List of Figures

	Figure 1: The BKL billiard of pure four-dimensional gravity. The figure represents the billiard region projected onto the hyperbolic plane. The particle geodesic is confined to the fundamental region enclosed by the three walls $1 2 1 α1(β) = 2β = 0,α2(β) = β − β = 0$ and $3 2 α3(β ) = β − β = 0$ , as indicated by the numbering in the figure. The two symmetry walls $α2(β) = 0$ and $α3(β ) = 0$ intersect at an angle of $π∕3$ , while the gravity wall $α1(β ) = 0$ intersects, respectively, at angles $0$ and $π∕2$ with the symmetry walls $α (β) = 0 2$ and $α (β) = 0 3$ . The particle has no direction of escape so the dynamics is chaotic.
	Figure 2: The figure on the left hand side displays the action of the modular group $P SL (2,ℤ )$ on the complex upper half plane $ℍ = {z ∈ ℂ \|ℑz > 0}$ . The two generators of $P SL (2,ℤ )$ are $S$ and $T$ , acting as follows on the coordinate $z ∈ ℍ : S(z) = − 1∕z; T(z) = z + 1$ , i.e., as an inversion and a translation, respectively. The shaded area indicates the fundamental domain $𝒟P SL (2,ℤ) = {z ∈ ℍ \| − 1∕2 ≤ ℜz ≤ 1∕2; \|z\| ≥ 1}$ for the action of $PSL (2,ℤ )$ on $ℍ$ . The figure on the right hand side displays the action of the “extended modular group” $P GL (2,ℤ )$ on $ℍ$ . The generators of $P GL (2,ℤ )$ are obtained by augmenting the generators of $P SL (2,ℤ )$ with the generator $s1$ , acting as $s1(z) = − ¯z$ on $ℍ$ . The additional two generators of $P GL (2,ℤ)$ then become: $s2 ≡ s1 ∘ T ; s3 ≡ s1 ∘ S$ , and their actions on $ℍ$ are $s2(z) = 1 − ¯z; s3(z ) = 1 ∕¯z$ . The new generator $s1$ corresponds to a reflection in the line $ℜz = 0$ , the generator $s2$ is in turn a reflection in the line $ℜz = 1∕2$ , while the generator $s 3$ is a reflection in the unit circle $\|z\| = 1$ . The fundamental domain of $P GL (2,ℤ )$ is $𝒟P GL (2,ℤ) = {z ∈ ℍ \|0 ≤ ℜz ≤ 1∕2; \|z\| ≥ 1}$ , corresponding to half the fundamental domain of $P SL (2,ℤ )$ . The “walls” $ℜz = 0,ℜz = 1∕2$ and $\|z\| = 1$ correspond, respectively, to the gravity wall $α1(β ) = 0$ , the symmetry wall $α2(β ) = 0$ and the symmetry wall $α3(β ) = 0$ of Figure 1.
	Figure 3: The equilateral triangle with its 3 axes of symmetries. The reflections $s1$ and $s2$ generate the entire symmetry group. We have pictured the vectors $α1$ and $α2$ orthogonal to the axes of reflection and chosen to make an obtuse angle. The shaded region ${w \|(w\|α1) ≥ 0} ∩ {w \|(w \|α2) ≥ 0}$ is a fundamental domain for the action of the group on the triangle. Note that the fundamental domain for the action of the group on the entire Euclidean plane extends indefinitely beyond the triangle but is, of course, still bounded by the two walls orthogonal to $α1$ and $α2$ .
	Figure 4: The Coxeter graph of the symmetry group $I (3) ≡ A 2 2$ of the equilateral triangle.
	Figure 5: The Coxeter graph of the dihedral group $I2(m )$ .
	Figure 6: The Coxeter graph of the symmetry group $H3$ of the regular icosahedron.
	Figure 7: The Coxeter graph of the affine Coxeter group $C+2$ corresponding to the group of isometries of the Euclidean plane.
	Figure 8: The Coxeter graph of the group $++ A 7$ .
	Figure 9: The Coxeter graph of $A+7$ .
	Figure 10: The Coxeter graph of $A7 × A1$ .
	Figure 11: The Coxeter graph of $A 8$ .
	Figure 12: The Coxeter graph of $D8$ .
	Figure 13: The Coxeter graph of $E8$ .
	Figure 14: The Coxeter graph of $+ E 7$ .
	Figure 15: This Coxeter graph corresponds to hyperbolic Coxeter groups for all values of $m$ and $n$ for which the associated bilinear form $B$ is not of positive definite or positive semidefinite type. This therefore gives rise to an infinite class of rank 3 hyperbolic Coxeter groups.
	Figure 16: The Dynkin diagrams corresponding to the finite Lie algebras $A2,B2$ and $G2$ and to the affine Kac–Moody algebras $A (22)$ and $A+1$ .
	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 $A1$ .
	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 $(2) A 2$ .
	Figure 19: The nonreduced $(BC )2$ - and $(BC )3$ -root systems. In each case, the highest root $θ$ is displayed.
	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 $α−2$ .
	Figure 21: The Dynkin diagram of $E10$ . Labels $1, ⋅⋅⋅ ,7$ and $10$ enumerate the nodes corresponding the regular $E8$ subalgebra discussed in the text.
	Figure 22: The Dynkin diagram of $+++ ℬ ≡ E 7$ . The root without number is the root denoted $¯α10$ in the text.
	Figure 23: $DE10 ≡ D++8$ regularly embedded in $E10$ . Labels $2,⋅⋅⋅ ,10$ represent the simple roots $α ,⋅⋅⋅ ,α 2 10$ of $E 10$ and the unlabeled node corresponds to the positive root $¯ β = 2α1 + 3α2 + 4α3 + 3α4 + 2 α5 + α6 + 2α10$ .
	Figure 24: The Dynkin diagram of the hyperbolic Kac–Moody algebra $++ A d− 2$ which controls the billiard dynamics of pure gravity in $D = d + 1$ dimensions. The nodes $s1,⋅⋅⋅ ,sd− 1$ represent the “symmetry walls” arising from the off-diagonal components of the spatial metric, and the node $r$ corresponds to a “curvature wall” coming from the spatial curvature. The horizontal line is the Dynkin diagram of the underlying $Ad−2$ -subalgebra and the two topmost nodes, $sd−2$ and $sd−1$ , give the affine- and overextension, respectively.
	Figure 25: The Dynkin diagram of $E10$ . Labels $m = 1,⋅⋅⋅ ,9$ enumerate the nodes corresponding to simple roots, $αm$ , of the $𝔰𝔩(10,ℝ )$ subalgebra and the exceptional node, labeled “ $10$ ”, is associated to the electric wall $α10 = e123$ .
	Figure 26: The $A 4$ Dynkin diagram.
	Figure 27: A Vogan diagram associated to $𝔰𝔲 (3,2)$ .
	Figure 28: Another Vogan diagram associated to $𝔰𝔲(3,2)$ .
	Figure 29: Yet another Vogan diagram associated to $𝔰𝔲 (3, 2)$ .
	Figure 30: The remaining Vogan diagrams associated to $𝔰𝔲(3,2)$ .
	Figure 31: The four Vogan diagrams associated to $𝔰𝔲(4,1)$ .
	Figure 32: The Vogan diagrams for $𝔰𝔲(5)$ and $𝔰𝔩(5, ℝ)$ .
	Figure 33: The Dynkin diagram of $E8$ . Seen as a Vogan diagram, it corresponds to the maximally compact form of $E8$ .
	Figure 34: Vogan diagrams of the two different noncompact real forms of $E8$ : $E8 (− 24)$ and $E8 (8)$ . The lower one corresponds to the split real form.
	Figure 35: The Vogan diagrams associated to a $𝔰𝔩(4ℝ )$ and $𝔰𝔩(2 ℍ)$ subalgebra.
	Figure 36: Tits–Satake diagrams for $𝔰𝔲 (3, 2)$ and $𝔰𝔲(4,1)$ .
	Figure 37: On the left, the Tits–Satake diagram of the real form $F 4(−20)$ . On the right, a non-admissible Tits–Satake diagram.
	Figure 38: Tits–Satake diagrams for the $𝔰𝔬(2p,2q + 1)$ Lie algebra with $q < p$ . If $p = q + 1$ , all nodes are white.
	Figure 39: Tits–Satake diagrams for the $𝔰𝔬(2p,2q + 1)$ Lie algebra with $q ≥ p$ . If $q = p$ , all nodes are white.
	Figure 40: Tits–Satake diagrams for the $𝔰𝔬(2p,2q)$ Lie algebra with $q < p − 1$ , $q = p − 1$ and $q = p$ .
	Figure 41: The Dynkin diagram of $(2)+ A2$ . Label $1$ denotes the simple root $ˆα (1)$ of the restricted root system of $𝔲3 = 𝔰𝔲(2,1)$ . Labels $2$ and $3$ correspond to the affine and overextended roots, respectively. The arrow points towards the short root which is normalized such that $(ˆα1\|ˆα1) = 1 2$ .
	Figure 42: The Dynkin diagram representing the restricted root system $Σ𝔰𝔬(8,24)$ of $𝔰𝔬(8,24)$ . Labels $1,⋅⋅ ⋅ ,7$ denote the long simple roots that are nondegenerate while the eighth simple root is short and has multiplicity 16.
	Figure 43: The Dynkin diagram representing the overextension $B++8$ of the restricted root system $Σ = B8$ of $𝔰𝔬(8,24)$ . Labels $− 1,0,1,⋅⋅⋅ ,7$ denote the long simple roots that are nondegenerate while the eighth simple root is short and has multiplicity 16.
	Figure 44: The Dynkin diagram of $𝔰𝔩(3, ℝ)$ .
	Figure 45: Level decomposition of the adjoint representation $ℛad = 8$ of $𝔰𝔩(3,ℝ)$ into representations of the subalgebra $𝔰𝔩(2,ℝ )$ . The labels $1$ and $2$ indicate the simple roots $α1$ and $α 2$ . Level zero corresponds to the horizontal axis where we find the adjoint representation $(0) ℛ ad = 30$ of $𝔰𝔩(2,ℝ )$ (red nodes) and the singlet representation $(0) ℛs = 10$ (green circle about the origin). At level one we find the two-dimensional representation $(1) ℛ = 21$ (green nodes). The black arrow denotes the negative level root $− α2$ and so gives rise to the level $ℓ = − 1$ representation $ℛ (−1) = 2 (− 1)$ . The blue arrows represent the fundamental weights $Λ1$ and $Λ2$ .
	Figure 46: The Dynkin diagram of the hyperbolic Kac–Moody algebra $AE3 ≡ A+1+$ . The labels indicate the simple roots $α ,α 1 2$ and $α 3$ . The nodes “2” and “3” correspond to the subalgebra $𝔯 = 𝔰𝔩(3,ℝ)$ with respect to which we perform the level decomposition.
	Figure 47: Level decomposition of the adjoint representation of $AE3$ . We have displayed the decomposition up to positive level $ℓ = 2$ . At level zero we have the adjoint representation $(0) ℛ 1 = 80$ of $𝔰𝔩(3,ℝ )$ and the singlet representation $(0) ℛ 2 = 10$ defined by the simple Cartan generator $∨ α 1$ . Ascending to level one with the root $α1$ (green vector) gives the lowest weight $Λ (1)$ of the representation $ℛ (1) = 61$ . The weights of $ℛ (1)$ labelled by white crosses are on the lightcone and so their norm squared is zero. At level two we find the lowest weight $Λ(2)$ (blue vector) of the 15-dimensional representation $ℛ (2) = 15 2$ . Again, the white crosses label weights that are on the lightcone. The three innermost weights are inside of the lightcone and the rings indicate that these all have multiplicity 2 as weights of $(2) ℛ$ . Since these also have multiplicity 2 as roots of $⋆ 𝔥 𝔤$ we find that the outer multiplicity of this representation is one, $μ(ℛ (2)) = 1$ .
	Figure 48: The representation $152$ of $𝔰𝔩(3,ℝ)$ appearing at level two in the decomposition of the adjoint representation of $AE3$ into representations of $𝔰𝔩(3,ℝ )$ . The lowest leftmost node is the lowest weight of the representation, corresponding to the real root $Λ(2) = 2α + α 1 2$ of $AE 3$ . This representation has outer multiplicity one.
	Figure 49: The Dynkin diagram of $E10$ . Labels $i = 1,⋅⋅⋅ ,9$ enumerate the nodes corresponding to simple roots $α i$ of the $𝔰𝔩(10,ℝ )$ subalgebra and “ $10$ ” labels the exceptional node.
	Figure 50: $(31,13)$ : The only allowed configuration for $n = 3$ .
	Figure 51: The configuration $(62,43)$ , dual to the Lie algebra $A1 ⊕ A1 ⊕ A1 ⊕ A1$ .
	Figure 52: The Fano Plane, $(73,73)$ , dual to the Lie algebra $A1 ⊕ A1 ⊕ A1 ⊕ A1 ⊕ A1 ⊕ A1 ⊕ A1$ .
	Figure 53: The simplest “magnetic configuration” $(61,16)$ , dual to the algebra $A1$ .The associated supergravity solution describes an $SM 5$ -brane, whose world volume is extended in the directions $x1,⋅⋅⋅ ,x6$ .
	Figure 54: $(10 ,10 ) 3 3 3$ : The Desargues configuration, dual to the Petersen graph.
	Figure 55: This is the so-called Petersen graph. It is the Dynkin diagram dual to the Desargues configuration, and is in fact a geometric configuration itself, denoted $(103,152)$ .
	Figure 56: An alternative drawing of the Petersen graph in the plane. This embedding reveals an $S3$ permutation symmetry about the central point.