We review the intimate connection between (super-)gravity close to a spacelike singularity (the "BKL-limit") and the theory of Lorentzian Kac-Moody algebras. We show that in this limit the gravitational theory can be reformulated in terms of billiard motion in a region of hyperbolic space, revealing that the dynamics is completely determined by a (possibly infinite) sequence of reflections, which are elements of a Lorentzian Coxeter group. Such Coxeter groups are the Weyl groups of infinite-dimensional Kac-Moody algebras, suggesting that these algebras yield symmetries of gravitational theories. Our presentation is aimed to be a self-contained and comprehensive treatment of the subject, with all the relevant mathematical background material introduced and explained in detail. We also review attempts at making the infinite-dimensional symmetries manifest, through the construction of a geodesic sigma model based on a Lorentzian Kac-Moody algebra. An explicit example is provided for the case of the hyperbolic algebra E10, which is conjectured to be an underlying symmetry of M-theory. Illustrations of this conjecture are also discussed in the context of cosmological solutions to eleven-dimensional supergravity.

中文翻译：

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类空奇点和引力的隐藏对称性。

我们回顾了近似于类空间奇点（“ BKL极限”）的（超）引力与洛伦兹Kac-Moody代数理论之间的紧密联系。我们证明在此极限下，可以根据双曲线空间区域中的台球运动来重新构造引力理论，这表明动力学完全由（可能是无限的）反射序列确定，这是洛伦兹·科克塞特群的元素。这样的Coxeter群是无限维Kac-Moody代数的Weyl群，表明这些代数产生了引力理论的对称性。我们的演讲旨在成为该主题的独立且全面的治疗方法，并介绍和详细解释所有相关的数学背景材料。我们还回顾了通过基于Lorentzian Kac-Moody代数的测地线sigma模型的构建来使无限维对称性表现出来的尝试。对于双曲代数E10的情况，提供了一个明确的示例，它被认为是M理论的基本对称性。还在十一维超重力的宇宙学解决方案的背景下讨论了这个猜想的例证。