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4 Lorentzian Kac–Moody Algebras

The explicit appearance of infinite crystallographic Coxeter groups in the billiard limit suggests that gravitational theories might be invariant under a huge symmetry described by Lorentzian Kac–Moody algebras (defined in Section 4.1). Indeed, there is an intimate connection between crystallographic Coxeter groups and Kac–Moody algebras. This connection might be familiar in the finite case. For instance, it is well known that the finite symmetry group A2 of the equilateral triangle (isomorphic to the group of permutations of 3 objects) and the corresponding hexagonal pattern of roots are related to the finite-dimensional Lie algebra 𝔰𝔩(3,ℝ ) (or 𝔰𝔲 (3 )). The group A2 is in fact the Weyl group of 𝔰𝔩(3,ℝ ) (see Section 4.7).

This connection is not peculiar to the Coxeter group A2 but is generally valid: Any crystallographic Coxeter group is the Weyl group of a Kac–Moody algebra traditionally denoted in the same way (see Section 4.7). This is the reason why it is expected that the Coxeter groups might signal a bigger symmetry structure. And indeed, there are indications that this is so since, as we shall discuss in Section 9, an attempt to reformulate the gravitational Lagrangians in a way that makes the conjectured symmetry manifest yields intriguing results.

The purpose of this section is to develop the mathematical concepts underlying Kac–Moody algebras and to explain the connection between Coxeter groups and Kac–Moody algebras. How this is relevant to gravitational theories will be discussed in Section 5.

 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
  4.6.1 Definition
  4.6.2 Real and imaginary roots
  4.6.3 Fundamental weights and the Weyl vector
  4.6.4 The generalized Casimir operator
  Note
 4.7 The Weyl group
  Examples
 4.8 Hyperbolic Kac–Moody algebras
  4.8.1 The fundamental domain ℱ
  4.8.2 Roots and the root lattice
  4.8.3 Examples
  The Kac–Moody Algebra A++1
  The Kac–Moody Algebra A (2)+ 2
 4.9 Overextensions of finite-dimensional Lie algebras
  4.9.1 Untwisted overextensions
  A special property of E10
  4.9.2 Root systems in Euclidean space
  4.9.3 Twisted overextensions
  Twisted overextensions associated with the (BC )-root systems
  Twisted overextensions associated with the highest short root
  4.9.4 Algebras of Gaberdiel–Olive–West type
 4.10 Regular subalgebras of Kac–Moody algebras
  4.10.1 Definitions
  4.10.2 Examples – Regular subalgebras of E10
  A9 ⊂ 𝓑 ⊂ E10
  DE10 ⊂ E10
  4.10.3 Further properties
  Comments

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