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

Lemma 6.34. Let M be a submanifold y = Π(x, a, b) that is solvable withrespect to the parameters (a, b). If a vector field that respects the separationbetween variables and parameters, namely of the form(6.35)L +L ∗ =n∑i=1X i (x, y) ∂∂x i+ m∑j=1Y j (x, y) ∂∂y j + p∑q=1259F q (a, b) ∂∂a q + m∑is tangent to M , then L is an infinitesimal Lie symmetry of ( E M)Corollary 6.36. Through the one-to-one correspondence (E ) ←→ M ofProposition 2.17, infinitesimal Lie symmetries of (E ) correspond to vectorfields L + L ∗ tangent to M .Definition 6.37. Let SYM(M ) denote the Lie algebra of vector fields L +L ∗ tangent to M . Let SYM(E ) denote the Lie algebra of infinitesimal Liesymmetries of (E ).In summary:(6.38)SYM(E ) ≃ SYM ( M (E ))j=1and SYM ( M ) ≃ SYM ( E M).G j (a, b) ∂∂b j6.39. Dual defining equations. As in §2.10, let M ⊂ K n x ×Km y ×Kp a ×Km bgiven by 0 = −y j + Π j (x, a, b) and assume if to be solvable with respectto the parameters. In particular, we can solve the b j , obtaining dual definingequations(6.40) b j = Π ∗j (a, x, y), j = 1, . . .,m,for some local K-analytic map map Π ∗ = (Π ∗1 , . . .,Π ∗m ) satisfying(6.41) b ≡ Π ∗( a, x, Π(x, a, b) ) and y ≡ Π ( x, a, Π ∗ (a, x, y) ) .6.42. An algorithm for the computation of SYM(M ). The tangency toM is expressed by applying the vector field (6.35) to 0 = −y j +Π j (x, a, b),which yields:(6.43)0 = −Y j (x, y) +n∑X i (x, y) Π j (x, a, b) +x ii=1+p∑q=1F q (a, b) Π j aq(x, a, b)m∑G l (a, b) Π j (x, a, b),b lfor j = 1, . . ., m and for (x, y, a, b) ∈ M . In fact, after replacing the variabley by Π(x, a, b), these equations should be interpreted as power seriesidentities in K{x, a, b}.l=1