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8.3 Level decomposition of AE3

The Kac–Moody algebra ++ AE3 = A1 is one of the simplest hyperbolic algebras and so provides a nice testing ground for investigating general properties of hyperbolic Kac–Moody algebras. From a physical point of view, it is the Weyl group of AE3 which governs the chaotic behavior of pure four-dimensional gravity close to a spacelike singularity [46Jump To The Next Citation Point], as we have explained. Moreover, as we saw in Section 3, the Weyl group of AE3 is isomorphic with the well-known arithmetic group P GL (2,ℤ ) which has interesting properties [75Jump To The Next Citation Point].

The level decomposition of 𝔤 = AE3 follows a similar route as for 𝔰𝔩(3,ℝ) above, but the result is much more complicated due to the fact that AE3 is infinite-dimensional. This decomposition has been treated before in [48Jump To The Next Citation Point]. Recall that the Cartan matrix for AE 3 is given by

( 2 − 2 0) ( ) − 2 2 − 1 , (8.30 ) 0 − 1 2
and the associated Dynkin diagram is given in Figure 46View Image.
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Figure 46: The Dynkin diagram of the hyperbolic Kac–Moody algebra AE3 ≡ A+1+. The labels indicate the simple roots α ,α 1 2 and α 3. The nodes “2” and “3” correspond to the subalgebra 𝔯 = 𝔰𝔩(3,ℝ) with respect to which we perform the level decomposition.

We see that there exist three rank 2 regular subalgebras that we can use for the decomposition: A2,A1 ⊕ A1 or + A 1. We will here focus on the decomposition into representations of 𝔯 = A2 = 𝔰𝔩(3, ℝ) because this is the one relevant for pure gravity in four dimensions [46]31. The level ℓ is then the coefficient in front of the simple root α1 in an expansion of an arbitrary root ⋆ γ ∈ 𝔥 𝔤, i.e.,

γ = ℓα1 + m2 α2 + m3 α3. (8.31 )

We restrict henceforth our analysis to positive levels only, ℓ ≥ 0. Before we begin, let us develop an intuitive idea of what to expect. We know that at each level we will have a set of finite-dimensional representations of the subalgebra 𝔯. The corresponding weight diagrams will then be represented in a Euclidean two-dimensional lattice in exactly the same way as in Figure 45View Image above. The level ℓ can be understood as parametrizing a third direction that takes us into the full three-dimensional root space of AE3. We display the level decomposition up to positive level two in Figure 47View Image32.

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Figure 47: Level decomposition of the adjoint representation of AE3. We have displayed the decomposition up to positive level ℓ = 2. At level zero we have the adjoint representation (0) ℛ 1 = 80 of 𝔰𝔩(3,ℝ ) and the singlet representation (0) ℛ 2 = 10 defined by the simple Cartan generator ∨ α 1. Ascending to level one with the root α1 (green vector) gives the lowest weight Λ (1) of the representation ℛ (1) = 61. The weights of ℛ (1) labelled by white crosses are on the lightcone and so their norm squared is zero. At level two we find the lowest weight Λ(2) (blue vector) of the 15-dimensional representation ℛ (2) = 15 2. Again, the white crosses label weights that are on the lightcone. The three innermost weights are inside of the lightcone and the rings indicate that these all have multiplicity 2 as weights of (2) ℛ. Since these also have multiplicity 2 as roots of ⋆ 𝔥 𝔤 we find that the outer multiplicity of this representation is one, μ(ℛ (2)) = 1.

From previous sections we recall that AE3 is hyperbolic so its root space is of Lorentzian signature. This implies that there is a lightcone in 𝔥⋆𝔤 whose origin lies at the origin of the root diagram for the adjoint representation of 𝔯 at level ℓ = 0. The lightcone separates real roots from imaginary roots and so it is clear that if a representation at some level ℓ intersects the walls of the lightcone, this means that some weights in the representation will correspond to imaginary roots of ⋆ 𝔥𝔤 but will be real as weights of ⋆ 𝔥 𝔯. On the other hand if a weight lies outside of the lightcone it will be real both as a root of 𝔥⋆𝔤 and as a weight of 𝔥⋆𝔯.

8.3.1 Level ℓ = 0

Consider first the representation content at level zero. Given our previous analysis we expect to find the adjoint representation of 𝔯 with the additional singlet representation from the Cartan generator α∨1. The Chevalley generators of 𝔯 are {e2,f2,e3,f3,α ∨2,α∨3 } and the generators associated to the root defining the level are {e1,f1,α ∨1}. As discussed previously, the additional Cartan generator α ∨ 1 that sits at the origin of the root space enlarges the subalgebra from 𝔰𝔩(3,ℝ ) to 𝔤𝔩(3,ℝ ). A canonical realisation of 𝔤𝔩(3,ℝ ) is obtained by defining the Chevalley generators in terms of the matrices i K j (i,j = 1,2,3) whose commutation relations are

[Kij, Kkl] = δkjKil − δilKkj. (8.32 )
All the defining Lie algebra relations of 𝔤𝔩(3,ℝ ) are then satisfied if we make the identifications
α ∨= K1 − K2 − K3 , 2 1 1∨ 21 21 3 e2 = K 31, f2 = K 22, α 2∨ = K 32 − K 12, (8.33 ) e3 = K 2, f3 = K 3, α 3 = K 3 − K 2.
Note that the trace K1 + K2 + K3 1 2 3 is equal to − 4 α∨ − 2α ∨− 3 α∨ 2 3 1. The generators e 1 and f 1 can of course not be realized in terms of the matrices i K j since they do not belong to level zero. The invariant bilinear form ( | ) at level zero reads
i k ik i k (K j|K l) = δlδj − δjδl , (8.34 )
where the coefficient in front of the second term on the right hand side is fixed to − 1 through the embedding of 𝔤𝔩(3,ℝ ) in AE 3.

The commutation relations in Equation (8.32View Equation) characterize the adjoint representation of 𝔤𝔩(3, ℝ) as was expected at level zero, which decomposes as the representation ℛ (0ad)⊕ ℛ (s0) of 𝔰𝔩(3,ℝ ) with (0) ℛ ad = 80 and (0) ℛs = 10.

8.3.2 Dynkin labels

It turns out that at each positive level ℓ, the weight that is easiest to identify is the lowest weight. For example, at level one, the lowest weight is simply α 1 from which one builds all the other weights by adding appropriate positive combinations of the roots α2 and α3. It will therefore turn out to be convenient to characterize the representations at each level by their (conjugate) Dynkin labels p2 and p3 defined as the coefficients of minus the (projected) lowest weight − ¯Λ (ℓlw) expanded in terms of the fundamental weights λ2 and λ3 of 𝔰𝔩(3,ℝ) (blue arrows in Figure 48View Image),

(ℓ) − ¯Λlw = p2λ2 + p3λ3. (8.35 )
Note that for any weight Λ we have the inequality
(Λ|Λ ) ≤ (Λ¯|¯Λ) (8.36 )
since ⊥ ⊥ (Λ|Λ) = (¯Λ |Λ¯) − |(Λ |Λ )|.

The Dynkin labels can be computed using the scalar product ( | ) in 𝔥⋆𝔤 in the following way:

(ℓ) (ℓ) p2 = − (α2|Λlw), p3 = − (α3|Λ lw ). (8.37 )
For the level zero sector we therefore have
80 : [p2,p3 ] = [1,1], (8.38 ) 10 : [p2,p3 ] = [0,0].

The module for the representation 80 is realized by the eight traceless generators Kij of 𝔰𝔩(3,ℝ ) and the module for the representation 10 corresponds to the “trace” α ∨1.

Note that the highest weight Λ hw of a given representation of 𝔯 is not in general equal to minus the lowest weight Λ of the same representation. In fact, − Λhw is equal to the lowest weight of the conjugate representation. This is the reason our Dynkin labels are really the conjugate Dynkin labels in standard conventions. It is only if the representation is self-conjugate that we have Λhw = − Λ. This is the case for example in the adjoint representation 80.

It is interesting to note that since the weights of a representation at level ℓ are related by Weyl reflections to weights of a representation at level − ℓ, it follows that the negative of a lowest weight (ℓ) Λ at level ℓ is actually equal to the highest weight (−ℓ) Λhw of the conjugate representation at level − ℓ. Therefore, the Dynkin labels at level ℓ as defined here are the standard Dynkin labels of the representations at level − ℓ.

8.3.3 Level ℓ = 1

We now want to exhibit the representation content at the next level ℓ = 1. A generic level one commutator is of the form [e ,[⋅⋅⋅[⋅⋅⋅ ]]] 1, where the ellipses denote (positive) level zero generators. Hence, including the generator e1 implies that we step upwards in root space, i.e., in the direction of the forward lightcone. The root vector e1 corresponds to a lowest weight of 𝔯 since it is annihilated by f2 and f3,

adf2(e1) = [f2,e1] = 0, ad (e ) = [f ,e] = 0, (8.39 ) f3 1 3 1
which follows from the defining relations of AE3.

Explicitly, the root associated to e1 is simply the root α1 that defines the level expansion. Therefore the lowest weight of this level one representation is

(1) Λ lw = ¯α1, (8.40 )
Although α1 is a real positive root of ⋆ 𝔥 𝔤, its projection ¯α(1) is a negative weight of ⋆ 𝔥𝔯. Note that since the lowest weight (1) Λ 1 is real, the representation ℛ (1) has outer multiplicity one, μ (ℛ (1)) = 1.

Acting on the lowest weight state with the raising operators of 𝔯 yields the six-dimensional representation (1) ℛ = 61 of 𝔰𝔩(3,ℝ ). The root α1 is displayed as the green vector in Figure 47View Image, taking us from the origin at level zero to the lowest weight of (1) ℛ. The Dynkin labels of this representation are

p2(ℛ (1)) = − (α2 |α1) = 2, p (ℛ (1)) = − (α |α ) = 0, (8.41 ) 3 3 1
which follows directly from the Cartan matrix of AE3. Three of the weights in ℛ (1) correspond to roots that are located on the lightcone in root space and so are null roots of 𝔥⋆ 𝔤. These are α1 + α2, α1 + α2 + α3 and α1 + 2α2 + α3 and are labelled with white crosses in Figure 47View Image. The other roots present in the representation, in addition to α1, are α1 + 2 α2 and α1 + 2α2 + 2α3, which are real. This representation therefore contains no weights inside the lightcone.

The 𝔤𝔩(3,ℝ )-generator encoding this representation is realized as a symmetric 2-index tensor ij E which indeed carries six independent components. In general we can easily compute the dimensionality of a representation given its Dynkin labels using the Weyl dimension formula which for 𝔰𝔩(3,ℝ ) takes the form [84]

( ) dΛ (𝔰𝔩(3,ℝ )) = (p2 + 1)(p3 + 1) 1(p2 + p3) + 1 . (8.42 ) hw 2
In particuar, for (p ,p ) = (2, 0) 2 3 this gives indeed d = 6 Λ(1h)w,1.

It is convenient to encode the Dynkin labels, and, consequently, the index structure of a given representation module, in a Young tableau. We follow conventions where the first Dynkin label gives the number of columns with 1 box and the second Dynkin label gives the number of columns with 2 boxes33. For the representation 61 the first Dynkin label is 2 and the second is 0, hence the associated Young tableau is

61 ⇐ ⇒ |--|-. (8.43 ) -----

At level ℓ = − 1 there is a corresponding negative generator Fij. The generators Eij and Fij transform contravariantly and covariantly, respectively, under the level zero generators, i.e.,

i kl k il l ki [K j,E ] = δjE + δjE , (8.44 ) [Kij,Fkl] = − δikFjl − δilFkj.
The internal commutator on level one can be obtained by first identifying
e1 ≡ E11, f1 ≡ F11, (8.45 )
and then by demanding ∨ [e1,f1] = α 1 we find
ij (i j) (ik) 1 2 3 [E ,Fkl] = 2δ(kK l) − δk δl (K 1 + K 2 + K 3), (8.46 )
which is indeed compatible with the realisation of α∨ 1 given in Equation (8.33View Equation). The Killing form at level 1 takes the form
(Fij|Ekl) = δ(kδl). (8.47 ) i j

8.3.4 Constraints on Dynkin labels

As we go to higher and higher levels it is useful to employ a systematic method to investigate the representation content. It turns out that it is possible to derive a set of equations whose solutions give the Dynkin labels for the representations at each level [47Jump To The Next Citation Point].

We begin by relating the Dynkin labels to the expansion coefficients ℓ,m2 and m3 of a root ⋆ γ ∈ 𝔥 𝔤, whose projection ¯γ onto ⋆ 𝔥 𝔯 is a lowest weight vector for some representation of 𝔯 at level ℓ. We let a = 2,3 denote indices in the root space of the subalgebra 𝔰𝔩(3,ℝ ) and we let i = 1,2,3 denote indices in the full root space of AE3. The formula for the Dynkin labels then gives

pa = − (αa|γ) = − ℓAa1 − m2Aa2 − m3Aa3, (8.48 )
where Aij is the Cartan matrix for AE3, given in Equation (8.30View Equation). Explicitly, we find the following relations between the coefficients m2, m3 and the Dynkin labels:
p = 2ℓ − 2m + m , 2 2 3 (8.49 ) p3 = m2 − 2m3.
These formulae restrict the possible Dynkin labels for each ℓ since the coefficients m2 and m3 must necessarily be non-negative integers. Therefore, by inverting Equation (8.49View Equation) we obtain two Diophantine equations that restrict the possible Dynkin labels,
m2 = 4ℓ − 2-p2 − 1p3 ≥ 0, 3 3 3 (8.50 ) m3 = 2ℓ − 1-p2 − 2p3 ≥ 0. 3 3 3
In addition to these constraints we can also make use of the fact that we are decomposing the adjoint representation of AE 3. Since the weights of the adjoint representation are the roots of the algebra we know that the lowest weight vector Λ must satisfy
(Λ|Λ) ≤ 2. (8.51 )
Taking Λ = ℓα1 + m2 α2 + m3 α3 then gives the following constraint on the coefficients ℓ,m2 and m3:
(Λ |Λ) = 2ℓ2 + 2m22 + 2m23 − 4ℓm2 − 2m2m3 ≤ 2. (8.52 )
We are interested in finding an equation for the Dynkin labels, so we insert Equation (8.50View Equation) into Equation (8.52View Equation) to obtain the constraint
2 2 2 p 2 + p3 + p2p3 − ℓ ≤ 3. (8.53 )
The inequalities in Equation (8.50View Equation) and Equation (8.53View Equation) are sufficient to determine the representation content at each level ℓ. However, this analysis does not take into account the outer multiplicities, which must be analyzed separately by comparing with the known root multiplicities of AE 3 as given in Table 38. We shall return to this issue later.

8.3.5 Level ℓ = 2

Let us now use these results to analyze the case for which ℓ = 2. The following equations must then be satisfied:

a8 − 2p2 − p3 ≥ 0, 4 − p2 − 2p3 ≥ 0, (8.54 ) p2+ p2+ p2p3 ≤ 7. 2 3
The only admissible solution is p2 = 2 and p3 = 1. This corresponds to a 15-dimensional representation 152 with the following Young tableau
|-|--|-| 61 ⇐ ⇒ |-|----. (8.55 ) ---
Note that p2 = p3 = 0 is also a solution to Equation (8.54View Equation) but this violates the constraint that m2 and m3 be integers and so is not allowed.

Moreover, the representation [p2,p3] = [0, 2] is also a solution to Equation (8.54View Equation) but has not been taken into account because it has vanishing outer multiplicity. This can be understood by examining Figure 48View Image a little closer. The representation [0,2] is six-dimensional and has highest weight 2λ3, corresponding to the middle node of the top horizontal line in Figure 48View Image. This weight lies outside of the lightcone and so is a real root of AE3. Therefore we know that it has root multiplicity one and may therefore only occur once in the level decomposition. Since the weight 2λ3 already appears in the larger representation 152 it cannot be a highest weight in another representation at this level. Hence, the representation [0, 2] is not allowed within AE3. A similar analysis reveals that also the representation [p ,p ] = [1,0] 2 3, although allowed by Equation (8.54View Equation), has vanishing outer multiplicity.

The level two module is realized by the tensor jk Ei whose index structure matches the Young tableau above. Here we have used the 𝔰𝔩(3,ℝ)-invariant antisymmetric tensor εabc to lower the two upper antisymmetric indices leading to a tensor Eijk with the properties

jk (jk) ik Ei = Ei , Ei = 0. (8.56 )
This corresponds to a positive root generator and by the Chevalley involution we have an associated negative root generator i F jk at level ℓ = − 2. Because the level decomposition gives a gradation of AE3 we know that all higher level generators can be obtained through commutators of the level one generators. More specifically, the level two tensor Eijk corresponds to the commutator
[Eij,Ekl ] = εmk (iE j)l + εml(iE j)k, (8.57 ) m m
where εijk is the totally antisymmetric tensor in three dimensions. Inserting the result p2 = 2 and p3 = 1 into Equation (8.50View Equation) gives m2 = 1 and m3 = 0, thus providing the explicit form of the root taking us from the origin of the root diagram in Figure 47View Image to the lowest weight of 15 2 at level two:
Λ (2) = 2α1 + α2. (8.58 )
This is a real root of AE3, (γ|γ) = 2, and hence the representation 152 has outer multiplicity one. We display the representation 152 of 𝔰𝔩(3,ℝ) in Figure 48View Image. The lower leftmost weight is the lowest weight (2) Λ. The expansion of the lowest weight (2) Λlw in terms of the fundamental weights λ2 and λ3 is given by the (conjugate) Dynkin labels
(2) − Λ hw = p2λ2 + p3λ3 = 2λ2 + λ3. (8.59 )
The three innermost weights all have multiplicity 2 as weights of 𝔰𝔩(3,ℝ ), as indicated by the black circles. These lie inside the lightcone of ⋆ 𝔥 𝔤 and so are timelike roots of AE3.
View Image

Figure 48: The representation 152 of 𝔰𝔩(3,ℝ) appearing at level two in the decomposition of the adjoint representation of AE3 into representations of 𝔰𝔩(3,ℝ ). The lowest leftmost node is the lowest weight of the representation, corresponding to the real root Λ(2) = 2α + α 1 2 of AE 3. This representation has outer multiplicity one.

8.3.6 Level ℓ = 3

We proceed quickly past level three since the analysis does not involve any new ingredients. Solving Equation (8.50View Equation) and Equation (8.53View Equation) for ℓ = 3 yields two admissible 𝔰𝔩(3,ℝ ) representations, 273 and 83, represented by the following Dynkin labels and Young tableaux:

|-|-|-|-| 27 : [p ,p ] = [2,2] ⇐ ⇒ |-|-|----, 3 2 3 ----- |-|-| (8.60 ) 83 : [p2,p3] = [1,1] ⇐ ⇒ |-|-. ---
The lowest weight vectors for these representations are
Λ (3)= 3α1 + 2α2, 15(3) (8.61 ) Λ 8 = 3α1 + 3α2 + α3.
The lowest weight vector for 273 is a real root of AE3, (Λ (3)|Λ(3)) = 2 27 27, while the lowest weight vectors for 83 is timelike, (3) (3) (Λ 8 |Λ 8 ) = − 4. This implies that the entire representation 83 lies inside the lightcone of 𝔥 ⋆𝔤. Both representations have outer multiplicity one.

Note that [0, 3] and [3,0] are also admissible solutions but have vanishing outer multiplicities by the same arguments as for the representation [0,2] at level 2.

8.3.7 Level ℓ = 4

At this level we encounter for the first time a representation with non-trivial outer multiplicity. It is a 15-dimensional representation with the following Young tableau structure:

|-|-|-| ¯ |-|-|-- 154 : [p2,p3] = [1,2] ⇐ ⇒ ----- . (8.62 )
The lowest weight vector is
Λ(14¯5)= 4α1 + 4α2 + α3, (8.63 )
which is an imaginary root of AE3,
(Λ(14¯5)|Λ (4¯1)5) = − 6. (8.64 )
From Table 38 we find that this root has multiplicity 5 as a root of AE3,
(4) mult(Λ ¯15) = 5. (8.65 )
In order for Equation (8.26View Equation) to make sense, this multiplicity must be matched by the total multiplicity of (4) Λ ¯15 as a weight of 𝔰𝔩(3,ℝ) representations at level four. The remaining representations at this level are
|-|-|-|--| 244 : [3,1] ⇐ ⇒ |-|------, --- |-| ¯34 : [0,1] ⇐ ⇒ |-, |-|-| (8.66 ) 64 : [2,0] ⇐ ⇒ -----, |-|-|-|--|-|-| |-|-|-|------- 424 : [2,3] ⇐ ⇒ ------- .
By drawing these representations explicitly, one sees that the root 4α1 + 4α2 + α3, representing the weight (4) Λ ¯15, also appears as a weight (but not as a lowest weight) in the representations 424 and 244. It occurs with weight multiplicity 1 in the 244 but with weight multiplicity 2 in the 424. Taking also into account the representation 1¯54 in which it is the lowest weight we find a total weight multiplicity of 4. This implies that, since in AE3
mult(4α1 + 4α2 + α3) = 5, (8.67 )
the outer multiplicity of 1¯54 must be 2, i.e.,
( ) μ Λ(2¯) = 2. (8.68 ) 15
When we go to higher and higher levels, the outer multiplicities of the representations located entirely inside the lightcone in 𝔥𝔤 increase exponentially.
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