The level decomposition of follows a similar route as for above, but the result is much more complicated due to the fact that is infinite-dimensional. This decomposition has been treated before in [48]. Recall that the Cartan matrix for is given by

and the associated Dynkin diagram is given in Figure 46. We see that there exist three rank 2 regular subalgebras that we can use for the
decomposition: or . We will here focus on the decomposition into
representations of because this is the one relevant for pure gravity in four
dimensions [46]^{31}.
The level is then the coefficient in front of the simple root in an expansion of an arbitrary root
, i.e.,

We restrict henceforth our analysis to positive levels only, . 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 . The corresponding weight diagrams will then be
represented in a Euclidean two-dimensional lattice in exactly the same way as in Figure 45 above.
The level can be understood as parametrizing a third direction that takes us into the full
three-dimensional root space of . We display the level decomposition up to positive level two in
Figure 47^{32}.

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

Consider first the representation content at level zero. Given our previous analysis we expect to find the adjoint representation of with the additional singlet representation from the Cartan generator . The Chevalley generators of are and the generators associated to the root defining the level are . As discussed previously, the additional Cartan generator that sits at the origin of the root space enlarges the subalgebra from to . A canonical realisation of is obtained by defining the Chevalley generators in terms of the matrices whose commutation relations are

All the defining Lie algebra relations of are then satisfied if we make the identifications Note that the trace is equal to . The generators and can of course not be realized in terms of the matrices since they do not belong to level zero. The invariant bilinear form at level zero reads where the coefficient in front of the second term on the right hand side is fixed to through the embedding of in .The commutation relations in Equation (8.32) characterize the adjoint representation of as was expected at level zero, which decomposes as the representation of with and .

It turns out that at each positive level , the weight that is easiest to identify is the lowest weight. For example, at level one, the lowest weight is simply from which one builds all the other weights by adding appropriate positive combinations of the roots and . It will therefore turn out to be convenient to characterize the representations at each level by their (conjugate) Dynkin labels and defined as the coefficients of minus the (projected) lowest weight expanded in terms of the fundamental weights and of (blue arrows in Figure 48),

Note that for any weight we have the inequality since .The Dynkin labels can be computed using the scalar product in in the following way:

For the level zero sector we therefore haveThe module for the representation is realized by the eight traceless generators of and the module for the representation corresponds to the “trace” .

Note that the highest weight of a given representation of is not in general equal to minus the lowest weight of the same representation. In fact, 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 . This is the case for example in the adjoint representation .

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

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

which follows from the defining relations of .Explicitly, the root associated to is simply the root that defines the level expansion. Therefore the lowest weight of this level one representation is

Although is a real positive root of , its projection is a negative weight of . Note that since the lowest weight is real, the representation has outer multiplicity one, .Acting on the lowest weight state with the raising operators of yields the six-dimensional representation of . The root is displayed as the green vector in Figure 47, taking us from the origin at level zero to the lowest weight of . The Dynkin labels of this representation are

which follows directly from the Cartan matrix of . Three of the weights in correspond to roots that are located on the lightcone in root space and so are null roots of . These are , and and are labelled with white crosses in Figure 47. The other roots present in the representation, in addition to , are and , which are real. This representation therefore contains no weights inside the lightcone.The -generator encoding this representation is realized as a symmetric 2-index tensor 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 takes the form [84]

In particuar, for this gives indeed . 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
boxes^{33}.
For the representation the first Dynkin label is 2 and the second is 0, hence the associated Young
tableau is

At level there is a corresponding negative generator . The generators and transform contravariantly and covariantly, respectively, under the level zero generators, i.e.,

The internal commutator on level one can be obtained by first identifying and then by demanding we find which is indeed compatible with the realisation of given in Equation (8.33). The Killing form at level 1 takes the form

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 [47].

We begin by relating the Dynkin labels to the expansion coefficients and of a root , whose projection onto is a lowest weight vector for some representation of at level . We let denote indices in the root space of the subalgebra and we let denote indices in the full root space of . The formula for the Dynkin labels then gives

where is the Cartan matrix for , given in Equation (8.30). Explicitly, we find the following relations between the coefficients and the Dynkin labels: These formulae restrict the possible Dynkin labels for each since the coefficients and must necessarily be non-negative integers. Therefore, by inverting Equation (8.49) we obtain two Diophantine equations that restrict the possible Dynkin labels, In addition to these constraints we can also make use of the fact that we are decomposing the adjoint representation of . Since the weights of the adjoint representation are the roots of the algebra we know that the lowest weight vector must satisfy Taking then gives the following constraint on the coefficients and : We are interested in finding an equation for the Dynkin labels, so we insert Equation (8.50) into Equation (8.52) to obtain the constraint The inequalities in Equation (8.50) and Equation (8.53) are sufficient to determine the representation content at each level . However, this analysis does not take into account the outer multiplicities, which must be analyzed separately by comparing with the known root multiplicities of as given in Table 38. We shall return to this issue later.

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

The only admissible solution is and . This corresponds to a 15-dimensional representation with the following Young tableau Note that is also a solution to Equation (8.54) but this violates the constraint that and be integers and so is not allowed.Moreover, the representation is also a solution to Equation (8.54) but has not been taken into account because it has vanishing outer multiplicity. This can be understood by examining Figure 48 a little closer. The representation is six-dimensional and has highest weight , corresponding to the middle node of the top horizontal line in Figure 48. This weight lies outside of the lightcone and so is a real root of . Therefore we know that it has root multiplicity one and may therefore only occur once in the level decomposition. Since the weight already appears in the larger representation it cannot be a highest weight in another representation at this level. Hence, the representation is not allowed within . A similar analysis reveals that also the representation , although allowed by Equation (8.54), has vanishing outer multiplicity.

The level two module is realized by the tensor whose index structure matches the Young tableau above. Here we have used the -invariant antisymmetric tensor to lower the two upper antisymmetric indices leading to a tensor with the properties

This corresponds to a positive root generator and by the Chevalley involution we have an associated negative root generator at level . Because the level decomposition gives a gradation of we know that all higher level generators can be obtained through commutators of the level one generators. More specifically, the level two tensor corresponds to the commutator where is the totally antisymmetric tensor in three dimensions. Inserting the result and into Equation (8.50) gives and , thus providing the explicit form of the root taking us from the origin of the root diagram in Figure 47 to the lowest weight of at level two: This is a real root of , , and hence the representation has outer multiplicity one. We display the representation of in Figure 48. The lower leftmost weight is the lowest weight . The expansion of the lowest weight in terms of the fundamental weights and is given by the (conjugate) Dynkin labels The three innermost weights all have multiplicity 2 as weights of , as indicated by the black circles. These lie inside the lightcone of and so are timelike roots of .We proceed quickly past level three since the analysis does not involve any new ingredients. Solving Equation (8.50) and Equation (8.53) for yields two admissible representations, and , represented by the following Dynkin labels and Young tableaux:

The lowest weight vectors for these representations are The lowest weight vector for is a real root of , , while the lowest weight vectors for is timelike, . This implies that the entire representation lies inside the lightcone of . Both representations have outer multiplicity one.Note that and are also admissible solutions but have vanishing outer multiplicities by the same arguments as for the representation at level 2.

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:

The lowest weight vector is which is an imaginary root of , From Table 38 we find that this root has multiplicity 5 as a root of , In order for Equation (8.26) to make sense, this multiplicity must be matched by the total multiplicity of as a weight of representations at level four. The remaining representations at this level are By drawing these representations explicitly, one sees that the root , representing the weight , also appears as a weight (but not as a lowest weight) in the representations and . It occurs with weight multiplicity 1 in the but with weight multiplicity 2 in the . Taking also into account the representation in which it is the lowest weight we find a total weight multiplicity of 4. This implies that, since in the outer multiplicity of must be 2, i.e., When we go to higher and higher levels, the outer multiplicities of the representations located entirely inside the lightcone in increase exponentially.http://www.livingreviews.org/lrr-2008-1 |
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