None of the algebras relevant to the truncated models turn out to be hyperbolic: They can be finite,
affine, or Lorentzian with infinite-volume Weyl chamber. Because of this, the solutions are non-chaotic.
After a finite number of collisions, they settle asymptotically into a definite Kasner regime (both in the
future and in the past).
In order to match diagonal Bianchi I cosmologies with the sigma model, one must truncate the
action in such a way that the sigma model metric
is diagonal. This will be the case if
the subalgebra
to which one truncates has no generator
with
. Indeed, recall from
Section 9 that the off-diagonal components of the metric are precisely the exponentials of the associated
sigma model fields. The set of simple roots of
should therefore not contain any root at level
zero.
Consider “electric” regular subalgebras of , for which the simple roots are all at level
one, where the 3-form electric field variables live. These roots can be parametrized by three
indices corresponding to the indices of the electric field, with
. We denote them
. For instance,
. In terms of the
-parametrization of [45
, 48
], one has
.
Now, for to be a regular subalgebra, it must fulfill, as we have seen, the condition that the difference
between any two of its simple roots is not a root of
:
for any pair
and
of simple roots of
. But one sees by inspection of the commutator of
with
in
Equation (8.78
) that
is a root of
if and only if the sets
and
have exactly two points in common. For instance, if
,
and
, the commutator of
with
produces the off-diagonal generator
corresponding to a level zero root of
. In order to fulfill the required condition, one must avoid
this case, i.e., one must choose the set of simple roots of the electric regular subalgebra
in
such a way that given a pair of indices
, there is at most one
such that the root
is a simple root of
, with
being the re-ordering of
such that
.
To each of the simple roots of
, one can associate the line
connecting the three
points
,
and
. If one does this, one sees that the above condition is equivalent to the following
statement: The set of points and lines associated with the simple roots of
must fulfill the third rule
defining a geometric configuration, namely, that two points determine at most one line. Thus,
this geometric condition has a nice algebraic interpretation in terms of regular subalgebras of
.
The first rule, which states that each line contains 3 points, is a consequence of the fact that the
-generators at level one are the components of a 3-index antisymmetric tensor. The second rule, that
each point is on
lines, is less fundamental from the algebraic point of view since it is not required to
hold for
to be a regular subalgebra. It was imposed in [61
] in order to allow for solutions
isotropic in the directions that support the electric field. We keep it here as it yields interesting
structure.
We have just shown that each geometric configuration with
defines a regular subalgebra
of
. In order to determine what this subalgebra
is, one needs, according to the theorem
recalled in Section 4, to compute the Cartan matrix
Using for instance the root parametrization of [45, 48] and the expression of the scalar product in terms
of this parametrization, one easily verifies that the scalar product is equal to
Because the geometric configurations have the property that the number of lines through any point is equal
to a constant , the number of lines parallel to any given line is equal to a number
that depends only
on the configuration and not on the line. This is in fact true in general and not only for
as can be
seen from the following argument. For a configuration with
points,
lines and
lines through
each point, any given line
admits
true secants, namely,
through each of its
points38.
By definition, these secants are all distinct since none of the lines that
intersects at one of its points,
say
, can coincide with a line that it intersects at another of its points, say
, since the only line
joining
to
is
itself. It follows that the total number of lines that
intersects is the
number of true secants plus
itself, i.e.,
. As a consequence, each line in the
configuration admits
parallel lines, which is then reflected by the
fact that each node in the associated Dynkin diagram has the same number
of adjacent
nodes.
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