In this section, we let be a complex, finite-dimensional, simple Lie algebra of rank
, with simple
roots
. As stated above, normalize the roots so that the long roots have length squared equal to
(the short roots, if any, have then length squared equal to
(or
for
)). The roots of
simply-laced algebras are regarded as long roots.
Let ,
be a positive root. One defines the height of
as
The standard overextensions are obtained by adding to the original roots of
the
roots
The matrix where
is a (generalized) Cartan matrix and defines
indeed a Kac–Moody algebra.
The root is called the affine root and the algebra
(
in Kac’s notations [116
]) with roots
is the untwisted affine extension of
. The root
is known as the overextended root.
One clearly has rank
rank
. The overextended root has vanishing scalar product with all
other simple roots except
. One has explicitly
and
, which
shows that the overextended root is attached to the affine root (and only to the affine root) with a single
link.
Of these Lorentzian algebras, the following ones are hyperbolic:
The algebras ,
and
are also denoted
,
and
, respectively.
Of these maximal rank hyperbolic algebras, plays a very special role. Indeed, one can verify that the
determinant of its Cartan matrix is equal to
. It follows that the lattice of
is self-dual, i.e., that
the fundamental weights belong to the root lattice of
. In view of the above theorem on roots
of hyperbolic algebras and of the hyperboliticity of
, the fundamental weights of
are actually (imaginary) roots since they are non-spacelike. The root lattice of
is the
only Lorentzian, even, self-dual lattice in 10 dimensions (these lattices exist only in 2 mod 8
dimensions).
In order to describe the “twisted” overextensions, we need to introduce the concept of a “root system”.
A root system in a real Euclidean space is by definition a finite subset
of nonzero elements of
obeying the following two conditions:
Since spans the vector space
, one can chose a basis
of elements of
within
.
This can furthermore be achieved in such a way the
enjoy the standard properties of simple roots of
Lie algebras so that in particular the concepts of positive, negative and highest roots can be
introduced [93
].
All the abstract root systems in Euclidean space have been classified (see, e.g., [93]) with the following
results:
It is sometimes convenient to rescale the roots by the factor so that the highest root
[93
] of the
-system has length 2 instead of 4.
We follow closely [95]. Twisted affine algebras are related to either the
-root systems or to
extensions by the highest short root (see [116
], Proposition 6.4).
These are the overextensions relevant for some of the gravitational billiards. The construction proceeds as
for the untwisted overextensions, but the starting point is now the root system with rescaled roots.
The highest root has length squared equal to 2 and has non-vanishing scalar product only with
(
). The overextension procedure (defined by the same formulas as in the untwisted case) yields
the algebra
, also denoted
.
There is an alternative overextension that can be defined by starting this time with the algebra
but taking one-half the highest root of
to make the extension (see [116
], formula in
Paragraph 6.4, bottom of page 84). The formulas for
and
are
and
(where
is now the highest root of
). The Dynkin diagram of
is dual to
that of
. (Duality amounts to reversing the arrows in the Dynkin diagram, i.e., replacing the
(generalized) Cartan matrix by its transpose.)
The algebras and
have rank
and are hyperbolic for
. The intermediate
affine algebras are in all cases the twisted affine algebras
. We shall see in Section 7 that by coupling
to three-dimensional gravity a coset model
, where the so-called restricted root system (see
Section 6) of the (real) Lie algebra
of the Lie group
is of
-type, one can realize all the
algebras.
We denote by the unique short root of heighest weight. It exists only for non-simply laced algebras and
has length
(or
for
). The twisted overextensions are defined as the standard overextensions
but one uses instead the highest short root
. The formulas for the affine and overextended roots
are
or
(We choose the overextended root to have the same length as the affine root and to be attached to it with a
single link. This choice is motivated by considerations of simplicity and yields the fourth rank ten
hyperbolic algebra when .)
The affine extensions generated by are respectively the twisted affine algebras
(
),
(
),
(
) and
(
). These twisted affine algebras are
related to external automorphisms of
,
,
and
, respectively (the same
holds for
above) [116
]. The corresponding twisted overextensions have the following
features.
We list in Table 14 the Dynkin diagrams of all twisted overextensions.
A satisfactory feature of the class of overextensions (standard and twisted) is that it is closed under
duality. For instance, is dual to
. In fact, one could get the twisted overextensions associated
with the highest short root from the standard overextensions precisely by requiring closure under duality. A
similar feature already holds for the affine algebras.
Note also that while not all hyperbolic Kac–Moody algebras are symmetrizable, the ones that are obtained through the process of overextension are.
One can further extend the overextended algebras to get “triple extensions” or “very extended algebras”.
This is done by adding a further simple root attached with a single link to the overextended root of
Section 4.9. For instance, in the case of , one gets
with the Dynkin diagram displayed in
Figure 20
. These algebras are Lorentzian, but not hyperbolic.
The very extended algebras belong to a more general class of algebras considered by Gaberdiel, Olive
and West in [86]. These are defined to be algebras with a connected Dynkin diagram that possesses at least
one node whose deletion yields a diagram with connected components that are of finite type except for at
most one of affine type. For a hyperbolic algebra, the deletion of any node should fulfill this condition. The
algebras of Gaberdiel, Olive and West are Lorentzian if not of finite or affine type [153, 86
]. They
include the overextensions of Section 4.9. The untwisted and twisted very extended algebras are
clearly also of this type, since removing the affine root gives a diagram with the requested
properties.
Higher order extensions with special additional properties have been investigated in [78].
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