As explained in Section 4.9.2, the restricted root system of the real form can be either reduced or
non-reduced. If it is reduced, it corresponds to one of the root systems of the finite-dimensional simple Lie
algebras. On the other hand, if the restricted root system is non-reduced, it is necessarily of
-type [93
] (see Figure 19
for a graphical presentation of the
root system).
By definition, the restricted Weyl group is the Coxeter group generated by the fundamental reflections,
Equation (4.55), with respect to the simple roots of the restricted root system. The restricted Weyl group
preserves multiplicities [93
].
Although multiplicities are an essential ingredient for describing the full symmetry , they turn out to be
irrelevant for the construction of the gravitational billiard. For this reason, it is useful to consider the
maximal split “subalgebra”
, which is defined as the real, semi-simple, split Lie algebra with the
same root system as the restricted root system as
(in the
-case, we choose for
definiteness the root system of
to be of
-type). The real rank of
coincides with the rank
of its complexification
, and one can find a Cartan subalgebra
of
, consisting of
all generators of
which are diagonalizable over the reals. This subalgebra
has the
same dimension as the maximal noncompact subalgebra
of the Cartan subalgebra
of
.
By construction, the root space decomposition of with respect to
provides the same root system
as the restricted root space decomposition of
with respect to
, except for multiplicities, which are
all trivial (i.e., equal to one) for
. In the
-case, there is also the possibility that
twice a root of
might be a root of
. It is only when
is itself split that
and
coincide.
One calls the “split symmetry algebra”. It contains as we shall see all the information about the
billiard region [95
]. How
can be embedded as a subalgebra of
is not a question that shall be of our
concern here.
The purpose of this section is to use the Iwasawa decomposition for described in Section 6.4.5 to
derive the scalar Lagrangian based on the coset space
. The important point is to understand
the origin of the similarities between the two Lagrangians in Equation (5.45
) and Equation (7.8
)
below.
The full algebra is subject to the root space decomposition
This implies that when constructing a Lagrangian based on the coset space , the only part
of
that will show up in the Borel gauge is the Borel subalgebra
More specifically, an (-dependent) element of the coset space
takes the form
By comparing Equation (7.8) with the corresponding expression (5.45
) for the split case, it is
clear why it is the maximal split subalgebra of the U-duality algebra that is relevant for the
gravitational billiard. Were it not for the additional sum over multiplicities, Equation (7.8
) would
exactly be the Lagrangian for the coset space
, where
is the maximal
compact subalgebra of
(note that
). Recall now that from the point of view of the
billiard, the positive roots correspond to walls that deflect the particle motion in the Cartan
subalgebra. Therefore, multiplicities of roots are irrelevant since these will only result in several walls
stacked on top of each other without affecting the dynamics. (In the
-case, the wall
associated with
is furthermore subdominant with respect to the wall associated with
when
both
and
are restricted roots, so one can keep only the wall associated with
.
This follows from the fact that in the
-case the root system of
is taken to be of
-type.)
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