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  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 to37. By inspection of the above scalar products, we come to the important conclusion that the Dynkin diagram of the regular, rank , Kac–Moody subalgebra associated with the geometric configuration is just its line incidence diagram. We shall call the Kac–Moody algebra the algebra “dual” to the geometric configuration .
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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