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
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
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
),
The Dynkin labels can be computed using the scalar product in
in the following way:
The 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
,
Explicitly, the root associated to is simply the root
that defines the level expansion. Therefore
the lowest weight of this level one representation is
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
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]
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 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.,
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
Let us now use these results to analyze the case for which . The following equations must then be
satisfied:
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
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:
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 ofhttp://www.livingreviews.org/lrr-2008-1 | ![]() This work is licensed under a Creative Commons Attribution-Noncommercial-No Derivative Works 2.0 Germany License. Problems/comments to |