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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 ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Figure 2:
The figure on the left hand side displays the action of the modular group ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Figure 3:
The equilateral triangle with its 3 axes of symmetries. The reflections ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Figure 4:
The Coxeter graph of the symmetry group ![]() |
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Figure 5:
The Coxeter graph of the dihedral group ![]() |
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Figure 6:
The Coxeter graph of the symmetry group ![]() |
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Figure 7:
The Coxeter graph of the affine Coxeter group ![]() |
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Figure 8:
The Coxeter graph of the group ![]() |
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Figure 9:
The Coxeter graph of ![]() |
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Figure 10:
The Coxeter graph of ![]() |
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Figure 11:
The Coxeter graph of ![]() |
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Figure 12:
The Coxeter graph of ![]() |
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Figure 13:
The Coxeter graph of ![]() |
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Figure 14:
The Coxeter graph of ![]() |
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Figure 15:
This Coxeter graph corresponds to hyperbolic Coxeter groups for all values of ![]() ![]() ![]() |
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Figure 16:
The Dynkin diagrams corresponding to the finite Lie algebras ![]() ![]() ![]() ![]() |
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Figure 17:
The Dynkin diagram of the hyperbolic Kac–Moody algebra ![]() ![]() |
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Figure 18:
The Dynkin diagram of the hyperbolic Kac–Moody algebra ![]() ![]() |
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Figure 19:
The nonreduced ![]() ![]() ![]() |
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Figure 20:
The Dynkin diagram of ![]() ![]() ![]() ![]() ![]() ![]() |
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Figure 21:
The Dynkin diagram of ![]() ![]() ![]() ![]() |
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Figure 22:
The Dynkin diagram of ![]() ![]() |
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Figure 23:
![]() ![]() ![]() ![]() ![]() ![]() |
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Figure 24:
The Dynkin diagram of the hyperbolic Kac–Moody algebra ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Figure 25:
The Dynkin diagram of ![]() ![]() ![]() ![]() ![]() ![]() |
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Figure 26:
The ![]() |
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Figure 27:
A Vogan diagram associated to ![]() |
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Figure 28:
Another Vogan diagram associated to ![]() |
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Figure 29:
Yet another Vogan diagram associated to ![]() |
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Figure 30:
The remaining Vogan diagrams associated to ![]() |
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Figure 31:
The four Vogan diagrams associated to ![]() |
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Figure 32:
The Vogan diagrams for ![]() ![]() |
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Figure 33:
The Dynkin diagram of ![]() ![]() |
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Figure 34:
Vogan diagrams of the two different noncompact real forms of ![]() ![]() ![]() |
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Figure 35:
The Vogan diagrams associated to a ![]() ![]() |
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Figure 36:
Tits–Satake diagrams for ![]() ![]() |
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Figure 37:
On the left, the Tits–Satake diagram of the real form ![]() |
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Figure 38:
Tits–Satake diagrams for the ![]() ![]() ![]() |
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Figure 39:
Tits–Satake diagrams for the ![]() ![]() ![]() |
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Figure 40:
Tits–Satake diagrams for the ![]() ![]() ![]() ![]() |
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Figure 41:
The Dynkin diagram of ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Figure 42:
The Dynkin diagram representing the restricted root system ![]() ![]() ![]() |
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Figure 43:
The Dynkin diagram representing the overextension ![]() ![]() ![]() ![]() |
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Figure 44:
The Dynkin diagram of ![]() |
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Figure 45:
Level decomposition of the adjoint representation ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Figure 46:
The Dynkin diagram of the hyperbolic Kac–Moody algebra ![]() ![]() ![]() ![]() |
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Figure 47:
Level decomposition of the adjoint representation of ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Figure 48:
The representation ![]() ![]() ![]() ![]() ![]() ![]() |
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Figure 49:
The Dynkin diagram of ![]() ![]() ![]() ![]() ![]() |
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Figure 50:
![]() ![]() |
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Figure 51:
The configuration ![]() ![]() |
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Figure 52:
The Fano Plane, ![]() ![]() |
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Figure 53:
The simplest “magnetic configuration” ![]() ![]() ![]() ![]() |
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Figure 54:
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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 ![]() |
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Figure 56:
An alternative drawing of the Petersen graph in the plane. This embedding reveals an ![]() |
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