The hyperbolic Kac–Moody algebras have been classified in [154] and exist only up to rank 10 (see
also [59]). 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 [116], 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 [116], 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 is
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 [116
], 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
and there are now two lengths for the simple roots. The scalar products are One may realize the simple roots as the linear forms in the three-dimensional space of the 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
.
http://www.livingreviews.org/lrr-2008-1 | ![]() This work is licensed under a Creative Commons Attribution-Noncommercial-No Derivative Works 2.0 Germany License. Problems/comments to |