
Abstract 
1 
Introduction 

1.1 
Cosmological billiards and hidden symmetries of
gravity 

1.2 
Outline of the paper 
2 
The BKL Phenomenon 

2.1 
The general
action 

2.2 
Hamiltonian description 

2.3 
Decoupling of spatial points close to a
spacelike singularity 

2.4 
Dynamics as a billiard in hyperbolic space 

2.5 
Rules
for deriving the wall forms from the Lagrangian – Summary 

2.6 
More on
the free motion: The Kasner solution 

2.7 
Chaos and billiard volume 

2.8 
A
note on the constraints 

2.9 
On the validity of the BKL conjecture – A status
report 
3 
Hyperbolic Coxeter Groups 

3.1 
Preliminary example: The BKL billiard
(vacuum gravity) 

3.2 
Coxeter groups – The general theory 

3.3 
Finite
Coxeter groups 

3.4 
Affine Coxeter groups 

3.5 
Lorentzian and hyperbolic
Coxeter groups 

3.6 
Crystallographic Coxeter groups 
4 
Lorentzian Kac–Moody
Algebras 

4.1 
Definitions 

4.2 
Roots 

4.3 
The Chevalley involution 

4.4 
Three
examples 

4.5 
The affine case 

4.6 
The invariant bilinear form 

4.7 
The
Weyl group 

4.8 
Hyperbolic Kac–Moody algebras 

4.9 
Overextensions of
finitedimensional Lie algebras 

4.10 
Regular subalgebras of Kac–Moody
algebras 
5 
Kac–Moody Billiards I – The Case of Split Real Forms 

5.1 
More on
Coxeter billiards 

5.2 
Dynamics in the Cartan subalgebra 

5.3 
Understanding
the emerging Kac–Moody algebra 

5.4 
Models associated with split real
forms 
6 
FiniteDimensional Real Lie Algebras 

6.1 
Definitions 

6.2 
A preliminary
example: 

6.3 
The compact and split real forms of a semisimple
Lie algebra 

6.4 
Classical decompositions 

6.5 
Vogan diagrams 

6.6 
Tits–Satake
diagrams 

6.7 
The real semisimple algebras 

6.8 
Summary – Tits–Satake
diagrams for noncompact real forms 
7 
Kac–Moody Billiards II – The Case
of NonSplit Real Forms 

7.1 
The restricted Weyl group and the maximal
split “subalgebra” 

7.2 
“Split symmetry controls chaos” 

7.3 
Models associated
with nonsplit real forms 
8 
Level Decomposition in Terms of Finite Regular
Subalgebras 

8.1 
A finitedimensional example: 

8.2 
Some formal
considerations 

8.3 
Level decomposition of 

8.4 
Level decomposition
of 
9 
Hidden Symmetries Made Manifest – InfiniteDimensional
Coset Spaces 

9.1 
Nonlinear sigma models on finitedimensional coset
spaces 

9.2 
Geodesic sigma models on infinitedimensional coset
spaces 

9.3 
Elevendimensional supergravity and 

9.4 
Further
comments 
10 
Cosmological Solutions from 

10.1 
Bianchi I models
and elevendimensional supergravity 

10.2 
Geometric configurations
and regular subalgebras of 

10.3 
Cosmological solutions with
electric flux 

10.4 
Cosmological solutions with magnetic flux 

10.5 
The
Petersen algebra and the Desargues configuration 

10.6 
Further
comments 
11 
Conclusions 
12 
Acknowledgements 
A 
Proof of Some Important
Properties of the Bilinear Form 
B 
Existence and “Uniqueness” of
the Aligned Compact Real Form 

References 

Index 

Footnotes 

Figures 

Tables 