − q = 2(β|α)βˆ•(α|α ). A standard result states that for semi-simple Lie algebras
∑ (α |β ) = (α|γ)(γ|β) ∈ β„š, (6.38 ) γ∈Δ
from which we notice that the roots are real when evaluated on an H β-generator,
α(H β) = (α|β). (6.39 )

An important consequence of this discussion is that in Equation (6.32View Equation), the structure constants of the commutations relations may all be chosen real. Thus, if we restrict ourselves to real scalars we obtain a real Lie algebra 𝔰0, which is called the split real form because it contains the maximal number of noncompact generators. This real form of 𝔀 reads explicitly

⊕ ⊕ 𝔰0 = ℝH α ⊕ ℝE α. (6.40 ) α∈ Δ α∈Δ
The signature of the Killing form on 𝔰0 (which is real) is easily computed. First, it is positive definite on the real linear span π”₯0 of the H α generators. Indeed,
∑ 2 B (H α, H α) = (α|α ) = (α|γ) > 0. (6.41 ) γ∈Δ
Second, the invariance of the Killing form fixes the normalization of the Eα generators to one,
B (E α,E −α) = 1, (6.42 )
B ([E α, E− α],H α) = (α |α ) = − B(E −α,[E α, H α]) = (α |α)B (Eα,E −α). (6.43 )
Moreover, one has B(g , g ) = 0 α β if α + β ⁄= 0. Indeed ad[g ] ad [g ] α β maps g μ into g μ+ α+β, i.e., in matrix terms ad[gα] ad [gβ] has zero elements on the diagonal when α + β ⁄= 0. Hence, the vectors E α + E− α are spacelike and orthogonal to the vectors E α − E −α, which are timelike. This implies that the signature of the Killing form is
( | | ) 1- || 1- || 2(dim 𝔰0 + rank 𝔰0)| , 2(dim 𝔰0 − rank𝔰0)| . (6.44 ) + −
The split real form 𝔰0 of 𝔀 is “unique”.

On the other hand, it is not difficult to check that the linear span

⊕ ⊕ ⊕ 𝔠0 = ℝ (iH α) ⊕ ℝ (E α − E −α) ⊕ ℝ i(E α + E− α) (6.45 ) α∈Δ α ∈Δ α∈Δ
also defines a real Lie algebra. An important property of this real form is that the Killing form is negative definite on it. Its signature is
(0|+, dim 𝔠0|− ). (6.46 )
This is an immediate consequence of the previous discussion and of the way 𝔠0 is constructed. Hence, this real Lie algebra is compact25. For this reason, 𝔠0 is called the “compact real form” of 𝔀. It is also “unique”.
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