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7.1 The restricted Weyl group and the maximal split “subalgebra”

Let 𝔲3 be any real form of the complex Lie algebra ℂ 𝔲3, θ its Cartan involution, and let
𝔲3 = 𝔨3 ⊕ 𝔭3 (7.1 )
be the corresponding Cartan decomposition. Furthermore, let
𝔥3 = 𝔱3 ⊕ 𝔞3 (7.2 )
be a maximal noncompact Cartan subalgebra, with 𝔱3 (respectively, 𝔞3) its compact (respectively, noncompact) part. The real rank of 𝔲3 is, as we have seen, the dimension of 𝔞3. Let now Δ denote the root system of ℂ 𝔲3, Σ the restricted root system and m λ the multiplicity of the restricted root λ.

As explained in Section 4.9.2, the restricted root system of the real form 𝔲3 can be either reduced or non-reduced. If it is reduced, it corresponds to one of the root systems of the finite-dimensional simple Lie algebras. On the other hand, if the restricted root system is non-reduced, it is necessarily of (BC )n-type [93Jump To The Next Citation Point] (see Figure 19View Image for a graphical presentation of the BC3 root system).

The restricted Weyl group

By definition, the restricted Weyl group is the Coxeter group generated by the fundamental reflections, Equation (4.55View Equation), with respect to the simple roots of the restricted root system. The restricted Weyl group preserves multiplicities [93Jump To The Next Citation Point].

The maximal split “subalgebra” 𝖋

Although multiplicities are an essential ingredient for describing the full symmetry 𝔲3, they turn out to be irrelevant for the construction of the gravitational billiard. For this reason, it is useful to consider the maximal split “subalgebra” 𝔣, which is defined as the real, semi-simple, split Lie algebra with the same root system as the restricted root system as 𝔲3 (in the (BC )n-case, we choose for definiteness the root system of 𝔣 to be of Bn-type). The real rank of 𝔣 coincides with the rank of its complexification 𝔣ℂ, and one can find a Cartan subalgebra 𝔥 𝔣 of 𝔣, consisting of all generators of 𝔥3 which are diagonalizable over the reals. This subalgebra 𝔥𝔣 has the same dimension as the maximal noncompact subalgebra 𝔞3 of the Cartan subalgebra 𝔥3 of 𝔲 3.

By construction, the root space decomposition of 𝔣 with respect to 𝔥𝔣 provides the same root system as the restricted root space decomposition of 𝔲3 with respect to 𝔞3, except for multiplicities, which are all trivial (i.e., equal to one) for 𝔣. In the (BC )n-case, there is also the possibility that twice a root of 𝔣 might be a root of 𝔲3. It is only when 𝔲3 is itself split that 𝔣 and 𝔲3 coincide.

One calls 𝔣 the “split symmetry algebra”. It contains as we shall see all the information about the billiard region [95Jump To The Next Citation Point]. How 𝔣 can be embedded as a subalgebra of 𝔲3 is not a question that shall be of our concern here.

The Iwasawa decomposition and scalar coset Lagrangians

The purpose of this section is to use the Iwasawa decomposition for 𝔲3 described in Section 6.4.5 to derive the scalar Lagrangian based on the coset space 𝒰3 ∕𝒦 (𝒰3). The important point is to understand the origin of the similarities between the two Lagrangians in Equation (5.45View Equation) and Equation (7.8View Equation) below.

The full algebra 𝔲3 is subject to the root space decomposition

⊕ 𝔲3 = g0 ⊕ gλ (7.3 ) λ∈Σ
with respect to the restricted root system. For each restricted root λ, the space g λ has dimension m λ. The nilpotent algebra 𝔫3 ⊂ 𝔲3, consisting of positive root generators only, is the direct sum
⊕ 𝔫3 = gλ (7.4 ) λ∈Σ+
over positive roots. The Iwasawa decomposition of the U-duality algebra 𝔲 3 reads
𝔲3 = 𝔨3 ⊕ 𝔞3 ⊕ 𝔫3 (7.5 )
(see Section 6.4.5). It is 𝔞3 that appears in Equation (7.5View Equation) and not the full Cartan subalgebra 𝔥3 since the compact part of 𝔥3 belongs to 𝔨3.

This implies that when constructing a Lagrangian based on the coset space 𝒰3∕𝒦 (𝒰3 ), the only part of 𝔲 3 that will show up in the Borel gauge is the Borel subalgebra

𝔟3 = 𝔞3 ⊕ 𝔫3. (7.6 )
Thus, there will be a number of dilatons equal to the dimension of 𝔞3, i.e., equal to the real rank of 𝔲3, and axion fields for the restricted roots (with multiplicities).

More specifically, an (x-dependent) element of the coset space 𝒰3∕𝒦 (𝒰3 ) takes the form

𝒱(x) = Exp [φ(x) ⋅ 𝔞3] Exp [χ (x) ⋅ 𝔫3 ], (7.7 )
where the dilatons φ and the axions χ are coordinates on the coset space, and where x denotes an arbitrary set of parameters on which the coset element might depend. The corresponding Lagrangian becomes
di∑m𝔞3 ∑ mu∑ltα [ ][ ] ℒ 𝒰 ∕𝒦(𝒰 ) = ∂xφ (i)(x )∂x φ(i)(x) + e2α(φ) ∂x χ(α)(x) + ⋅⋅⋅ ∂xχ (α) (x ) + ⋅⋅⋅ , (7.8 ) 3 3 i=1 +s =1 [sα] [sα] α∈Σ α
where the sums over sα = 1,⋅⋅⋅ ,mult α are sums over the multiplicities of the positive restricted roots α.

By comparing Equation (7.8View Equation) with the corresponding expression (5.45View Equation) for the split case, it is clear why it is the maximal split subalgebra of the U-duality algebra that is relevant for the gravitational billiard. Were it not for the additional sum over multiplicities, Equation (7.8View Equation) would exactly be the Lagrangian for the coset space ℱ∕𝒦 (ℱ ), where 𝔨𝔣 = Lie 𝒦(ℱ ) is the maximal compact subalgebra of 𝔣 (note that 𝔨𝔣 ⁄= 𝔨3). Recall now that from the point of view of the billiard, the positive roots correspond to walls that deflect the particle motion in the Cartan subalgebra. Therefore, multiplicities of roots are irrelevant since these will only result in several walls stacked on top of each other without affecting the dynamics. (In the (BC )n-case, the wall associated with 2λ is furthermore subdominant with respect to the wall associated with λ when both λ and 2λ are restricted roots, so one can keep only the wall associated with λ. This follows from the fact that in the (BC ) n-case the root system of 𝔣 is taken to be of Bn-type.)

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