Figure 24

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.