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4.8 Hyperbolic Kac–Moody algebras

Hyperbolic Kac–Moody algebras are by definition Lorentzian Kac–Moody algebras with the property that removing any node from their Dynkin diagram leaves one with a Dynkin diagram of affine or finite type. The Weyl group of hyperbolic Kac–Moody algebras is a crystallographic hyperbolic Coxeter group (as defined in Section 3.5). Conversely, any crystallographic hyperbolic Coxeter group is the Weyl group of at least one hyperbolic Kac–Moody algebra. Indeed, consider one of the lattices preserved by the Coxeter group as constructed in Section 3.6. The matrix with entries equal to the dij of that section is the Cartan matrix of a Kac–Moody algebra that has this given Coxeter group as Weyl group.

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 ++ E10 ≡ E 8, ++ BE10 ≡ B 8, ++ DE10 ≡ D 8 and (2)+ CE10 ≡ A15, BE10 and CE10 being dual to each other and possessing the same Weyl group (the notation will be explained below).

4.8.1 The fundamental domain 𝓕

For a hyperbolic Kac–Moody algebra, the fundamental weights Λi are timelike or null and lie within the (say) past lightcone. Similarly, the fundamental Weyl chamber ℱ defined by {v ∈ ℱ ⇔ (v|αi) ≥ 0} 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 αK of the algebra fulfill (αK|Λi) ≥ 0 (with, for any K, strict inequality for at least one i) 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 − ℱ. Any imaginary root is Weyl-conjugated to one that lies in − ℱ.

4.8.2 Roots and the root lattice

We have mentioned that not all points on the root lattice Q 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 (αi|αi) = 2 for all simple roots αi. Then, any point α on the root lattice Q with (α|α) ≤ 2 is a root (note that (α|α) is even). In particular, the set of real roots is the set of points on the root lattice with (α |α ) = 2, while the set of imaginary roots is the set of points on the root lattice (minus the origin) with (α |α ) ≤ 0.

For a proof, see [116Jump To The Next Citation Point], 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 Q. Let a be the smallest length squared of the simple roots, a = mini(αi|αi). Then we have:

For a proof, we refer again to [116Jump To The Next Citation Point], Chapter 5.

We shall illustrate these theorems in the examples below. Note that it follows in particular from the theorems that if α is an imaginary root, all its integer multiples are also imaginary roots.

4.8.3 Examples

We discuss here briefly two examples, namely ++ A1, for which all simple roots have equal length, and A (22)+, with respective Dynkin diagrams shown in Figures 17View Image and 18View Image.

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Figure 17: The Dynkin diagram of the hyperbolic Kac–Moody algebra A++ 1. This algebra is obtained through a standard overextension of the finite Lie algebra A1.
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Figure 18: The Dynkin diagram of the hyperbolic Kac–Moody algebra (2)+ A2. This algebra is obtained through a Lorentzian extension of the twisted affine Kac–Moody algebra (2) A 2.

The Kac–Moody Algebra A++ 1

This is the algebra associated with vacuum four-dimensional Einstein gravity and the BKL billiard. We encountered its Weyl group P GL (2,ℤ) already in Section 3.1.1. The algebra is also denoted AE3, or H3. The Cartan matrix is

( ) 2 − 2 0 ( − 2 2 − 1) . (4.63 ) 0 − 1 2
As it follows from our analysis in Section 3.1.1, the simple roots may be identified with the following linear forms αi(β ) in the three-dimensional space of the βi’s,
α (β) = 2β1, α (β ) = β2 − β1, α (β) = β3 − β2 (4.64 ) 1 2 3
with scalar product
∑ ( ∑ ) ( ∑ ) (F |G ) = FiGi − 1- Fi Gi (4.65 ) i 2 i i
for two linear forms F = Fiβi and G = Gi βi. It is sometimes convenient to analyze the root system in terms of an “affine” level ℓ that counts the number of times the root α3 occurs: The root k α1 + m α2 + ℓα3 has by definition level ℓ13. We shall consider here only positive roots for which k,m, ℓ ≥ 0.

Applying the first theorem, one easily verifies that the only positive roots at level zero are the roots kα1 + m α2, |k − m | ≤ 1 (k,m ≥ 0) of the affine subalgebra A+1. When k = m, the root is imaginary and has length squared equal to zero. When |k − m | = 1, the root is real and has length squared equal to two.

Similarly, the only roots at level one are (m + a)α1 + m α2 + α3 with 2 a ≤ m, i.e., √ -- √ -- − [ m ] ≤ a ≤ [ m ]. Whenever √-- m is an integer, the roots √ -- (m ± m )α1 + m α2 + α3 have squared length equal to two and are real. The roots (m + a )α1 + m α2 + α3 with a2 < m are imaginary and have squared length equal to 2(a2 + 1 − m ) ≤ 0. In particular, the root m (α + α ) + α 1 2 3 has length squared equal to 2(1 − m ). Of all the roots at level one with m > 1, these are the only ones that are in the fundamental domain − ℱ (i.e., that fulfill (β |αi) ≤ 0). When m = 1, none of the level-1 roots is in − ℱ and is either in the Weyl orbit of α1 + α2, or in the Weyl orbit of α3.

We leave it to the reader to verify that the roots at level two that are in the fundamental domain − ℱ take the form (m − 1)α1 + m α2 + 2α3 and m (α1 + α2 ) + 2 α3 with m ≥ 4. Further information on the roots of A++1 may be found in [116Jump To The Next Citation Point], Chapter 11, page 215.

The Kac–Moody Algebra (2)+ A 2

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

( ) 2 − 4 0 ( − 1 2 − 1) , (4.66 ) 0 − 1 2
and there are now two lengths for the simple roots. The scalar products are
1- (α1 |α1 ) = 2, (α1 |α2 ) = − 1 = (α2 |α1 ), (α2|α2) = 2. (4.67 )
One may realize the simple roots as the linear forms
1 2 1 3 2 α1 (β) = β , α2(β) = β − β , α3(β) = β − β (4.68 )
in the three-dimensional space of the βi’s with scalar product Equation (4.65View Equation).

The real roots, which are Weyl conjugate to one of the simple roots α1 or α2 (α3 is in the same Weyl orbit as α2), 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 α1 should be a multiple of 4. 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 ≤ 0.

We define again the level ℓ as counting the number of times the root α3 occurs. The positive roots at level zero are the positive roots of the twisted affine algebra A (22), namely, α1 and (2m + a)α1 + m α2, m = 1,2,3,⋅⋅⋅, with a = − 2,− 1,0,1,2 for m odd and a = − 1,0,1 for m odd. Although belonging to the root lattice and of length squared equal to two, the vectors (2m ± 2)α1 + m α2 are not long real roots when m is even because they fail to satisfy the condition that the coefficient (2m ± 2) of α 1 is a multiple of 4. The roots at level zero are all real, except when a = 0, in which case the roots m(2α1 + α2) have zero norm.

To get the long real roots at level one, we first determine the vectors α = α3 + kα1 + m α2 of squared length equal to two. The condition (α|α) = 2 easily leads to m = p2 for some integer p ≥ 0 and 2 k = 2p ± 2p = 2p(p ± 1). Since k is automatically a multiple of 4 for all p = 0,1,2,3,⋅⋅⋅, the corresponding vectors are all long real roots. Similarly, the short real roots at level one are found to be (2p2 + 1)α1 + (p2 + p + 1)α2 + α3 and (2p2 + 4p + 3)α1 + (p2 + p + 1)α2 + α3 for p a non-negative integer.

Finally, the imaginary roots at level one in the fundamental domain − ℱ read (2m − 1 )α + m α + α 1 2 3 and 2m α1 + m α2 + α3 where m is an integer greater than or equal to 2. The first roots have length squared equal to 5 − 2m + 2, the second have length squared equal to − 2m + 2.


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