Travaux sur les symétries de Lie des équations aux ... - DMA - Ens

To conclude, we replace X xx so obtained in (8.27) 7 : this yields (8.18) 8 .We replace Y 1yand Y 2 1 yfrom (8.32) in (8.27) 2 4 and in (8.27) 5 : this yields(8.18) 4 and this yields (8.18) 9 . Thanks to (8.18) 2 (got) we observe that(8.34) P x = P + r Y 1xx + r Y 2x + r X xx = P.Differentiating (8.18) 4 (got) and (8.18) 9 (got) with respect to x then yields(8.18) 7 and (8.18) 12 . We replace Y 1yand Y 2 1 yfrom (8.18) 2 5 (got) and (8.18) 11(got) in (8.27) 2 : this yields (8.18) 6 . Finally, to obtain the very last (8.18) 13 ,we differentiate (8.18) 10 (got) with respect to x.The proof of Proposition 8.16 is complete.We claim that the bound dimSYM(E 4 ) 5 is attained for themodel (8.4). Indeed, with 0 = r = s and 0 = g 1 = g 2 = h (whencek ∗ = 0) (8.24) is Y 1y= 0 and then the seven equations (8.27) are:2⎧0 = −Yx 2 + 2x Yx 1 ,0 = −Y 2y − 2x Y 21 y + 2 X + 2 Y 2 x 1 + 2x Y 1y 1,⎪⎨ 0 = −Y 2y − X 2 x + 2 Y 1y 1,(8.35) 0 = −X y 1,0 = −X y 2,⎪⎩0 = −Yxx,10 = −2 Y 1xy + X xx, 1having the general solution⎧X = a − d + e x,⎪⎨(8.36)Y 1 = b + d x + 2e y 1 ,⎪⎩Y 2 = c + 2a y 1 + 3e y 2 + d xx.depending on five parameters a, b, c, d, e ∈ K. Five generators of SYM(E 4 )are:⎧D := x∂ x + 2y 1 ∂ y 1 + 3y 2 ∂ y 2,(8.37)The commutator table⎪⎨L 1 := −∂ x + x∂ y 1 + xx∂ y 2,L 1 ′ := ∂ x + 2y 1 ∂ y 2,L 2 := ∂ y 1,⎪⎩L 3 := ∂ y 2.273