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

254of (y k ) 2 and of (y k ) 3 , we get the linear system(5.19)⎧Y x i 1x i 2 = P ( )Y x l⎪⎨ δi k 1Y x i 2y + δi k 2Y x i 1y − X kx i 1x i 2= P ( Y y , X l 2, X )l , Y , Yx l 1 x lδ k,ki 1 ,i 2Y yy − δi k 1X kx⎪⎩i 2y − δk i 2X kx i 1y = P( Y y , X l 2, X )lx l 1 y , X l , Y , Y x lδ k,ki 1 ,i 2Xyy k = P( Y y , X l 2, X lx l 1 y , X l , Y , Y x l),upon which obvious linear combinations yield a known generalization ofProposition 5.9.∑Proposition 5.20. ([Su2001, GM2003a]) An infinitesimal Lie symmetrynk=1 X k ∂ + Y ∂ is uniquely determined by the ∂x k ∂y n2 + 4n + 3 initialTaylor coefficients:(5.21) X l (0), Y (0), X l 2(0), X lx l 1 y (0), Y x l(0), Y y(0), Y x y(0), Y l yy (0).The bound dimSYM(E 2 ) n 2 + 4n + 3 is attained with F i1 ,i 2= 0,whence all P = 0 and(5.22)⎧A := ∂ y , E := y ∂ y ,⎪⎨ B i := ∂ x i, F i := y ∂ x i,C i := x i ∂ y , G i := x i( )x 1 ∂ x 1 + · · · + x n ∂ x n + y ∂ y + xy ∂y ,⎪⎩D i,k := x i ∂ x k, H := y ( )x 1 ∂ x 1 + · · · + x n ∂ x n + y ∂ y .are infinitesimal generators of the group PGL n+2 (K) = Aut(P n+1 (K)) ofprojective transformations(5.23)(x, y) ↦→(α1 x 1 + · · · + α n x n + βy + γλ 1 x 1 + · · · + λ n x n + µy + ν ,)δ 1 x 1 + · · · + δ n x n + ηy + ǫλ 1 x 1 + · · · + λ n x n + µy + ν ,stabilizing the collections of all affine planes of K n+1 , namely the solutionsof the model equation y x i 1x i 2 = 0. The model Lie algebra pgl n+2 (K) ≃sl n+2 (K) is simple, hence rigid.Theorem 5.24. The bound dim SYM(E 2 ) n 2 +4n+3 is attained if andonly if (E 2 ) is equivalent, through a diffeomorphism (x i , y) ↦→ (X k , Y ), toY X k 1X k 2 = 0.The proof, similar to that of Theorem 5.13, is skipped.The study of (E 3 ) also leads to the model algebra pgl n+2 (K) ≃ sl n+2 (K)and an analog to Theorem 5.13 holds. Details are similar.§6. TRANSFER OF LIE SYMMETRIES TO THE PARAMETER SPACE6.1. Stabilization of foliations. As announced in §2.38, we now transferthe theory of Lie symmetries to submanifolds of solutions.