List of Figures

View Image 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.
View Image 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.
View Image 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.
View Image Figure 4:
The Coxeter graph of the symmetry group I (3) ≡ A 2 2 of the equilateral triangle.
View Image Figure 5:
The Coxeter graph of the dihedral group I2(m ).
View Image Figure 6:
The Coxeter graph of the symmetry group H3 of the regular icosahedron.
View Image Figure 7:
The Coxeter graph of the affine Coxeter group C+2 corresponding to the group of isometries of the Euclidean plane.
View Image Figure 8:
The Coxeter graph of the group ++ A 7.
View Image Figure 9:
The Coxeter graph of A+7.
View Image Figure 10:
The Coxeter graph of A7 × A1.
View Image Figure 11:
The Coxeter graph of A 8.
View Image Figure 12:
The Coxeter graph of D8.
View Image Figure 13:
The Coxeter graph of E8.
View Image Figure 14:
The Coxeter graph of + E 7.
View Image 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.
View Image 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.
View Image 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.
View Image 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.
View Image Figure 19:
The nonreduced (BC )2- and (BC )3-root systems. In each case, the highest root θ is displayed.
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.
View Image Figure 21:
The Dynkin diagram of E10. Labels 1, ⋅⋅⋅ ,7 and 10 enumerate the nodes corresponding the regular E8 subalgebra discussed in the text.
View Image Figure 22:
The Dynkin diagram of +++ ℬ ≡ E 7. The root without number is the root denoted ¯α10 in the text.
View Image 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.
View Image 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.
View Image 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.
View Image Figure 26:
The A 4 Dynkin diagram.
View Image Figure 27:
A Vogan diagram associated to 𝔰𝔲 (3,2).
View Image Figure 28:
Another Vogan diagram associated to 𝔰𝔲(3,2).
View Image Figure 29:
Yet another Vogan diagram associated to 𝔰𝔲 (3, 2).
View Image Figure 30:
The remaining Vogan diagrams associated to 𝔰𝔲(3,2).
View Image Figure 31:
The four Vogan diagrams associated to 𝔰𝔲(4,1).
View Image Figure 32:
The Vogan diagrams for 𝔰𝔲(5) and 𝔰𝔩(5, ℝ).
View Image Figure 33:
The Dynkin diagram of E8. Seen as a Vogan diagram, it corresponds to the maximally compact form of E8.
View Image 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.
View Image Figure 35:
The Vogan diagrams associated to a 𝔰𝔩(4ℝ ) and 𝔰𝔩(2 ℍ) subalgebra.
View Image Figure 36:
Tits–Satake diagrams for 𝔰𝔲 (3, 2) and 𝔰𝔲(4,1).
View Image 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.
View Image Figure 38:
Tits–Satake diagrams for the 𝔰𝔬(2p,2q + 1) Lie algebra with q < p. If p = q + 1, all nodes are white.
View Image Figure 39:
Tits–Satake diagrams for the 𝔰𝔬(2p,2q + 1) Lie algebra with q ≥ p. If q = p, all nodes are white.
View Image Figure 40:
Tits–Satake diagrams for the 𝔰𝔬(2p,2q) Lie algebra with q < p − 1, q = p − 1 and q = p.
View Image 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.
View Image 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.
View Image 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.
View Image Figure 44:
The Dynkin diagram of 𝔰𝔩(3, ℝ).
View Image 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.
View Image 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.
View Image 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.
View Image 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.
View Image 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.
View Image Figure 50:
(31,13): The only allowed configuration for n = 3.
View Image Figure 51:
The configuration (62,43), dual to the Lie algebra A1 ⊕ A1 ⊕ A1 ⊕ A1.
View Image Figure 52:
The Fano Plane, (73,73), dual to the Lie algebra A1 ⊕ A1 ⊕ A1 ⊕ A1 ⊕ A1 ⊕ A1 ⊕ A1.
View Image 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.
View Image Figure 54:
(10 ,10 ) 3 3 3: The Desargues configuration, dual to the Petersen graph.
View Image 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).
View Image Figure 56:
An alternative drawing of the Petersen graph in the plane. This embedding reveals an S3 permutation symmetry about the central point.