The hyperbolic Kac–Moody algebras have been classified in  and exist only up to rank 10 (see also ). In rank 10, there are four possibilities, known as , , and , and being dual to each other and possessing the same Weyl group (the notation will be explained below).
For a hyperbolic Kac–Moody algebra, the fundamental weights are timelike or null and lie within the (say) past lightcone. Similarly, the fundamental Weyl chamber defined by also lies within the past lightcone and is a fundamental region for the action of the Weyl group on the Tits cone, which coincides in fact with the past light cone. All these properties carries over from our discussion of hyperbolic Coxeter groups in Section 3.
The positive imaginary roots of the algebra fulfill (with, for any , strict inequality for at least one ) and hence, since they are non-spacelike, must lie in the future light cone. Recall indeed that the scalar product of two non-spacelike vectors with the same time orientation is non-positive. For this reason, it is also of interest to consider the action of the Weyl group on the future lightcone, obtained from the action on the past lightcone by mere changes of signs. A fundamental region is clearly given by . Any imaginary root is Weyl-conjugated to one that lies in .
We have mentioned that not all points on the root lattice of a Kac–Moody algebras are actually roots. For hyperbolic algebras, there exists a simple criterion which enables one to determine whether a point on the root lattice is a root or not. We give it first in the case where all simple roots have equal length squared (assumed equal to two).
Theorem: Consider a hyperbolic Kac–Moody algebra such that for all simple roots . Then, any point on the root lattice with is a root (note that is even). In particular, the set of real roots is the set of points on the root lattice with , while the set of imaginary roots is the set of points on the root lattice (minus the origin) with .
For a proof, see , Chapter 5.
The version of this theorem applicable to Kac–Moody algebras with different simple root lengths is the following.
Theorem: Consider a hyperbolic algebra with root lattice . Let be the smallest length squared of the simple roots, . Then we have:
For a proof, we refer again to , Chapter 5.
We shall illustrate these theorems in the examples below. Note that it follows in particular from the theorems that if is an imaginary root, all its integer multiples are also imaginary roots.
We discuss here briefly two examples, namely , for which all simple roots have equal length, and , with respective Dynkin diagrams shown in Figures 17 and 18.
This is the algebra associated with vacuum four-dimensional Einstein gravity and the BKL billiard. We encountered its Weyl group already in Section 3.1.1. The algebra is also denoted , or . The Cartan matrix is13. We shall consider here only positive roots for which .
Applying the first theorem, one easily verifies that the only positive roots at level zero are the roots , () of the affine subalgebra . When , the root is imaginary and has length squared equal to zero. When , the root is real and has length squared equal to two.
Similarly, the only roots at level one are with , i.e., . Whenever is an integer, the roots have squared length equal to two and are real. The roots with are imaginary and have squared length equal to . In particular, the root has length squared equal to . Of all the roots at level one with , these are the only ones that are in the fundamental domain (i.e., that fulfill ). When , none of the level-1 roots is in and is either in the Weyl orbit of , or in the Weyl orbit of .
We leave it to the reader to verify that the roots at level two that are in the fundamental domain take the form and with . Further information on the roots of may be found in , Chapter 11, page 215.
This is the algebra associated with the Einstein–Maxwell theory (see Section 7). The notation will be explained in Section 4.9. The Cartan matrix is
The real roots, which are Weyl conjugate to one of the simple roots or ( is in the same Weyl orbit as ), divide into long and short real roots. The long real roots are the vectors on the root lattice with squared length equal to two that fulfill the extra condition in the theorem. This condition expresses here that the coefficient of should be a multiple of . The short real roots are the vectors on the root lattice with length squared equal to one-half. The imaginary roots are all the vectors on the root lattice with length squared .
We define again the level as counting the number of times the root occurs. The positive roots at level zero are the positive roots of the twisted affine algebra , namely, and , , with for odd and for odd. Although belonging to the root lattice and of length squared equal to two, the vectors are not long real roots when is even because they fail to satisfy the condition that the coefficient of is a multiple of . The roots at level zero are all real, except when , in which case the roots have zero norm.
To get the long real roots at level one, we first determine the vectors of squared length equal to two. The condition easily leads to for some integer and . Since is automatically a multiple of for all , the corresponding vectors are all long real roots. Similarly, the short real roots at level one are found to be and for a non-negative integer.
Finally, the imaginary roots at level one in the fundamental domain read and where is an integer greater than or equal to . The first roots have length squared equal to , the second have length squared equal to .
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