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4.9 Overextensions of finite-dimensional Lie algebras

An interesting class of Lorentzian Kac–Moody algebras can be constructed by adding simple roots to finite-dimensional simple Lie algebras in a particular way which will be described below. These are called “overextensions”.

In this section, we let 𝔤 be a complex, finite-dimensional, simple Lie algebra of rank r, with simple roots α1,⋅⋅⋅ ,αr. As stated above, normalize the roots so that the long roots have length squared equal to 2 (the short roots, if any, have then length squared equal to 1 (or 2 ∕3 for G2)). The roots of simply-laced algebras are regarded as long roots.

Let ∑ α = i niαi, ni ≥ 0 be a positive root. One defines the height of α as

∑ ht(α) = ni. (4.69 ) i
Among the roots of 𝔤, there is a unique one that has highest height, called the highest root. We denote it by θ. It is long and it fulfills the property that (θ|αi) ≥ 0 for all simple roots αi, and
(αi|θ)- (θ|αi)- 2 (θ|θ) ∈ ℤ, 2 (αi |αi) ∈ ℤ (4.70 )
(see, e.g., [85]). We denote by V the r-dimensional Euclidean vector space spanned by αi (i = 1,⋅⋅⋅ ,r). Let M 2 be the two-dimensional Minkowski space with basis vectors u and v so that (u|u) = (v|v) = 0 and (u|v) = 1. The metric in the space V ⊕ M2 has clearly Minkowskian signature (− ,+, +,⋅⋅⋅ ,+ ) so that any Kac–Moody algebra whose simple roots span V ⊕ M2 is necessarily Lorentzian.

4.9.1 Untwisted overextensions

The standard overextensions 𝔤++ are obtained by adding to the original roots of 𝔤 the roots

α0 = u − θ, α−1 = − u − v.

The matrix (αi|αj)- Aij = 2 (αi|αi) where i,j = − 1,0,1,⋅⋅⋅ ,r is a (generalized) Cartan matrix and defines indeed a Kac–Moody algebra.

The root α0 is called the affine root and the algebra 𝔤+ (𝔤 (1) in Kac’s notations [116Jump To The Next Citation Point]) with roots α0, α1,⋅⋅⋅ ,αr is the untwisted affine extension of 𝔤. The root α −1 is known as the overextended root. One clearly has rank(𝔤++ ) = rank(𝔤) + 2. The overextended root has vanishing scalar product with all other simple roots except α0. One has explicitly (α−1|α− 1) = 2 = (α0 |α0 ) and (α−1|α0) = − 1, 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 B++ 8, D++ 8 and E++ 8 are also denoted BE 10, DE 10 and E 10, respectively.

A special property of E 10

Of these maximal rank hyperbolic algebras, E10 plays a very special role. Indeed, one can verify that the determinant of its Cartan matrix is equal to − 1. It follows that the lattice of E10 is self-dual, i.e., that the fundamental weights belong to the root lattice of E10. In view of the above theorem on roots of hyperbolic algebras and of the hyperboliticity of E10, the fundamental weights of E10 are actually (imaginary) roots since they are non-spacelike. The root lattice of E10 is the only Lorentzian, even, self-dual lattice in 10 dimensions (these lattices exist only in 2 mod 8 dimensions).

4.9.2 Root systems in Euclidean space

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 V is by definition a finite subset Δ of nonzero elements of V obeying the following two conditions:

Δ spans V, (4.71 ) { A α,β = 2(α|β) ∈ ℤ, ∀α, β ∈ Δ : (β|β) (4.72 ) β − Aβ,αα ∈ Δ.
The elements of Δ are called the roots. From the definition one can prove the following properties [93Jump To The Next Citation Point]:

  1. If α ∈ Δ, then − α ∈ Δ.
  2. If α ∈ Δ, then the only elements of Δ proportional to α are ± 12α, ± α, ±2 α. If only ± α occurs (for all roots α), the root system is said to be reduced (proper in “Araki terminology” [5Jump To The Next Citation Point]).
  3. If α, β ∈ Δ, then 0 ≤ A α,β A β,α ≤ 4, i.e., A α,β = 0, ±1, ±2, ±3, ±4; the last occurrence appearing only for β = ±2 α, i.e., for nonreduced systems. (The proof of this point requires the use of the Schwarz inequality.)
  4. If α, β ∈ Δ are not proportional to each other and (α|α) ≤ (β|β) then A α,β = 0, ±1. Moreover if (α|β) ⁄= 0, then (β|β) = (α|α), 2(α|α), or 3(α |α ).
  5. If α, β ∈ Δ, but α − β ⁄∈ Δ ∪ 0, then (α |β ) ≤ 0 and, as a consequence, if α, β ∈ Δ but α ± β ⁄∈ Δ ∪ 0 then (α|β) = 0. That (α|β) ≤ 0 can be seen as follows. Clearly, α and β can be assumed to be linearly independent14. Now, assume (α |β) > 0. By the previous point, A α,β = 1 or A β,α = 1. But then either α − A α,ββ = α − β ∈ Δ or − (β − A β,αα ) = α − β ∈ Δ by (4.72View Equation), contrary to the assumption. This proves that (α|β) ≤ 0.

Since Δ spans the vector space V, one can chose a basis {αi} of elements of V within Δ. This can furthermore be achieved in such a way the αi 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 [93Jump To The Next Citation Point].

All the abstract root systems in Euclidean space have been classified (see, e.g., [93Jump To The Next Citation Point]) with the following results:

View Image

Figure 19: The nonreduced (BC )2- and (BC )3-root systems. In each case, the highest root θ is displayed.

It is sometimes convenient to rescale the roots by the factor √ -- (1∕ 2) so that the highest root θ = 2(α1 + α2 + ⋅⋅⋅ + αr) [93Jump To The Next Citation Point] of the (BC )-system has length 2 instead of 4.

4.9.3 Twisted overextensions

We follow closely [95Jump To The Next Citation Point]. Twisted affine algebras are related to either the (BC )-root systems or to extensions by the highest short root (see [116Jump To The Next Citation Point], Proposition 6.4).

Twisted overextensions associated with the (BC )-root systems

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 (BC ) r root system with rescaled roots. The highest root has length squared equal to 2 and has non-vanishing scalar product only with αr ((αr|θ) = 1). The overextension procedure (defined by the same formulas as in the untwisted case) yields the algebra (BC )+r+, also denoted A (22r)+.

There is an alternative overextension A (2)′+ 2r that can be defined by starting this time with the algebra Cr but taking one-half the highest root of Cr to make the extension (see [116Jump To The Next Citation Point], formula in Paragraph 6.4, bottom of page 84). The formulas for α0 and α−1 are 2α0 = u − θ and 2α −1 = − u − v (where θ is now the highest root of Cr). The Dynkin diagram of A(2)′+ 2r is dual to that of (2)+ A2r. (Duality amounts to reversing the arrows in the Dynkin diagram, i.e., replacing the (generalized) Cartan matrix by its transpose.)

The algebras A (22r)+ and ′ A (22)r+ have rank r + 2 and are hyperbolic for r ≤ 4. The intermediate affine algebras are in all cases the twisted affine algebras (2) A 2r. We shall see in Section 7 that by coupling to three-dimensional gravity a coset model 𝒢∕𝒦 (𝒢), where the so-called restricted root system (see Section 6) of the (real) Lie algebra 𝔤 of the Lie group 𝒢 is of (BC )r-type, one can realize all the A (2)+ 2r algebras.

Twisted overextensions associated with the highest short root

We denote by θs the unique short root of heighest weight. It exists only for non-simply laced algebras and has length 1 (or 2∕3 for G2). The twisted overextensions are defined as the standard overextensions but one uses instead the highest short root θs. The formulas for the affine and overextended roots are

α0 = u − θs, α− 1 = − u − 1v, (𝔤 = Br,Cr, F4) 2


α0 = u − θs, α −1 = − u − 1v, (𝔤 = G2). 3

(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 𝔤 = C 8.)

The affine extensions generated by α0, ⋅⋅⋅ ,αr are respectively the twisted affine algebras (2) D r+1 (𝔤 = Br), A (22)r−1 (𝔤 = Cr), E(62) (𝔤 = F4) and D (43) (𝔤 = G2). These twisted affine algebras are related to external automorphisms of D r+1, A 2r−1, E 6 and D 4, respectively (the same holds for (2) A2r above) [116Jump To The Next Citation Point]. The corresponding twisted overextensions have the following features.

We list in Table 14 the Dynkin diagrams of all twisted overextensions.

Table 14: Twisted overextended Kac–Moody algebras.


Dynkin diagram




(2)′+ A2



A(22n)+ (n ≥ 2)



(2)′+ A2n (n ≥ 2)



A(22n)+−1 (n ≥ 3)



(2)+ Dn+1



E(2)+ 6



(3)+ D4


A satisfactory feature of the class of overextensions (standard and twisted) is that it is closed under duality. For instance, A(22r)+−1 is dual to B+r+. 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.

4.9.4 Algebras of Gaberdiel–Olive–West type

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 E10, one gets E11 with the Dynkin diagram displayed in Figure 20View Image. These algebras are Lorentzian, but not hyperbolic.

View Image

Figure 20: The Dynkin diagram of E11. Labels 0,− 1 and − 2 enumerate the nodes corresponding, respectively, to the affine root α0, the overextended root α−1 and the “very extended” root α−2.

The very extended algebras belong to a more general class of algebras considered by Gaberdiel, Olive and West in [86Jump To The Next Citation Point]. 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 [15386Jump To The Next Citation Point]. 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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