Travaux sur les symÃ©tries de Lie des Ã©quations aux ... - DMA - Ens

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252The bound dim SYM(E 1 ) 8 is attained with F = 0, whence all P = 0and⎧A := ∂ y , E := y ∂ y ,⎪⎨ B := ∂ x , F := y ∂ x ,(5.11)C := x∂ y , G := xx∂ x + xy ∂ y ,⎪⎩D := x∂ x , H := xy ∂ x + yy ∂ y .are infinitesimal generators of the group PGL 3 (K) = Aut(P 2 (K)) of projectivetransformations(5.12) (x, y) ↦→( )αx + βy + γλx + µy + ν , δx + ηy + ǫλx + µy + ν ,stabilizing the collections of all affine lines of K 2 , namely the solutions ofthe model equation y xx = 0. The model Lie algebra pgl 3 (K) ≃ sl 3 (K) issimple.Theorem 5.13. The bound dim SYM(E 1 ) 8 is attained if and only if(E 1 ) is equivalent, through a diffeomorphism (x, y) ↦→ (X, Y ), to Y XX = 0.Proof. The statement is well known ([Lie1883, EL1890, Tr1896, Se1931,Ca1932a, Ol1986, HK1989, Ib1992, Ol1995, Sh1997, Su2001, N2003,Me2004]). We provide a (new?) proof which has the advantage to enjoydirect generalizations to all PDE systems whose model Lie algebras aresemisimple, for instance (E 2 ), (E 3 ) and (E 5 ).The Lie brackets between the eight generators (5.11) are:A B C D E F G HA 0 0 0 0 A B C D + 2EB 0 0 A B 0 0 E + 2D FC 0 −A 0 −C C D − E 0 GD 0 −B C 0 0 −F G 0E −A 0 −C 0 0 F 0 HF −B 0 −D + E F −F 0 H 0G −C −E − 2D 0 −G 0 H 0 0H −D − 2E −F −G 0 −H 0 0 0Table 2.
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 ,
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Joël M E R K E RÉcole Normale Sup
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§1. INTRODUCTIONSeveral physically
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e a collection of m analytic second
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7where the indices j, l 1 vary in {
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This phenomenon could be explained
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This yields the prolongation of the
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X(x, y) and Y (x, y) such that it m
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2.23. Compatibility conditions for
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This lemma is left to the reader; a
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simplification nor any reordering:(
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aleza, y por otra, las organizacion
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computational level (differential-g
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For instance, in the case m = 2, by
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in (3.11). The second equation that
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Similarly, the second equation take
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order derivatives of X and of the Y
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Lemma 3.32. The system yxx j = F j
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Lemma 3.45. The following quadratic
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often be denoted by the sign “·
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1Now, taking account of the factor
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conditions, totally equivalent to t
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43remaining terms afterwards:(3.71)
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Here, the sign ≡ precisely means:
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Next, replacing plainly (3.64) in (
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49the order of §3.73. We get:(3.89
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51Multiplying by −2 and reorganiz
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⎧Θ l 1y l 2 = −Ll 1l 1 ,l 1 ,y
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+ 1 ∑4 δj l 1Hl k 2L k k,kk− 2
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+ 1 ∑2 δj l 1Hl k 3M l2 ,kk+ 1 4
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In the hardest techical part of thi
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− 1 3+ 1 3− 1 4+ 1 4∑Hl k 1H
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− 1 3∑kL l 1l1 ,k,x Hk l 1+ 1 3
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Here, the sign ≡ means “modulo
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Thirdly, put j := l 2 in (3.108) wi
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69correct. We get:(4.29)0 =?== −
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Next, apply the operator ∑ k Ll 2
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73Writing term by term the substrac
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of the terms of the subgoal (4.29):
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77+ 1 2∑kH k l 1 ,y l 2 Hl 2k15
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79= − X xx Yx j + Y jm∑+++++l 1
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Our first task is to compute the de
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Replacing this expression of A k in
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85have the continuation(5.17) −y
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87and where thirdly (we are nearly
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89obtain:(5.23)III :=m∑m∑l 1 =1
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[GTW1989] GRISSOM, C.; THOMPSON, G.
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93Nonalgebraizable real analytic tu
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and of CR dimension m = n − d ≥
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coordinates z ′ = 2i ln(z/z p ),
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{x ∈ K n : |x| < ρ} for some ρ
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(2) A K-algebraic inversion mapping
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(1) The complex Lie algebra Hol(M,
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Proposition 3.1. Let t ↦→ G i (
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The first relation gives nothing, s
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with |β∗ k | ≥ 1 and integers
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Clearly, the left hand side is an a
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field X 1 ′ := h ∗ (X 1 ). Taki
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which yields after differentiating
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117Im w∆ n(ρ 4 )∆ n(ρ 1 )∆
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[∂ i,ei H ei (t ′ )] t ′ =He
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p = (w p , z p , ζ p , ξ p ) ∈
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Finite nondegeneracy is interesting
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such that |ω j i ∗ ,β∗ i(t,
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for k even, we have Γ k (z (k) ) =
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that for every local holomorphic se
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h(t) = ˜H(t, J l 0µ 0¯h(0)) with
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infinite families of pairwise non b
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functions w(z)). The coefficients R
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Then the Lie criterion states that
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y k ′ = λk k y k and w ′ = µ
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The last statements of Corollaries
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[Sha2000] SHAFIKOV, R.: Analytic co
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Here x = (x 1 , . . ., x n ) ∈ K
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Consequently, for the case κ = 2,
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are devoted to provide a general on
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for l = 1, . . .,m such that the lo
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Lemma 8.1. The following conditions
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155namely X ←→ X + X ←→ X .
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In terms of Sussmann’s approach [
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x κ−1 χ κ−1 + O(|x| κ ) + O
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Observe that these expressions are
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where the first term I involves onl
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I 5 = ∑ i 1I 6 = ∑ i 1 ,i 2I 7
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U 2 U κ−1 , we obtain the five f
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and U 2 U κ−1 , we obtain the fi
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following equations for j = 1, . .
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linear equations:(7.28) ⎧0 = Rx j
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175R j containing at least one part
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177[3] :∑σ∈S κ−1κδ l,....
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the values of the partial derivativ
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also appears some derivatives Q l
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[5] F. ENGEL; LIE, S.: Theorie der
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185Nonrigid sphericalreal analytic
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origin:{ ( ∣ ∣ )AJ 4 1∣∣∣
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I 2 characterizes equivalence to w
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y means of a fundamental, elementar
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2) When M is Levi nondegenerate at
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195holding in C { z ′ , z ′ , w
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Fortunately for our present purpose
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We notice passim that S ≡ Q x (no
Page 201 and 202: for a certain local K-analytic new
Page 203 and 204: Conversely, if y xx (x) = F ( x, y(
Page 205 and 206: 205But we may also express the dual
Page 207 and 208: Then thanks to a straightforward ap
Page 209 and 210: and we then expand carefully the re
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Page 213 and 214: 213+T ah3Q 2 a Q b∆(aa|b)∆(b|bb
Page 215 and 216: 215Vanishing Hachtroudi curvaturean
Page 217 and 218: to fix the ideas, it will be assume
Page 219 and 220: then by replacing the(so obtained)v
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Page 227 and 228: Then by differentiating with respec
Page 229 and 230: ∂Here, the coefficients of the2 T
Page 231 and 232: 231for (7.28):( )∂□ ·∂a □
Page 233 and 234: 233Lie symmetriesof partial differe
Page 235 and 236: Differentiating the first equation
Page 237 and 238: Example 1.21. (Continued) With n =
Page 239 and 240: defined by the graphed equations:((
Page 241 and 242: for some two local analytic maps Π
Page 243 and 244: Proof. Let l = l ( x i , y j , yβ(
Page 245 and 246: complementary views on the same obj
Page 247 and 248: Classification problem 3.11. Classi
Page 249 and 250: Definition 4.10. L is an infinitesi
Page 251: Convention 5.2. The letters R will
Page 255 and 256: Restarting from §4.1, let ϕ a Lie
Page 257 and 258: Here, R j l 1 ,kare universal polyn
Page 259 and 260: Lemma 6.34. Let M be a submanifold
Page 261 and 262: where the expressions D β,β1 are
Page 263 and 264: Then the sum L + L is tangent to M
Page 265 and 266: Lemma 7.11. ([CM1974, BER1999, Me20
Page 267 and 268: where(7.33) ∆ := ∑1kn∂ 2∂x
Page 269 and 270: with r, s being unspecified functio
Page 271 and 272: and moreover, all other, higher ord
Page 273 and 274: To conclude, we replace X xx so obt
Page 275 and 276: Definition 8.43. A real analytic hy
Page 277 and 278: Lemma 8.59. The function b depends
Page 279 and 280: where the letter r denotes an unspe
Page 281 and 282: (8.80) ⎧Y 1,1,1 = Y x 1 x 1 x 1 +
Page 283 and 284: (X 1 , X 2 , Y , Xy 1,X 2x, Y 2 x 1
Page 285 and 286: espect to x 1 and (8.96) 2 with res
Page 287 and 288: For the two unknowns Xyy 1 and X 2x
Page 289 and 290: Redeveloping the determinant, the v
Page 291 and 292: 291are obtained by equating to zero
Page 293 and 294: Let (z 0 , c 0 ) = (x 0 , y 0 , a 0
Page 295 and 296: we deduce that Γ 5 is submersive (
Page 297 and 298: 11.4. Regularity and jet parametriz
Page 299 and 300: Cramer’s rule, we get(11.13) ⎧
Page 301 and 302: in K[a, z] n+m and in K[x, c] p+m .
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of the cancellation properties (11.
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305II: Explicit prolongations of in
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Define then the transformed jet ϕ
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Let λ ∈ N be an arbitrary intege
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311[Ol1986], [BK1989]):(Y(1.31)⎧
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+ [Y y 2 − 2 X xy ] y 1 y 2 + [
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for some nonnegative integers A, B,
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Once the correct theorem is formula
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By the classical formula for the de
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The induction formulas are⎧(3.4)
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We have to compute:( ) ⎛ ⎞∑
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Next, we gather the underlined term
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+ ∑k 1 ,k 2 ,k 3 ,k 4[−δ k 1,k
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Correspondingly, we identify the se
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must be equal to the number of indi
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Y i1 ,i 2 ,i 3 ,i 4written in one o
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335conclusion, we have shown that (
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337Theorem 3.73. For every κ 1 an
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Theorem 3.79. For every integer κ
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+++m∑l 1 ,l 2 =1m∑l 1 ,l 2 ,l 3
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343form:(4.12)κ+1Yκ j = Y jx κ +
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eing related to the number 2 in the
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Appying the chain rule, we may comp
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Secondly:(5.3)Y j i 1 ,i 2= Y jx i
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As explained before the statement o
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g j i 1 ,...,i λand similarly for
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with respect to the variables x i 1
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Assuming F = F(x, y x ) to be indep
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359(III’) ⎧0 = ∑ (δ k 2)j 3H
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Open problem 1.17. For n = 2 establ
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⎧∣ ⎨X 1 xX 1+ y x 1 y x 1 ·1
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eplacing the third column by the se
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⎧ ∣ ∣⎫ ⎨ ∣∣∣∣∣
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e written under the specific form:(
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Indeed, the collection of equations
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3.12. Principal unknowns. As there
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375Sixthly:(3.22) ⎧δ k 1j 1Θ n+
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are a consequence of (I’), (I”)
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379IV: BibliographyREFERENCES[Ar198
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[G1989] GARDNER, R.B.: The method o
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[Me2005a] MERKER, J.: On the local
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