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

Among the roots of , there is a unique one that has highest height, called the highest root. We denote it by . It is long and it fulfills the property that for all simple roots , and (see, e.g., [85]). We denote by the -dimensional Euclidean vector space spanned by (). Let be the two-dimensional Minkowski space with basis vectors and so that and . The metric in the space has clearly Minkowskian signature so that any Kac–Moody algebra whose simple roots span is necessarily Lorentzian.

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:

The elements of are called the roots. From the definition one can prove the following properties [93]:

- If , then .
- If , then the only elements of proportional to are , , . If only occurs (for all roots ), the root system is said to be reduced (proper in “Araki terminology” [5]).
- If , , then , i.e., ; the last occurrence appearing only for , i.e., for nonreduced systems. (The proof of this point requires the use of the Schwarz inequality.)
- If , are not proportional to each other and then . Moreover if , then .
- If , , but , then and, as a consequence, if ,
but then . That can be seen as follows. Clearly,
and can be assumed to be linearly independent
^{14}. Now, assume . By the previous point, or . But then either or by (4.72), contrary to the assumption. This proves that .

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:

- The most general root system is obtained by taking a union of irreducible root systems. An irreducible root system is one that cannot be decomposed into two disjoint nonempty orthogonal subsets.
- The irreducible reduced root systems are simply the root systems of finite-dimensional simple Lie algebras ( with , with , with , with , , , , and ).
- Irreducible nonreduced root systems are all given by the so-called -systems. A -system is obtained by combining the root system of the algebra with the root system of the algebra in such a way that the long roots of are the short roots of . There are in that case three different root lengths. Explicitly is given by the unit vectors and their opposite along the Cartesian axis of an -dimensional Euclidean space, the vectors obtained by multiplying the previous vectors by 2 and the diagonal vectors , with and . The case is pictured in Figure 19. The Dynkin diagram of is the Dynkin diagram of with a double circle over the simple short root, say , to indicate that is also a root.

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.

- The overextensions have rank and are hyperbolic for .
- The overextensions have rank and are hyperbolic for . The last hyperbolic case, , yields the algebra , also denoted . It is the fourth rank-10 hyperbolic algebra, besides , and .
- The overextensions (rank 6) and (rank 4) are hyperbolic.

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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