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8.1 A finite-dimensional example: 𝖘𝖑(3, ℝ)

The rank 2 Lie algebra 𝔤 = 𝔰𝔩(3,ℝ ) is characterized by the Cartan matrix
( ) A [𝔰𝔩(3,ℝ )] = 2 − 1 , (8.1 ) − 1 2
whose Dynkin diagram is displayed in Figure 44View Image.
View Image

Figure 44: The Dynkin diagram of 𝔰𝔩(3, ℝ).

Recall from Section 6 that 𝔰𝔩(3,ℝ ) is the split real form of 𝔰𝔩(3,ℂ) ≡ A2, and is thus defined through the same Chevalley–Serre presentation as for 𝔰𝔩(3,ℂ), but with all coefficients restricted to the real numbers.

The Cartan generators {h1,h2} will indifferently be denoted by { α∨,α ∨} 1 2. As we have seen, they form a basis of the Cartan subalgebra 𝔥, while the simple roots {α1,α2 }, associated with the raising operators e1 and e2, form a basis of the dual root space ⋆ 𝔥. Any root ⋆ γ ∈ 𝔥 can thus be decomposed in terms of the simple roots as follows,

γ = mα1 + ℓα2, (8.2 )
and the only values of (m, n ) are (1,0), (0,1), (1,1) for the positive roots and minus these values for the negative ones.

The algebra 𝔰𝔩(3, ℝ) defines through the adjoint action a representation of 𝔰𝔩(3,ℝ ) itself, called the adjoint representation, which is eight-dimensional and denoted 8. The weights of the adjoint representation are the roots, plus the weight (0,0) which is doubly degenerate. The lowest weight of the adjoint representation is

Λ ð”¤ = − α1 − α2, (8.3 )
corresponding to the generator [f1,f2]. We display the weights of the adjoint representation in Figure 45View Image.
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Figure 45: Level decomposition of the adjoint representation ℛad = 8 of 𝔰𝔩(3,ℝ) into representations of the subalgebra 𝔰𝔩(2,ℝ ). The labels 1 and 2 indicate the simple roots α1 and α 2. Level zero corresponds to the horizontal axis where we find the adjoint representation (0) ℛ ad = 30 of 𝔰𝔩(2,ℝ ) (red nodes) and the singlet representation (0) ℛs = 10 (green circle about the origin). At level one we find the two-dimensional representation (1) ℛ = 21 (green nodes). The black arrow denotes the negative level root − α2 and so gives rise to the level ℓ = − 1 representation ℛ (−1) = 2 (− 1). The blue arrows represent the fundamental weights Λ1 and Λ2.

The idea of the level decomposition is to decompose the adjoint representation into representations of one of the regular 𝔰𝔩(2,ℝ )-subalgebras associated with one of the two simple roots α1 or α2, i.e., either {e1,α∨1,f1} or {e2,α ∨2,f2}. For definiteness we choose the level to count the number ℓ of times the root α2 occurs, as was anticipated by the notation in Equation (8.2View Equation). Consider the subspace of the adjoint representation spanned by the vectors with a fixed value of ℓ. This subspace is invariant under the action of the subalgebra ∨ {e1,α1,f1}, which only changes the value of m. Vectors at a definite level transform accordingly in a representation of the regular 𝔰𝔩(2,ℝ)-subalgebra

∨ 𝔯 ≡ ℝe1 ⊕ ℝα 1 ⊕ ℝf1. (8.4 )

Let us begin by analyzing states at level ℓ = 0, i.e., with weights of the form γ = m α1. We see from Figure 45View Image that we are restricted to move along the horizontal axis in the root diagram. By the defining Lie algebra relations we know that adf1(f1) = 0, implying that (0) Λad = − α1 is a lowest weight of the 𝔰𝔩(2,ℝ )-representation. Here, the superscript 0 indicates that this is a level ℓ = 0 representation. The corresponding complete irreducible module is found by acting on f1 with e1, yielding

∨ ∨ [e1,f1] = α1 , [e1,α1] = − 2e1, [e1,e1] = 0. (8.5 )
We can then conclude that Λ(a0d)= − α1 is the lowest weight of the three-dimensional adjoint representation 30 of 𝔰𝔩(2,ℝ ) with weights (0) (0) {Λ ad ,0,− Λad}, where the subscript on 30 again indicates that this representation is located at level ℓ = 0 in the decomposition. The module for this representation is ℒ (Λ(0)) = span {f ,α∨,e } ad 1 1 1.

This is, however, not the complete content at level zero since we must also take into account the Cartan generator α ∨2 which remains at the origin of the root diagram. We can combine α ∨2 with α∨1 into the vector

h = α ∨1 + 2 α∨2, (8.6 )
which constitutes the one-dimensional singlet representation 10 of 𝔯 since it is left invariant under all generators of 𝔯,
∨ [e1,h ] = [f1,h] = [α1,h] = 0, (8.7 )
as follows trivially from the Chevalley relations. Thus level zero contains the representations 3 0 and 10.

Note that the vectors at level 0 not only transform in a (reducible) representation of 𝔰𝔩(2,ℝ), but also form a subalgebra since the level is additive under taking commutators. The algebra in question is 𝔤𝔩(2,ℝ ) = 𝔰𝔩(2,ℝ ) ⊕ ℝ. Accordingly, if the generator α ∨2 is added to the subalgebra 𝔯, through the combination in Equation (8.6View Equation), so as to take the entire ℓ = 0 subspace, 𝔯 is enlarged from 𝔰𝔩(2,ℝ ) to 𝔤𝔩(2,ℝ ), the generator h being somehow the “trace” part of 𝔤𝔩(2, ℝ). This fact will prove to be important in subsequent sections.

Let us now ascend to the next level, ℓ = 1. The weights of 𝔯 at level 1 take the general form γ = m α1 + α2 and the lowest weight is Λ (1) = α2, which follows from the vanishing of the commutator

[f1,e2] = 0. (8.8 )
Note that m ≥ 0 whenever ℓ > 0 since m α1 + ℓα2 is then a positive root. The complete representation is found by acting on the lowest weight (1) Λ with e1 and we get that the commutator [e1,e2] is allowed by the Serre relations, while [e1,[e1,e2]] is killed, i.e.,
[e1,e2] ⁄= 0, [e1,[e1,e2]]] = 0. (8.9 )
The non-vanishing commutator eθ ≡ [e1,e2] is the vector associated with the highest root θ of 𝔰𝔩(3,ℝ ) given by
θ = α + α . (8.10 ) 1 2
This is just the negative of the lowest weight Λ ð”¤. The only representation at level one is thus the two-dimensional representation 21 of 𝔯 with weights {Λ (1),θ}. The decomposition stops at level one for 𝔰𝔩(3,ℝ ) because any commutator with two e2’s vanishes by the Serre relations. The negative level representations may be found simply by applying the Chevalley involution and the result is the same as for level one.

Hence, the total level decomposition of 𝔰𝔩(3,ℝ ) in terms of the subalgebra 𝔰𝔩(2, ℝ) is given by

8 = 30 ⊕ 10 ⊕ 21 ⊕ 2(−1). (8.11 )
Although extremely simple (and familiar), this example illustrates well the situation encountered with more involved cases below. In the following analysis we will not mention the negative levels any longer because these can always be obtained simply through a reflection with respect to the ℓ = 0 “hyperplane”, using the Chevalley involution.
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