One can classify the various theories through the number of supersymmetries that they possess in
spacetime dimensions. All
-forms can be dualized to scalars or to 1-forms in four
dimensions so the theories all take the form of pure supergravities coupled to collections of Maxwell
multiplets. The analysis performed for the split forms in Section 5.3 were all concerned with the
cases of
or
supergravity in
. We consider all pure four-dimensional
supergravities (
) as well as pure
supergravity coupled to
Maxwell
multiplets.
As we have pointed out, the main new feature in the non-split cases is the possible appearance of
so-called twisted overextensions. These arise when the restricted root system of is of non-reduced type
hence yielding a twisted affine Kac–Moody algebra in the affine extension of
. It turns
out that the only cases for which the restricted root system is of non-reduced (
-type)
is for the pure
and
supergravities. The example of
was already
discussed in detail before, where it was found that the U-duality algebra is
whose restricted root system is
, thus giving rise to the twisted overextension
.
It turns out that for the
case the same twisted overextension appears. This is due
to the fact that the U-duality algebra is
which has the same restricted root
system as
, namely
. Hence,
controls the BKL-limit also for this
theory.
The case of follows along similar lines. In
the non-split form
of
appears, whose maximal split subalgebra is
. However, the relevant Kac–Moody
algebra is not
but rather
because the restricted root system of
is
.
In Table 37 we display the algebraic structure for all pure supergravities in four dimensions as well as
for supergravity with
Maxwell multiplets. We give the relevant U-duality algebras
, the
restricted root systems
, the maximal split subalgebras
and, finally, the resulting overextended
Kac–Moody algebras
.
Let us end this section by noting that the study of real forms of hyperbolic Kac–Moody algebras has been pursued in [17].
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