100 Years of Relativity Space-Time Structure: Einstein and Beyond ...

Gravitational Billiards, Dualities and Hidden Symmetries 63and F ij = F ji . One verifies that all algebra relations are satisfied with(a (ij) ≡ (a ij + a ji )/2)and the identifications[K i j, E kl ] = δ k j Eil + δ l j Eki[K i j, F kl ] = −δk i F jl − δl i F kj[E ij , F kl ] = 2δ (i(k Kj) l) − δ (i (k δj) l K1 1 + K 2 2 + K 3 3)〈F ij |E kl 〉 = δ (ki δ l)j (63)e 3 = E 33 , f 3 = F 33 (64)As one proceeds to higher levels, the classification of sl(3, R) representationsbecomes rapidly more involved due to the exponential increase inthe number of representations with level l. Generally, the representationsthat can occur at level l + 1 must be contained in the product of the level-lrepresentations with the level-one representation (0, 2). Working out theseproducts is elementary, but cumbersome. For instance, the level-two generatorE ab|jk ≡ ε abi E i jk , with labels (1, 2), is straightforwardly obtained bycommuting two level-one elements[E ij , E kl ] = ε mk(i E m j)l + ε ml(i E mj)k(65)A more economical way to identify the relevant representations is to workout the relation between Dynkin labels and the associated highest weights,using the fact that the highest weights of the adjoint representation arethe roots. More precisely, the highest weight vectors being (as exemplifiedabove at level 1) of the ‘lowering type’, the corresponding highest weightsare negative roots, say Λ = −α. Working out the associated Dynkin labelsone obtainsp 1 ≡ p = n − 2m , p 2 ≡ q = 2l + m − 2n (66)As indicated, we shall henceforth use the notation [p 1 , p 2 ] ≡ [p, q] for theDynkin labels. This formula is restrictive because all the integers enteringit must be non-negative. Inverting this relation we getm = 2 3 l − 2 3 p − 1 3 qn = 4 3 l − 1 3 p − 2 3 q (67)with n ≥ 2m ≥ 0. A further restriction derives from the fact that thehighest weight must be a root of AE 3 , viz. its square must be smaller orequal to 2:Λ 2 = 2 3(p 2 + q 2 + pq − l 2) ≤ 2 (68)