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8 Level Decomposition in Terms of Finite Regular Subalgebras

We have shown in the previous sections that Weyl groups of Lorentzian Kac–Moody algebras naturally emerge when analysing gravity in the extreme BKL regime. This has led to the conjecture that the corresponding Kac–Moody algebra is in fact a symmetry of the theory (most probably enlarged with new fields) [46Jump To The Next Citation Point]. The idea is that the BKL analysis is only the “revelator” of that huge symmetry, which would exist independently of that limit, without making the BKL truncations. Thus, if this conjecture is true, there should be a way to rewrite the gravity Lagrangians in such a way that the Kac–Moody symmetry is manifest. This conjecture itself was made previously (in this form or in similar ones) by other authors on the basis of different considerations [113Jump To The Next Citation Point139167Jump To The Next Citation Point]. To explore this conjecture, it is desirable to have a concrete method of dealing with the infinite-dimensional structure of a Lorentzian Kac–Moody algebra 𝔤. In this section we present such a method.

The method by which we shall deal with the infinite-dimensional structure of a Lorentzian Kac–Moody algebra 𝔤 is based on a certain gradation of 𝔤 into finite-dimensional subspaces 𝔤ℓ. More precisely, we will define a so-called level decomposition of the adjoint representation of 𝔤 such that each level ℓ corresponds to a finite number of representations of a finite regular subalgebra 𝔯 of 𝔤. Generically the decomposition will take the form of the adjoint representation of 𝔯 plus a (possibly infinite) number of additional representations of 𝔯. This type of expansion of 𝔤 will prove to be very useful when considering sigma models invariant under 𝔤 for which we may use the level expansion to consistently truncate the theory to any finite level ℓ (see Section 9).

We begin by illustrating these ideas for the finite-dimensional Lie algebra 𝔰𝔩(3,ℝ) after which we generalize the procedure to the indefinite case in Sections 8.2, 8.3 and 8.4.

 8.1 A finite-dimensional example: 𝔰𝔩(3,ℝ )
 8.2 Some formal considerations
  8.2.1 Gradation
  8.2.2 Weights of 𝔤 and weights of 𝔯
  8.2.3 Outer multiplicity
 8.3 Level decomposition of AE3
  8.3.1 Level ℓ = 0
  8.3.2 Dynkin labels
  8.3.3 Level ℓ = 1
  8.3.4 Constraints on Dynkin labels
  8.3.5 Level ℓ = 2
  8.3.6 Level ℓ = 3
  8.3.7 Level ℓ = 4
 8.4 Level decomposition of E10
  8.4.1 Decomposition with respect to 𝔰𝔩(10,ℝ)
  Algebraic structure at low levels
  8.4.2 “Gradient representations”
  8.4.3 Decomposition with respect to 𝔰𝔬(9,9) and 𝔰𝔩(2, ℝ) ⊕ 𝔰𝔩(9, ℝ)

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