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

Assuming that dimSYM(E 1 ) = 8, taking account of (5.7), after makingsome linear combinations, there must exist eight generators of the form⎧A ′ := ∂ y + O(1), E ′ := y ∂ y + O(2),⎪⎨ B ′ := ∂ x + O(1), F ′ := y ∂ x + O(2),(5.14)C ′ := x∂ y + O(2), G ′ := xx∂ x + xy ∂ y + O(3),⎪⎩D ′ := x∂ x + O(2), H ′ := xy ∂ x + yy ∂ y + O(3).To insure that the Lie brackets between these vector fields are small perturbationsof the model ones, we can in advance replace (x, y) by (εx, εy), sothat y xx = ε F ( εx, εy, y x)is an O(ε), hence all the remainders O(1), O(2)and O(3) above are also O(ε). It follows that the structure constants forA ′ , . . .,H ′ are ε-close to those of Table 2.Theorem 5.15. ([OV1994]) Every semisimple Lie algebra over R or C isrigid: small deformations of the structure constants just give isomorphic Liealgebras.Consequently, there exists a change of basis close to the identity leadingto new generators A ′′ , B ′′ , . . .,G ′′ , H ′′ having exactly the same structureconstants as in Table 2. Then A ′′ (0) and B ′′ (0) are still linearly independent.Since [ A ′′ , B ′′] = [A, B] = 0, there exist local coordinates (X, Y ) centeredat 0 in which A ′′ = ∂ X and B ′′ = ∂ Y . Since [ A ′′ , C ′′] = [A, C] = 0and [ B ′′ , C ′′] = [B, C] = A, it follows that C ′′ = X∂ Y . The tangency to0 = −Y 2 + F(X, Y, Y 1 ) (with F(0) = 0) of ( ∂ X) (2)= ∂ X , of ( ∂ Y) (2)= ∂ Yand of ( X∂ Y) (2)= X∂ Y + ∂ Y1 yields F = 0.Open question 5.16. Does this proof generalize to y x κ+1 =F ( x, y, y x , . . ., y x κ)?5.17. Complete system of second order. We now summarize a generalizationto (E 2 ). According to Section 7 below, one may assume that thesubmanifold of solutions is y = b + ∑ ni=1 ai[ x i + O(|x| 2 ) + O(a) + O(b) ] (,whence y x i 1x i 2 = F i1 ,i 2 x i , y,(y x k)with) F(x, y, 0) ≡ 0. Applying to theskeleton 0 = −y i1 ,i 2+ F i1 ,i 2 x i , y, y k a second prolongation L (2) havingcoefficients Y i1 given by (3.9)(II) and Y i1 ,i 2given by (3.20)(II), we get(5.18) 0 = −Y i1 ,i 2+n∑k=1[ ] Xk∂F i1 ,i 2+ [ Y ] ∂F i1 ,i 2∂x k ∂y+n∑k=1253[Yk]∂F i1 ,i 2∂y k.Replacing y i1 ,i 2everywhere by F i1 ,i 2= y 1 R + · · · + y n R, developping inpowers of the pure jet variables y l and picking the coefficients of cst., of y k ,