In Sections 3 and 4, we develop the mathematical tools necessary for apprehending those aspects of Coxeter groups and Kac–Moody algebras that are needed in the BKL analysis. First, in Section 3, we provide a primer on Coxeter groups (which are the mathematical structures that make direct contact with the BKL billiards). We then move on to Kac–Moody algebras in Section 4, and we discuss, in particular, some prominent features of hyperbolic Kac–Moody algebras.
In Section 5 we then make use of these mathematical concepts to relate the BKL billiards to Lorentzian Kac–Moody algebras. We show that there is a simple connection between the relevant Kac–Moody algebra and the U-duality algebras that appear upon toroidal dimensional reduction to three dimensions, when these U-duality algebras are split real forms. The Kac–Moody algebra is then just the standard overextension of the U-duality algebra in question.
To understand the non-split case requires an understanding of real forms of finite-dimensional semi-simple Lie algebras. This mathematical material is developed in Section 6. Here, again, we have tried to be both rather complete and explicit through the use of many examples. We have followed a pedagogical approach privileging illustrative examples over complete proofs (these can be found in any case in the references given in the text). We explain the complementary Vogan and Tits–Satake approaches, where maximal compact and maximal noncompact Cartan subalgebras play the central roles, respectively. The concepts of restricted root systems and of the Iwasawa decomposition, central for understanding the emergence of the billiard, have been given particular attention. For completeness we provide tables listing all real forms of finite Lie algebras, both in terms of Vogan diagrams and in terms of Tits–Satake diagrams. In Section 7 we use these mathematical developments to relate the Kac–Moody billiards in the non-split case to the U-duality algebras appearing in three dimensions.
Up to (and including) Section 7, the developments present well-established results. With Section 8 we initiate a journey into more speculative territory. The presence of hyperbolic Weyl groups suggests that the corresponding infinite-dimensional Kac–Moody algebras might, in fact, be true underlying symmetries of the theory. How this conjectured symmetry should actually act on the physical fields is still unclear, however. We explore one approach in which the symmetry is realized nonlinearly on a -dimensional sigma model based on , which is the case relevant to eleven-dimensional supergravity. To this end, in Section 8 we introduce the concept of a level decomposition of some of the relevant Kac–Moody algebras in terms of finite regular subalgebras. This is necessary for studying the sigma model approach to the conjectured infinite-dimensional symmetries, a task undertaken in Section 9. We show that the sigma model for spectacularly reproduces important features of eleven-dimensional supergravity. However, we also point out important limitations of the approach, which probably does not constitute the final word on the subject.
In Section 10 we show that the interpretation of eleven-dimensional supergravity in terms of a manifestly -invariant sigma model sheds interesting and useful light on certain cosmological solutions of the theory. These solutions were derived previously but without the Kac–Moody algebraic understanding. The sigma model approach also suggests a new method of uncovering novel solutions. Finally, in Section 11 we present a concluding discussion and some suggestions for future research.
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