Living Reviews in Relativity

"Spacelike Singularities and Hidden Symmetries of Gravity"
by
Marc Henneaux and Daniel Persson and Philippe Spindel  

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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 D = 4 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 finite-dimensional 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 Finite-Dimensional Real Lie Algebras
6.1 Definitions
6.2 A preliminary example: 𝔰𝔩(2,ℂ )
6.3 The compact and split real forms of a semi-simple Lie algebra
6.4 Classical decompositions
6.5 Vogan diagrams
6.6 Tits–Satake diagrams
6.7 The real semi-simple algebras 𝔰𝔬(k,l)
6.8 Summary – Tits–Satake diagrams for non-compact real forms
7 Kac–Moody Billiards II – The Case of Non-Split Real Forms
7.1 The restricted Weyl group and the maximal split “subalgebra”
7.2 “Split symmetry controls chaos”
7.3 Models associated with non-split real forms
8 Level Decomposition in Terms of Finite Regular Subalgebras
8.1 A finite-dimensional example: 𝔰𝔩(3,ℝ )
8.2 Some formal considerations
8.3 Level decomposition of AE 3
8.4 Level decomposition of E10
9 Hidden Symmetries Made Manifest – Infinite-Dimensional Coset Spaces
9.1 Nonlinear sigma models on finite-dimensional coset spaces
9.2 Geodesic sigma models on infinite-dimensional coset spaces
9.3 Eleven-dimensional supergravity and ℰ10∕𝒦 (ℰ10)
9.4 Further comments
10 Cosmological Solutions from E10
10.1 Bianchi I models and eleven-dimensional supergravity
10.2 Geometric configurations and regular subalgebras of E10
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
Open References References
Index
Footnotes
Figures
Tables