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

More documents

Recommendations

Info

326the fourth lines of (3.18), we may permute freely certain indices in some ofthe terms inside the brackets. This yields the passage from lines 2, 3 and 4of (3.18) to lines 2, 3 and 4 of (3.20).It remains to explain how we pass from the fifth (last) line of (3.18) to thefifth (last) line of (3.20). The bracket in the fifth line of (3.18) contains threeterms: [−T 1 − T 2 − T 3 ]. The term T 3 involves the product δ k 1,k 3i 2 , i 1, which werewrite as δ k 3,k 1i 1 , i 2, in order that i 1 appears before i 2 . Then, we rewrite the threeterms in the new order [−T 2 − T 3 − T 1 ], which yields:(3.22)∑ [−δ k 1,k 3i 1 , i 2X k 2y − δ k 3,k 1i 1 , i 2X k 2y − δ k 2,k 3i 1 , i 2X k 1yk 1 ,k 2 ,k 3]y k1 y k2 ,k 3.It remains to observe that we can permute k 2 and k 3 in the first term −T 2 ,which yields the last line of (3.20). The detailed proof is complete.3.23. Final perfect expression of Y i1 ,i 2 ,i 3. Thanks to similar (longer) computations,we have obtained an expression of Y i1 ,i 2 ,i 3which we consider tobe in final harmonious shape. Without copying the intermediate steps, let uswrite down the result. The comments which are necessary to read it and tointerpret it start just below.Y i1 ,i 2 ,i 3= Y x i 1x i 2x i 3 + ∑ k 1[δ k 1i 1Y x i 2x i 3y + δ k 1i 2Y x i 1x i 3y + δ k 1i 3Y x i 1x i 2y − X k 1x i 1x i 2x i 3]y k1 +(3.24)+ ∑ k 1 ,k 2[δ k 1,k 2i 1 , i 2Y x i 3y 2 + δ k 1,k 2i 3 , i 1Y x i 2y 2 + δ k 1,k 2i 2 , i 3Y x i 1y 2−]−δ k 1i 1X k 2x i 2x i 3y − δk 1i 2X k 2x i 1x i 3y − δk 1i 3X k 2yx i 1x i 2y k1 y k2 ++ ∑ []δ k 1,k 2 ,k 3i 1 , i 2 , i 3Y y 3 − δ k 1,k 2i 1 , i 2X k 3− δ k 1,k 2x i 3y 2 i 1 , i 3X k 3− δ k 1,k 2x i 2y 2 i 2 , i 3X k 3yx i 1y 2 k1 y k2 y k3 +k 1 ,k 2 ,k 3+ ∑ [−δ k 1,k 2 ,k 3i 1 , i 2 , i 3X k 4]yy 3 k1 y k2 y k3 y k4 +k 1 ,k 2 ,k 3 ,k 4+ ∑ k 1 ,k 2[δ k 1,k 2i 1 , i 2Y x i 3y + δ k 1,k 2i 3 , i 1Y x i 2y + δ k 1,k 2i 2 , i 3Y x i 1y −]−δ k 1i 1X k 2x i 2x − i 3 δk 1i 2X k 2x i 1x − i 3 δk 1i 3X k 2yx i 1x i 2 k1 ,k 2++ ∑ [δ k 1,k 2 ,k 3i 1 , i 2 , i 3Y y 2 + δ k 3,k 1 ,k 2i 1 , i 2 , i 3Y y 2 + δ k 2,k 3 ,k 1i 1 , i 2 , i 3Y y 2−k 1 ,k 2 ,k 3−δ k 1,k 2i 1 , i 2X k 3x i 3y − δk 3,k 1i 1 , i 2X k 2x i 3y − δk 2,k 3i 1 , i 2X k 1x i 3y −−δ k 1,k 2i 1 , i 3X k 3x i 2y − δk 3,k 1i 1 , i 3X k 2x i 2y − δk 2,k 3i 1 , i 3X k 1x i 2y −−δ k 1,k 2i 2 , i 3X k 3x i 1y − δk 3,k 1i 2 , i 3X k 2x i 1y − δk 2,k 3i 2 , i 3X k 1x i 1y]y k1 y k2 ,k 3+
+ ∑k 1 ,k 2 ,k 3 ,k 4[−δ k 1,k 2 ,k 3i 1 , i 2 , i 3X k 4y 2 − δ k 2,k 3 ,k 1i 1 , i 2 , i 3X k 4y 2 − δ k 3,k 2 ,k 1i 1 , i 2 , i 3X k 4y 2 −−δ k 3,k 4 ,k 1i 1 , i 2 , i 3X k 2y 2 − δ k 3,k 1 ,k 4i 1 , i 2 , i 3X k 2y 2 − δ k 1,k 3 ,k 4i 1 , i 2 , i 3X k 2+ ∑ [−δ k 1,k 2 ,k 3i 1 , i 2 , i 3X k 4y − δ k 2,k 3 ,k 1i 1 , i 2 , i 3X k 4y − δ k 3,k 1 ,k 2i 1 , i 2 , i 3X k 4yk 1 ,k 2 ,k 3 ,k 4+ ∑k 1 ,k 2 ,k 3[δ k 1,k 2 ,k 3i 1 , i 2 , i 3Y y − δ k 1,k 2i 1 , i 2X k 3x i 3 − δk 1,k 2i 1 , i 3X k 3x i 2 − δk 1,k 2i 2 , i 3X k 3x i 1327]yy 2 k1 y k2 y k3 ,k 4+]y k1 ,k 2y k3 ,k 4+]y k1 ,k 2 ,k 3++ ∑ [−δ k 1,k 2 ,k 3i 1 , i 2 , i 3X k 4y − δ k 4,k 1 ,k 2i 1 , i 2 , i 3X k 3y − δ k 3,k 4 ,k 1i 1 , i 2 , i 3X k 2y − δ k 2,k 3 ,k 4i 1 , i 2 , i 3X k 1yk 1 ,k 2 ,k 3 ,k 43.25. Comments, analysis and induction. First of all, by comparing thisexpression of Y i1 ,i 2 ,i 3with the expression (2.8) of Y 3 , we easily guess a partof the (inductional) dictionary beween the cases n = 1 and the case n 1.For instance, the three monomials [·](y 1 ) 3 , [·] y 1 y 2 and [·] (y 1 ) 2 y 2 in Y 3 arereplaced in Y i1 ,i 2 ,i 3by the following three sums:(3.26) ∑∑∑[·]y k1 y k2 y k3 , [·] y k1 y k2 ,k 3, and [·]y k1 y k2 y k3 ,k 4.k 1 ,k 2 ,k 3 k 1 ,k 2 ,k 3 k 1 ,k 2 ,k 3 ,k 4Similar formal correspondences may be observed for all the monomials ofY 1 , Y i1 , of Y 2 , Y i1 ,i 2and of Y 3 , Y i1 ,i 2 ,i 3. Generally and inductively speaking,the monomial(3.27) [·](y λ1 ) µ1 · · ·(y λd ) µ dappearing in the expression (2.25) of Y κ should be replaced by a certainmultiple sum generalizing (3.26). However, it is necessary to think, to pauseand to search for an appropriate formalism before writing down the desiredmultiple sum.The jet variable y λ1 should be replaced by a jet variable corresponding toa λ 1 -th partial derivative, say y k1 ,...,k λ1, where k 1 , . . ., k λ1 = 1, . . .,n. Forthe moment, to simplify the discussion, we leave out the presence of a sumof the form ∑ k 1 ,...,k λ1. The µ 1 -th power (y λ1 ) µ 1should be replaced not by(y k1 ,...,k λ1) µ1,but by a product of µ1 different jet variables y k1 ,...,k λ1of lengthλ, with all indices k α = 1, . . .,n being distinct. This rule may be confirmedby inspecting the expressions of Y i1 , of Y i1 ,i 2and of Y i1 ,i 2 ,i 3. So y k1 ,...,k λ1should be developed as a product of the form(3.28) y k1 ,...,k λ1y kλ1 +1,...,k 2λ1 · · · y k(µ1 −1)λ 1 +1,...,k µ1 λ 1,where(3.29) k 1 , . . .,k λ1 , . . .,k µ1 λ 1= 1, . . ., n.]y k1 y k2 ,k 3 ,k 4.
Page 1 and 2:
Joël M E R K E RÉcole Normale Sup
Page 3 and 4:
§1. INTRODUCTIONSeveral physically
Page 5 and 6:
e a collection of m analytic second
Page 7 and 8:
7where the indices j, l 1 vary in {
Page 9 and 10:
This phenomenon could be explained
Page 11 and 12:
This yields the prolongation of the
Page 13 and 14:
X(x, y) and Y (x, y) such that it m
Page 15 and 16:
2.23. Compatibility conditions for
Page 17 and 18:
This lemma is left to the reader; a
Page 19 and 20:
simplification nor any reordering:(
Page 21 and 22:
aleza, y por otra, las organizacion
Page 23 and 24:
computational level (differential-g
Page 25 and 26:
For instance, in the case m = 2, by
Page 27 and 28:
in (3.11). The second equation that
Page 29 and 30:
Similarly, the second equation take
Page 31 and 32:
order derivatives of X and of the Y
Page 33 and 34:
Lemma 3.32. The system yxx j = F j
Page 35 and 36:
Lemma 3.45. The following quadratic
Page 37 and 38:
often be denoted by the sign “·
Page 39 and 40:
1Now, taking account of the factor
Page 41 and 42:
conditions, totally equivalent to t
Page 43 and 44:
43remaining terms afterwards:(3.71)
Page 45 and 46:
Here, the sign ≡ precisely means:
Page 47 and 48:
Next, replacing plainly (3.64) in (
Page 49 and 50:
49the order of §3.73. We get:(3.89
Page 51 and 52:
51Multiplying by −2 and reorganiz
Page 53 and 54:
⎧Θ l 1y l 2 = −Ll 1l 1 ,l 1 ,y
Page 55 and 56:
+ 1 ∑4 δj l 1Hl k 2L k k,kk− 2
Page 57 and 58:
+ 1 ∑2 δj l 1Hl k 3M l2 ,kk+ 1 4
Page 59 and 60:
In the hardest techical part of thi
Page 61 and 62:
− 1 3+ 1 3− 1 4+ 1 4∑Hl k 1H
Page 63 and 64:
− 1 3∑kL l 1l1 ,k,x Hk l 1+ 1 3
Page 65 and 66:
Here, the sign ≡ means “modulo
Page 67 and 68:
Thirdly, put j := l 2 in (3.108) wi
Page 69 and 70:
69correct. We get:(4.29)0 =?== −
Page 71 and 72:
Next, apply the operator ∑ k Ll 2
Page 73 and 74:
73Writing term by term the substrac
Page 75 and 76:
of the terms of the subgoal (4.29):
Page 77 and 78:
77+ 1 2∑kH k l 1 ,y l 2 Hl 2k15
Page 79 and 80:
79= − X xx Yx j + Y jm∑+++++l 1
Page 81 and 82:
Our first task is to compute the de
Page 83 and 84:
Replacing this expression of A k in
Page 85 and 86:
85have the continuation(5.17) −y
Page 87 and 88:
87and where thirdly (we are nearly
Page 89 and 90:
89obtain:(5.23)III :=m∑m∑l 1 =1
Page 91 and 92:
[GTW1989] GRISSOM, C.; THOMPSON, G.
Page 93 and 94:
93Nonalgebraizable real analytic tu
Page 95 and 96:
and of CR dimension m = n − d ≥
Page 97 and 98:
coordinates z ′ = 2i ln(z/z p ),
Page 99 and 100:
{x ∈ K n : |x| < ρ} for some ρ
Page 101 and 102:
(2) A K-algebraic inversion mapping
Page 103 and 104:
(1) The complex Lie algebra Hol(M,
Page 105 and 106:
Proposition 3.1. Let t ↦→ G i (
Page 107 and 108:
The first relation gives nothing, s
Page 109 and 110:
with |β∗ k | ≥ 1 and integers
Page 111 and 112:
Clearly, the left hand side is an a
Page 113 and 114:
field X 1 ′ := h ∗ (X 1 ). Taki
Page 115 and 116:
which yields after differentiating
Page 117 and 118:
117Im w∆ n(ρ 4 )∆ n(ρ 1 )∆
Page 119 and 120:
[∂ i,ei H ei (t ′ )] t ′ =He
Page 121 and 122:
p = (w p , z p , ζ p , ξ p ) ∈
Page 123 and 124:
Finite nondegeneracy is interesting
Page 125 and 126:
such that |ω j i ∗ ,β∗ i(t,
Page 127 and 128:
for k even, we have Γ k (z (k) ) =
Page 129 and 130:
that for every local holomorphic se
Page 131 and 132:
h(t) = ˜H(t, J l 0µ 0¯h(0)) with
Page 133 and 134:
infinite families of pairwise non b
Page 135 and 136:
functions w(z)). The coefficients R
Page 137 and 138:
Then the Lie criterion states that
Page 139 and 140:
y k ′ = λk k y k and w ′ = µ
Page 141 and 142:
The last statements of Corollaries
Page 143 and 144:
[Sha2000] SHAFIKOV, R.: Analytic co
Page 145 and 146:
Here x = (x 1 , . . ., x n ) ∈ K
Page 147 and 148:
Consequently, for the case κ = 2,
Page 149 and 150:
are devoted to provide a general on
Page 151 and 152:
for l = 1, . . .,m such that the lo
Page 153 and 154:
Lemma 8.1. The following conditions
Page 155 and 156:
155namely X ←→ X + X ←→ X .
Page 157 and 158:
In terms of Sussmann’s approach [
Page 159 and 160:
x κ−1 χ κ−1 + O(|x| κ ) + O
Page 161 and 162:
Observe that these expressions are
Page 163 and 164:
where the first term I involves onl
Page 165 and 166:
I 5 = ∑ i 1I 6 = ∑ i 1 ,i 2I 7
Page 167 and 168:
U 2 U κ−1 , we obtain the five f
Page 169 and 170:
and U 2 U κ−1 , we obtain the fi
Page 171 and 172:
following equations for j = 1, . .
Page 173 and 174:
linear equations:(7.28) ⎧0 = Rx j
Page 175 and 176:
175R j containing at least one part
Page 177 and 178:
177[3] :∑σ∈S κ−1κδ l,....
Page 179 and 180:
the values of the partial derivativ
Page 181 and 182:
also appears some derivatives Q l
Page 183 and 184:
[5] F. ENGEL; LIE, S.: Theorie der
Page 185 and 186:
185Nonrigid sphericalreal analytic
Page 187 and 188:
origin:{ ( ∣ ∣ )AJ 4 1∣∣∣
Page 189 and 190:
I 2 characterizes equivalence to w
Page 191 and 192:
y means of a fundamental, elementar
Page 193 and 194:
2) When M is Levi nondegenerate at
Page 195 and 196:
195holding in C { z ′ , z ′ , w
Page 197 and 198:
Fortunately for our present purpose
Page 199 and 200:
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
Page 211 and 212:
explicit computation, so let us rew
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
Page 221 and 222:
and also in addition the fact that
Page 223 and 224:
and this defines without ambiguity
Page 225 and 226:
the leaves of this local foliation
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 and 252:
Convention 5.2. The letters R will
Page 253 and 254:
Assuming that dimSYM(E 1 ) = 8, tak
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 .
Page 303 and 304: of the cancellation properties (11.
Page 305 and 306: 305II: Explicit prolongations of in
Page 307 and 308: Define then the transformed jet ϕ
Page 309 and 310: Let λ ∈ N be an arbitrary intege
Page 311 and 312: 311[Ol1986], [BK1989]):(Y(1.31)⎧
Page 313 and 314: + [Y y 2 − 2 X xy ] y 1 y 2 + [
Page 315 and 316: for some nonnegative integers A, B,
Page 317 and 318: Once the correct theorem is formula
Page 319 and 320: By the classical formula for the de
Page 321 and 322: The induction formulas are⎧(3.4)
Page 323 and 324: We have to compute:( ) ⎛ ⎞∑
Page 325: Next, we gather the underlined term
Page 329 and 330: Correspondingly, we identify the se
Page 331 and 332: must be equal to the number of indi
Page 333 and 334: Y i1 ,i 2 ,i 3 ,i 4written in one o
Page 335 and 336: 335conclusion, we have shown that (
Page 337 and 338: 337Theorem 3.73. For every κ 1 an
Page 339 and 340: Theorem 3.79. For every integer κ
Page 341 and 342: +++m∑l 1 ,l 2 =1m∑l 1 ,l 2 ,l 3
Page 343 and 344: 343form:(4.12)κ+1Yκ j = Y jx κ +
Page 345 and 346: eing related to the number 2 in the
Page 347 and 348: Appying the chain rule, we may comp
Page 349 and 350: Secondly:(5.3)Y j i 1 ,i 2= Y jx i
Page 351 and 352: As explained before the statement o
Page 353 and 354: g j i 1 ,...,i λand similarly for
Page 355 and 356: with respect to the variables x i 1
Page 357 and 358: Assuming F = F(x, y x ) to be indep
Page 359 and 360: 359(III’) ⎧0 = ∑ (δ k 2)j 3H
Page 361 and 362: Open problem 1.17. For n = 2 establ
Page 363 and 364: ⎧∣ ⎨X 1 xX 1+ y x 1 y x 1 ·1
Page 365 and 366: eplacing the third column by the se
Page 367 and 368: ⎧ ∣ ∣⎫ ⎨ ∣∣∣∣∣
Page 369 and 370: e written under the specific form:(
Page 371 and 372: Indeed, the collection of equations
Page 373 and 374: 3.12. Principal unknowns. As there
Page 375 and 376: 375Sixthly:(3.22) ⎧δ k 1j 1Θ n+
Page 377 and 378:
are a consequence of (I’), (I”)
Page 379 and 380:
379IV: BibliographyREFERENCES[Ar198
Page 381 and 382:
[G1989] GARDNER, R.B.: The method o
Page 383 and 384:
[Me2005a] MERKER, J.: On the local
show all

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

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?