(ed.). Gravitational waves (IOP, 2001)(422s).

260 Infinite-dimensional symmetries in gravitytransformation generated by f 1 , which is clearly nonlinear and non-local, becauseone has to perform an integration to calculate explicitly the dual potential.We then evaluate also the action on ˜λ. From the definition of (14.82), andobserving that δλ = 0, it follows that˜λ −1 δ ˜λ =− 1 2 −1 δ (14.113)14.4.2 Action of SL(2, R) MM on λ, BExactly the same analysis has to be done for the other group, with generators(e 0 , h 0 , f 0 )e 0 :⇒ e 0 (B) = c 0 (14.114)h 0 :⇒ h 0 (B) = 2B (14.115)f 0 :⇒ f 0 (B) = 2φ 0 (14.116)withρɛ µν ∂ ν φ 0 =− 2 B 2 ∂ µ B 2 + ρ∂ µ( ρ). (14.117)14.4.3 The affine Kac–Moody SL(2, R) algebraThe transformations we have just derived are to be identified with an affineSL(2, R) Kac–Moody algebra. The latter is characterized by the Cartan matrix( )2 −2A ij =(14.118)−2 2and the standard Chevalley–Serre presentation defining the algebra which can beread off from the Cartan matrix:[h i , h j ] = 0[h i , e j ] = A ij e j[h i , f j ] =−A ij f j[e i , f j ] = δ ij h j[e i [e i [e i , e j ]]] = 0[ f i [ f i [ f i , f j ]]] = 0.Here i = j = 0, 1; note that there is no summation on repeated indices andthat the first relation defines the Cartan subalgebra. To see the relation with theSL(2, R) transformations dealt with before, we make the identificationse 1 = T0 + , f 1 = T0 − , h 1 = T0 3 (14.119)e 0 = T − 1 , f 0 = T − + , h 0 = c − T0 3 . (14.120)