Recall from Section 6 that is the split real form of , and is thus defined through the same Chevalley–Serre presentation as for , but with all coefficients restricted to the real numbers.

The Cartan generators will indifferently be denoted by . As we have seen, they form a basis of the Cartan subalgebra , while the simple roots , associated with the raising operators and , form a basis of the dual root space . Any root can thus be decomposed in terms of the simple roots as follows,

and the only values of are , , for the positive roots and minus these values for the negative ones.The algebra defines through the adjoint action a representation of itself, called the adjoint representation, which is eight-dimensional and denoted . The weights of the adjoint representation are the roots, plus the weight which is doubly degenerate. The lowest weight of the adjoint representation is

corresponding to the generator . We display the weights of the adjoint representation in Figure 45.The idea of the level decomposition is to decompose the adjoint representation into representations of one of the regular -subalgebras associated with one of the two simple roots or , i.e., either or . For definiteness we choose the level to count the number of times the root occurs, as was anticipated by the notation in Equation (8.2). Consider the subspace of the adjoint representation spanned by the vectors with a fixed value of . This subspace is invariant under the action of the subalgebra , which only changes the value of . Vectors at a definite level transform accordingly in a representation of the regular -subalgebra

Let us begin by analyzing states at level , i.e., with weights of the form . We see from Figure 45 that we are restricted to move along the horizontal axis in the root diagram. By the defining Lie algebra relations we know that , implying that is a lowest weight of the -representation. Here, the superscript indicates that this is a level representation. The corresponding complete irreducible module is found by acting on with , yielding

We can then conclude that is the lowest weight of the three-dimensional adjoint representation of with weights , where the subscript on again indicates that this representation is located at level in the decomposition. The module for this representation is .This is, however, not the complete content at level zero since we must also take into account the Cartan generator which remains at the origin of the root diagram. We can combine with into the vector

which constitutes the one-dimensional singlet representation of since it is left invariant under all generators of , as follows trivially from the Chevalley relations. Thus level zero contains the representations and .Note that the vectors at level 0 not only transform in a (reducible) representation of , but also form a subalgebra since the level is additive under taking commutators. The algebra in question is . Accordingly, if the generator is added to the subalgebra , through the combination in Equation (8.6), so as to take the entire subspace, is enlarged from to , the generator being somehow the “trace” part of . This fact will prove to be important in subsequent sections.

Let us now ascend to the next level, . The weights of at level 1 take the general form and the lowest weight is , which follows from the vanishing of the commutator

Note that whenever since is then a positive root. The complete representation is found by acting on the lowest weight with and we get that the commutator is allowed by the Serre relations, while is killed, i.e., The non-vanishing commutator is the vector associated with the highest root of given by This is just the negative of the lowest weight . The only representation at level one is thus the two-dimensional representation of with weights . The decomposition stops at level one for because any commutator with two ’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 in terms of the subalgebra is given by

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 “hyperplane”, using the Chevalley involution.http://www.livingreviews.org/lrr-2008-1 |
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