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,
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
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
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
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
Hence, the total level decomposition of in terms of the subalgebra
is given by
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