LiePseudo–Groups olver/t_/psgparis.pdf · une jungle presque impénétrable; mais cell-ci menace de se refermer sur les sentiers déja tracés, si l’on ne procède bientot

Lie Pseudo–Groups

Peter J. Olver

University of Minnesota

http://www.math.umn.edu/! olver

Paris, December, 2009

Sur la theorie, si importante sans doute, mais

pour nous si obscure, des "groupes de Lie infinis#,

nous ne savons rien que ce qui se trouve dans les

memoires de Cartan, premiere exploration a travers

une jungle presque impenetrable; mais cell-ci menace

de se refermer sur les sentiers deja traces, si l’on

ne procede bientot a un indispensable travail de

defrichement.

— Andre Weil, 1947

What’s the Di!culty with Infinite–Dimensional Groups?

• Lie invented Lie groups to study symmetry and solution ofdi!erential equations.

$ In Lie’s time, there were no abstract Lie groups. All groupswere realized by their action on a space.

% Therefore, Lie saw no essential distinction between finite-dimensional and infinite-dimensional group actions.

However, with the advent of abstract Lie groups, the twosubjects have gone in radically di!erent directions.

& The general theory of finite-dimensional Lie groups has beenrigorously formalized and applied.

' But there is still no generally accepted abstract object thatrepresents an infinite-dimensional Lie pseudo-group!

Ehresmann’s Trinity

1953:

• Lie Pseudo-groups

• Jets

• Groupoids


1953:

• Lie pseudo-groups

• Jets

• Groupoids


1953:


• Jets

• Groupoids


1953:


• Jets

• Groupoids

Lie Pseudo-groups — History

• Lie, Medolaghi, Vessiot

• E. Cartan

• Ehresmann, Libermann

• Kuranishi, Spencer, Singer, Sternberg, Guillemin,Kumpera, . . .

Lie Pseudo-groups — Applications

• Relativity

• Noether’s (Second) Theorem

• Gauge theory and field theories:Maxwell, Yang–Mills, conformal, string, . . .

• Fluid mechanics, metereology:Navier–Stokes, Euler, boundary layer, quasi-geostropic, . . .

• Linear and linearizable PDEs

• Solitons (in 2 + 1 dimensions): K–P, Davey-Stewartson, . . .

• Kac–Moody

• Morphology and shape recognition

• Control theory

• Geometric numerical integration

• Lie groups !

Lie Pseudo-groups — Moving Frames

$ Motivation: To develop an algorithmic invariant calculus for Lie group andpseudo-group actions. Classify and construct di!erential invariants— including their generators and syzygies — invariant di!erentialforms, invariant di!erential operators, invariant di!erential equa-tions, invariant variational problems, etc.

% Tools: The equivariant approach to moving frames — which can beimplemented for arbitrary Lie group and most Lie pseudo-groupactions — along with the induced invariant variational bicomplex.

& Additional benefits: A new, elementary approach to the structure theoryfor Lie pseudo-groups, including explicit construction of Maurer–Cartan forms and direct, elementary determination of structureequations from the infinitesimal generators.

=( PJO, Fels, Pohjanpelto, Cheh, Itskov, Valiquette

Lie Pseudo-groups—Further applications

• Symmetry groups of di!erential equations

• Vessiot group splitting; explicit solutions

• Gauge theories

• Calculus of variations

• Invariant geometric flows

• Computer vision and mathematical morphology

• Geometric numerical integration

Pseudo-groupsM — analytic (smooth) manifold

Definition. A pseudo-group is a collection of local analytic

di!eomorphisms ! : dom! ) M * M such that

• Identity : 1M + G

• Inverses : !!1 + G

• Restriction: U ) dom! =( ! | U + G

• Continuation: dom! =!

U! and ! | U! + G =( ! + G

• Composition: im! ) dom" =( " "! + G

The Di"eomorphism Pseudo-group

M — m-dimensional manifold

D = D(M) — pseudo-group ofall local analytic di!eomorphisms

Z = !(z)

!"

#

z = (z1, . . . , zm) — source coordinates

Z = (Z1, . . . , Zm) — target coordinates

!"

#

L"(!) = " "! — left action

R"(!) = ! ""!1 — right action

Jets

For 0 , n , -:

Given a smooth map ! : M * M , written in local coordinates as

Z = !(z), let jn!|z denote its n–jet at z + M , i.e., its nth

order Taylor polynomial or series based at z.

Jn(M,M) is the nth order jet bundle, whose points are the jets.

Local coordinates on Jn(M,M):

(z, Z(n)) = ( . . . za . . . Zb . . . ZbA . . . ), Zb

A =#kZb

#za1 · · · #zak

Di"eomorphism Jets

The nth order di!eomorphism jet bundle is the subbundle

D(n) = D(n)(M) ) Jn(M,M)

consisting of nth order jets of local di!eomorphisms ! : M * M .

The Inverse Function Theorem tells us that D(n) is definedby the non-vanishing of the Jacobian determinant:

det(Zab ) = det( #Za/#zb ) .= 0

$ D(n) forms a groupoid undercomposition of Taylor polynomials/series.

Groupoid StructureDouble fibration:

D(n)

!!

"!(n) #

#$" (n)

M M

!(n)(z, Z(n)) = z — source map

" (n)(z, Z(n)) = Z — target map

You are only allowed to multiply h(n) · g(n) if

!(n)(h(n)) = " (n)(g(n))

$ Composition of Taylor polynomials/series is well-definedonly when the source of the second matches the target ofthe first.

One-dimensional case: M = R

Source coordinate: x Target coordinate: X

Local coordinates on D(n)(R)

g(n) = (x,X,Xx, Xxx, Xxxx, . . . , Xn)

Di!eomorphism jet:

X[[h ]] = X +Xx h+ 12 Xxx h

2 + 16 Xxxx h

3 + · · ·

=( Taylor polynomial/series at a source point x

Groupoid multiplication of di!eomorphism jets:

(X,X,XX,XXX, . . . ) · (x,X,Xx, Xxx, . . . )

= (x,X,XX Xx,XX Xxx +XXX X2x, . . . )

=( Composition of Taylor polynomials/series

• The groupoid multiplication (or Taylor composition) is onlydefined when the source coordinate X of the first multipli-cand matches the target coordinate X of the second.

• The higher order terms are expressed in terms of Bell polyno-mials according to the general Faa–di–Bruno formula.

Pseudo-group Jets

Any pseudo-group G ) D defines

a Lie sub-groupoid G(n) ) D(n).

Definition. G is regular if, for n # 0, its jets ! : G(n) * M

form an embedded subbundle of ! : D(n) * M and the

projection %n+1n : G(n+1) * G(n) is a fibration.

Definition. A regular, analytic pseudo-group G is called a

Lie pseudo-group of order n# / 1 if every local di!eomor-

phism ! + D satisfying jn!! ) G(n!) belongs it: ! + G.

In local coordinates, G(n!) ) D(n!) forms a system ofdi!erential equations

F (n!)(z, Z(n!)) = 0

called the determining system of the pseudo-group. The Liecondition requires that every local solution to the determiningsystem belongs to the pseudo-group.

What about integrability/involutivity?

Lemma. In the analytic category, for su"ciently large n # 0the determining system G(n) ) D(n) of a regular pseudo-group is an involutive system of partial di!erential equations.

Proof : Cartan–Kuranishi + local solvability.


F (n!)(z, Z(n!)) = 0




Proof : Cartan–Kuranishi + local solvability.


F (n!)(z, Z(n!)) = 0




Proof : regularity + Cartan–Kuranishi + local solvability.

Lie Completion of a Pseudo-group

Definition. The Lie completion G 0 G of a regular pseudo-group is defined as the space of all analytic di!eomorphisms! that solve the determining system G(n!).

Theorem. G and G have the same di!erential invariants, thesame invariant di!erential forms, etc.

$ Thus, for local geometry, there is no loss in generality assum-ing all (regular) pseudo-groups are Lie pseudo-groups!

Lie Completion of a Pseudo-group

Definition. The Lie completion G 0 G of a regular pseudo-group is defined as the space of all analytic di!eomorphisms! that solve the determining system G(n!).

Theorem. G and G have the same di!erential invariants, thesame invariant di!erential forms, etc.

$ Thus, for local geometry, there is no loss in generality assum-ing all (regular) pseudo-groups are Lie pseudo-groups!

A Non-Lie Pseudo-group

X = !(x) Y = !(y) where ! + D(R)

On the o!-diagonal set M = { (x, y) |x .= y }, the pseudo-group G is regular of order 1, and G(1) ) D(1) is defined by thefirst order determining system

Xy = Yx = 0 Xx, Yy .= 0

The general solution to the determining system G(1) formsthe Lie completion G:

X = !(x) Y = "(y) where !," + D(R)

Structure of Lie Pseudo-groups

Recall:

The structure of a finite-dimensional Lie groupG is specified by its Maurer–Cartan forms — a basisµ1, . . . , µr for the right-invariant one-forms:

dµk =$

i<j

Ckij µ

i 1 µj

What should be the Maurer–Cartan forms of aLie pseudo-group?

Cartan: Use exterior di!erential systems andprolongation to determine the structure equations.

I propose a direct approach based on the followingobservation:

The Maurer–Cartan forms for a Lie group and henceLie pseudo-group can be identified with the right-invariant one-forms on the jet groupoid G(#).

The structure equations can be determined immedi-ately from the infinitesimal determining equations.















The structure equations can be determined immediatelyfrom the infinitesimal determining equations.





The structure equations can be determined immediatelyfrom the infinitesimal determining equations.

The Variational Bicomplex

The di!erential one-forms on an infinite jet bundle split intotwo types:

• horizontal forms

• contact forms

Consequently, the exterior derivative on D(#) splits

d = dM + dG

into horizontal (manifold) and contact (group) compo-nents, leading to the variational bicomplex structure onthe algebra of di!erential forms on D(#).

For the di!eomorphism jet bundle

D(#) ) J#(M,M)

Local coordinates:

z1, . . . zm

% &' (, Z1, . . . Zm

% &' (, . . . Zb

A, . . .% &' (

source target jet

Horizontal forms:dz1, . . . , dzm

Basis contact forms:

#bA = dGZb

A = dZbA 2

m$

a=1

ZaA,a dz

a


Local coordinates on D(#)(R)

(x,X,Xx, Xxx, Xxxx, . . . , Xn, . . . )

Horizontal form:dx

Contact forms:# = dX 2Xx dx

#x = dXx 2Xxx dx

#xx = dXxx 2Xxxx dx

...

Maurer–Cartan Forms

Definition. The Maurer–Cartan forms for the di!eomorphismpseudo-group are the right-invariant one-forms on thedi!eomorphism jet groupoid D(#).

Key observation: Since the right action only a!ects sourcecoordinates, the target coordinate functions Za

are right-invariant.

Thus, when we decompose

dZa = &a + µa

horizontal contact

both components &a, µa are right-invariant one forms.






dZa = &a + µa

horizontal contact







dZa = &a + µa

horizontal contact


Invariant horizontal forms:

&a = dM Za =m$

b=1

Zab dz

b

Dual invariant total di!erentiation operators:

DZa =m$

b=1

(Zab )

!1Dzb

Thus, the invariant contact forms µbA are obtained by invariant

di!erentiation of the order zero contact forms:

µb = dGZb = #b = dZb 2m$

a=1

Zba dz

a

µbA = D

AZµ

b = DZa1 · · · DZanµb b = 1, . . . ,m, #A / 0


Contact forms:

# = dX 2Xx dx

#x = Dx# = dXx 2Xxx dx

#xx = D2x# = dXxx 2Xxxx dx

Right-invariant horizontal form:

& = dM X = Xx dx

Invariant di!erentiation:

DX =1

Xx

Dx

Invariant contact forms:

µ = # = dX 2Xx dx

µX = DXµ =#x

Xx

=dXx 2Xxx dx

Xx

µXX = D2Xµ =

Xx#xx 2Xxx#x

X3x

=Xx dXxx 2Xxx dXx + (X2

xx 2XxXxxx) dx

X3x

...

µn = DnXµ

The Structure Equations for

the Di"eomorphism Pseudo–group

dµbA =

)Cb,B,C

A,c,d µcB 1 µd

C

Formal Maurer–Cartan series:

µb[[H ]] =$

A

1

A!µbAHA

H = (H1, . . . , Hm) — formal parameters

dµ[[H ]] = 3µ[[H ]] 1 (µ[[H ]]2 dZ )

d& = 2 dµ[[ 0 ]] = 3µ[[ 0 ]] 1 &

The Structure Equations for

the Di"eomorphism Pseudo–group

dµbA =

)Cb,B,C

A,c,d µcB 1 µd

C

Formal Maurer–Cartan series:

µb[[H ]] =$

A

1

A!µbAHA

H = (H1, . . . , Hm) — formal parameters

dµ[[H ]] = 3µ[[H ]] 1 (µ[[H ]]2 dZ )

d& = 2 dµ[[ 0 ]] = 3µ[[ 0 ]] 1 &


Structure equations:

d& = µX 1 & dµ[[H ]] =dµ

dH[[H ]] 1 (µ[[H ]]2 dZ)

where

& = Xx dx = dX 2 µ

µ[[H ]] = µ+ µX H + 12 µXX H2 + · · ·

µ[[H ]]2 dZ = 2& + µX H + 12 µXX H2 + · · ·

dµ

dH[[H ]] = µX + µXX H + 1

2 µXXX H2 + · · ·

In components:

d& = µ1 1 &

dµn = 2µn+1 1 & +n!1$

i=0

*n

i

+

µi+1 1 µn!i

= & 1 µn+1 2[ n+1

2 ]$

j=1

n2 2j + 1

n+ 1

*n+ 1

j

+

µj 1 µn+1!j.

=( Cartan

The Maurer–Cartan Forms

for a Lie Pseudo-group

The Maurer–Cartan forms for a pseudo-group G ) Dare obtained by restricting the di!eomorphismMaurer–Cartan forms &a, µb

A to G(#) ) D(#).

$ $ The resulting one-forms are no longer linearlyindependent, but the dependencies can bedetermined directly from the infinitesimalgenerators of G.

The Maurer–Cartan Forms

for a Lie Pseudo-group

The Maurer–Cartan forms for a pseudo-group G ) Dare obtained by restricting the di!eomorphismMaurer–Cartan forms &a, µb

A to G(#) ) D(#).

$ $ The resulting one-forms are no longer linearlyindependent, but the dependencies can bedetermined directly from the infinitesimalgenerators of G.

Infinitesimal Generators

g — Lie algebra of infinitesimal generators ofthe pseudo-group G

z = (x, u) — local coordinates on M

Vector field:

v =m$

a=1

'a(z)#

#za=

p$

i=1

(i#

#xi+

q$

$=1

)$#

#u$

Vector field jet:jnv 42* '(n) = ( . . . 'bA . . . )

'bA =##A'b

#zA=

#k'b

#za1 · · · #zak

The infinitesimal generators of G are the solutions to theinfinitesimal determining equations

L(z, '(n)) = 0 (5)

obtained by linearizing the nonlinear determining equations atthe identity.

• If G is the symmetry group of a system of di!erential equa-tions, then (5) is the (involutive completion of) the usualLie determining equations for the symmetry group.

The infinitesimal generators of G are the solutions to theinfinitesimal determining equations

L(z, '(n)) = 0 (5)

obtained by linearizing the nonlinear determining equations atthe identity.

• If G is the symmetry group of a system of di!erential equa-tions, then (5) is the (involutive completion of) the usualLie determining equations for the symmetry group.

Theorem. The Maurer–Cartan forms on G(#) satisfy theinvariantized infinitesimal determining equations

L( . . . Za . . . µbA . . . ) = 0 ($ $)

obtained from the infinitesimal determining equations

L( . . . za . . . 'bA . . . ) = 0 ($)

by replacing

• source variables za by target variables Za

• derivatives of vector field coe"cients 'bA byright-invariant Maurer–Cartan forms µb

A

The Structure Equationsfor a Lie Pseudo-group

Theorem. The structure equations for the pseudo-group

G are obtained by restricting the universal di!eomorphism

structure equations

dµ[[H ]] = 3µ[[H ]] 1 (µ[[H ]]2 dZ )

to the solution space of the linear algebraic system

L( . . . Za, . . . µbA, . . . ) = 0.

Comparison of StructureEquations

If the action is transitive, then our structure equations are isomorphicto Cartan’s. However, this is not true for intransitive pseudo-groups.Whose structure equations are “correct”?

• To find the Cartan structure equations, one first needs to work in anadapted coordinate chart, which requires identification of the invariantson M . Ours can be found in any system of local coordinates.

• Cartan’s procedure for identifying the invariant forms is recursive, and noteasy to implement. Ours follow immediately from the structure equationsfor the di!eomorphism pseudo-group using merely linear algebra.

• For finite-dimensional intransitive Lie group actions, Cartan’s pseudo-group structure equations do not coincide with the standard Maurer–Cartan equations. Ours do (upon restriction to a source fiber).

• Cartan’s structure equations for isomorphic pseudo-groups can be non-isomorphic. Ours are always isomorphic.

























Lie–Kumpera Example

X = f(x) U =u

f $(x)

Infinitesimal generators:

v = (#

#x+ )

#

#u= ((x)

#

#x2 ($(x)u

#

#u

Linearized determining system

(x = 2)

u(u = 0 )u =

)

u

Maurer–Cartan forms:

& =u

Udx = fx dx, * = Ux dx+

U

udu =

2u fxx dx+ fx du

fx2

µ = dX 2U

udx = df 2 fx dx, + = dU 2 Ux dx2

U

udu = 2

u

fx2( dfx 2 fxx dx )

µX =du

u2

dU 2 Ux dx

U=

dfx 2 fxx dx

fx, µU = 0

+X =U

u(dUx 2 Uxx dx)2

Ux

u(dU 2 Ux dx)

= 2u

fx3( dfxx 2 fxxx dx ) +

u fxxfx

4( dfx 2 fxx dx )

+U = 2du

u+

dU 2 Ux dx

U= 2

dfx 2 fxx dx

fx

First order linearized determining system:

(x = 2)

u(u = 0 )u =

)

u

First order Maurer–Cartan determining system:

µX = 2+

UµU = 0 +U =

+

U

Substituting into the full di!eomorphism structure equationsyields the (first order) structure equations:

dµ = 2 d& =+ 1 &

U, d+ = 2 +X 1 & 2

+ 1 *

U

d+X = 2 +XX 1 & 2+X 1 (* + 2 +)

U

Symmetry Groups—Review

System of di!erential equations:

$%(x, u(n)) = 0, + = 1, 2, . . . , k

By a symmetry, we mean a transformation that maps solutionsto solutions.

Lie: To find the symmetry group of the di!erential equations,work infinitesimally.

The vector field

v =p$

i=1

(i(x, u)#

#xi+

q$

$=1

)$(x, u)#

#u$

is an infinitesimal symmetry if its flow exp(tv) is a one-parameter symmetry group of the di!erential equation.

We prolong v to the jet space whose coordinates are thederivatives appearing in the di!erential equation:

v(n) =p$

i=1

(i#

#xi+

q$

$=1

n$

#J=0

)J$

#

#u$J

where

)J$ = DJ

*

)$ 2p$

i=1

u$i (i

+

+p$

i=1

u$J,i (i

DJ — total derivativesInfinitesimal invariance criterion:

v(n)($%) = 0 whenever $ = 0.

Infinitesimal determining equations:

L(x, u; ((n),)(n)) = 0

$ $ We can determine the structure of the symmetry groupwithout solving the determining equations!

The Korteweg–deVries equation

ut + uxxx + uux = 0

Symmetry generator:

v = *(t, x, u)#

#t+ ((t, x, u)

#

#x+ )(t, x, u)

#

#uProlongation:

v(3) = v + )t #

#ut

+ )x #

#ux

+ · · · + )xxx #

#uxxx

where)t = )t + ut)u 2 ut*t 2 u2

t*u 2 ux(t 2 utux(u

)x = )x + ux)u 2 ut*x 2 utux*u 2 ux(x 2 u2x(u

)xxx = )xxx + 3ux)u + · · ·

Infinitesimal invariance:

v(3)(ut + uxxx + uux) = )t + )xxx + u)x + ux ) = 0

on solutions

Infinitesimal determining equations:

*x = *u = (u = )t = )x = 0

) = (t 223 u*t )u = 2 2

3 *t = 2 2 (x

*tt = *tx = *xx = · · · = )uu = 0

General solution:

* = c1 + 3c4t, ( = c2 + c3t+ c4x, ) = c3 2 2c4u.

Basis for symmetry algebra gKdV :

v1 = #t,

v2 = #x,

v3 = t #x + #u,

v4 = 3 t #t + x #x 2 2u#u.

The symmetry group GKdV is four-dimensional

(x, t, u) 42* (,3 t+ a,,x+ c t+ b,,!2 u+ c)

v1 = #t, v2 = #x,

v3 = t #x + #u, v4 = 3 t#t + x #x 2 2u #u.

Commutator table:

v1 v2 v3 v4

v1 0 0 0 v1

v2 0 0 v1 3v2

v3 0 2v1 0 22v3

v4 2v1 23v2 2v3 0

Entries: [vi,vj] =$

k

Ckijvk. Ck

ij — structure constants of g

Di!eomorphism Maurer–Cartan forms:

µt, µx, µu, µtT , µt

X , µtU , µx

T , . . . , µuU , µt

TT , µTTX, . . .

Infinitesimal determining equations:*x = *u = (u = )t = )x = 0

) = (t 223 u *t )u = 2 2

3 *t = 2 2 (x

*tt = *tx = *xx = · · · = )uu = 0

Maurer–Cartan determining equations:

µtX = µt

U = µxU = µu

T = µuX = 0,

µu = µxT 2 2

3 UµtT , µu

U = 2 23 µ

tT = 2 2µx

X,

µtTT = µt

TX = µtXX = · · · = µu

UU = . . . = 0.

Basis (dimGKdV = 4):

µ1 = µt, µ2 = µx, µ3 = µu, µ4 = µtT .

Substituting into the full di!eomorphism structure equationsyields the structure equations for gKdV :

dµ1 = 2µ1 1 µ4,

dµ2 = 2µ1 1 µ3 2 23 U µ1 1 µ4 2 1

3 µ2 1 µ4,

dµ3 = 23 µ

3 1 µ4,

dµ4 = 0.

dµi = Cijk µ

j 1 µk

Basis (dimGKdV = 4):

µ1 = µt, µ2 = µx, µ3 = µu, µ4 = µtT .

Substituting into the full di!eomorphism structure equationsyields the structure equations for gKdV :

dµ1 = 2µ1 1 µ4,

dµ2 = 2µ1 1 µ3 2 23 U µ1 1 µ4 2 1

3 µ2 1 µ4,

dµ3 = 23 µ

3 1 µ4,

dµ4 = 0.

dµi = Cijk(Z)µj 1 µk

Essential Invariants

• The pseudo-group structure equations live on the bundle" : G(#) * M , and the structure coe"cients Ci

jk con-structed above may vary from point to point.

& In the case of a finite-dimensional Lie group action,G(#) 6 G 7M , and this means the basis of Maurer–Cartanforms on each fiber of G(#) is varying with the target pointZ + M . However, we can always make a Z–dependentchange of basis to make the structure coe"cients constant.

$ However, for infinite-dimensional pseudo-groups, it may notbe possible to find such a change of Maurer–Cartan basis,leading to the concept of essential invariants.

Kadomtsev–Petviashvili

(KP) Equation

(ut +32 uux +

14 uxxx )x ±

34 uyy = 0

Symmetry generators:

vf = f(t) #t +23 y f

$(t) #y +,

13 x f

$(t)8 29 y

2f $$(t)-#x

+,2 2

3 u f$(t) + 2

9 x f$$(t)8 4

27 y2f $$$(t)

-#u,

wg = g(t) #y 823 y g

$(t) #x 849 y g

$$(t) #u,

zh = h(t) #x +23 h

$(t) #u.

=( Kac–Moody loop algebra A(1)4

Navier–Stokes Equations

#u

#t+ u ·3u = 23p+ +$u, 3 · u = 0.

Symmetry generators:

v# = #(t) · #x+#$(t) · #

u2#$$(t) · x #p

v0 = #t

s = x · #x+ 2 t #t 2 u · #

u2 2 p #p

r = x 1 #x+ u 1 #

u

wh = h(t) #p

Action of Pseudo-groups on Submanifolds

a.k.a. Solutions of Di"erential Equations

G — Lie pseudo-group acting on p-dimensional submanifolds:

N = {u = f(x)} ) M

For example, G may be the symmetry group of asystem of di!erential equations

$(x, u(n)) = 0

and the submanifolds are the graphs of solutions u = f(x).

Goal: Understand G–invariant objects (moduli spaces)

Prolongation

Jn = Jn(M,p) — nth order submanifold jet bundle

Local coordinates :

z(n) = (x, u(n)) = ( . . . xi . . . u$J . . . )

Prolonged action of G(n) on submanifolds:

(x, u(n)) 42* (X, .U (n))

Coordinate formulae:

.U$J = F$

J (x, u(n), g(n))

=( Implicit di!erentiation.

Di"erential Invariants

A di!erential invariant is an invariant function I : Jn * R

for the prolonged pseudo-group action

I(g(n) · (x, u(n))) = I(x, u(n))

=( curvature, torsion, . . .

Invariant di!erential operators:

D1, . . . ,Dp =( arc length derivative

• If I is a di!erential invariant, so is DjI.

I(G) — the algebra of di!erential invariants

The Basis Theorem

Theorem. The di!erential invariant algebra I(G) is locallygenerated by a finite number of di!erential invariants

I1, . . . , I&and p = dimS invariant di!erential operators

D1, . . . ,Dp

meaning that every di!erential invariant can be locallyexpressed as a function of the generating invariants andtheir invariant derivatives:

DJI! = Dj1Dj2

· · · DjnI!.

=( Lie groups: Lie, Ovsiannikov

=( Lie pseudo-groups: Tresse, Kumpera, Kruglikov–Lychagin,Munoz–Muriel–Rodrıguez, Pohjanpelto–O

Key Issues

• Minimal basis of generating invariants: I1, . . . , I&

• Commutation formulae for

the invariant di!erential operators:

[Dj,Dk ] =p$

i=1

Y ijk Di

=( Non-commutative di!erential algebra

• Syzygies (functional relations) among

the di!erentiated invariants:

%( . . . DJI! . . . ) 9 0

=( Codazzi relations

Computing Di"erential Invariants

% The infinitesimal method:

v(I) = 0 for every infinitesimal generator v + g

=( Requires solving di!erential equations.

& Moving frames.

• Completely algebraic.

• Can be adapted to arbitrary group and pseudo-group actions.

• Describes the complete structure of the di!erential invariantalgebra I(G) — using only linear algebra & di!erentiation!

• Prescribes di!erential invariant signatures for equivalence andsymmetry detection.

Moving Frames

In the finite-dimensional Lie group case, a moving frame is

defined as an equivariant map

-(n) : Jn 2* G

However, we do not have an appropriate abstract object to

represent our pseudo-group G.

Consequently, the moving frame will be an equivariant section

-(n) : Jn 2* H(n)

of the pulled-back pseudo-group jet groupoid:

G(n) H(n)

% %M & Jn.

Moving Frames for Pseudo–Groups

Definition. A (right) moving frame of order n is a right-

equivariant section -(n) : V n * H(n) defined on an open

subset V n ) Jn.

=( Groupoid action.

Proposition. A moving frame of order n exists if and only if

G(n) acts freely and regularly.

Moving Frames for Pseudo–Groups

Definition. A (right) moving frame of order n is a right-

equivariant section -(n) : V n * H(n) defined on an open

subset V n ) Jn.

=( Groupoid action.

Proposition. A moving frame of order n exists if and only if

G(n) acts freely and regularly.

FreenessFor Lie group actions, freeness means trivial isotropy:

Gz = { g + G | g · z = z } = {e}.

For infinite-dimensional pseudo-groups, this definition cannot work, and onemust restrict to the transformation jets of order n, using the nth orderisotropy subgroup:

G(n)z(n) =

/g(n) + G(n)

z

000 g(n) · z(n) = z(n)1

Definition. At a jet z(n) + Jn, the pseudo-group G acts

• freely if G(n)z(n) = {1(n)

z }

• locally freely if• G(n)

z(n) is a discrete subgroup of G(n)z

• the orbits have dimension rn = dimG(n)z

=( Kumpera’s growth bounds on Spencer cohomology.

Persistence of Freeness

Theorem. If n / 1 and G(n) acts (locally) freely

at z(n) + Jn, then it acts (locally) freely at any

z(k) + Jk with 2%kn(z

(k)) = z(n) for all k > n.

TheNormalization Algorithm

To construct a moving frame :I. Compute the prolonged pseudo-group action

u$K 42* U$K = F$

K(x, u(n), g(n))

by implicit di!erentiation.

II. Choose a cross-section to the pseudo-group orbits:

u$"

J"= c!, . = 1, . . . , rn = fiber dim G(n)

III. Solve the normalization equations

U$"

J"= F$"

J"(x, u(n), g(n)) = c!

for the nth order pseudo-group parameters

g(n) = -(n)(x, u(n))

IV. Substitute the moving frame formulas into the un-

normalized jet coordinates u$K = F$K(x, u(n), g(n)).

The resulting functions form a complete system of nth order

di!erential invariants

I$K(x, u(n)) = F$K(x, u(n), -(n)(x, u(n)))

Invariantization

A moving frame induces an invariantization process, denoted /,

that projects functions to invariants, di!erential operators

to invariant di!erential operators; di!erential forms to

invariant di!erential forms, etc.

Geometrically, the invariantization of an object is the unique

invariant version that has the same cross-section values.

Algebraically, invariantization amounts to replacing the group

parameters in the transformed object by their moving frame

formulas.

Invariantization

In particular, invariantization of the jet coordinates leads to a

complete system of functionally independent di!erential

invariants: /(xi) = Hi /(u$J) = I$J

• Phantom di!erential invariants: I$"

J"= c!

• The non-constant invariants form a complete system of

functionally independent di!erential invariants

• Replacement Theorem

I( . . . xi . . . u$J . . . ) = /( I( . . . xi . . . u$J . . . ) )

= I( . . . Hi . . . I$J . . . )

The Invariant Variational Bicomplex

$ Di!erential functions =( di!erential invariants

/(xi) = Hi /(u$J) = I$J

$ Di!erential forms =( invariant di!erential forms

/( dxi) = 0i /(1$K) = 2$K

$ Di!erential operators =( invariant di!erential operators

/ (Dxi ) = Di

Recurrence Formulae

$ $ Invariantization and di!erentiation

do not commute$ $

The recurrence formulae connect the di!erentiated invariants

with their invariantized counterparts:

DiI$J = I$J,i +M$

J,i

=( M$J,i — correction terms

& Once established, the recurrence formulae completely

prescribe the structure of the di!erential invariant

algebra I(G) — thanks to the functional independence

of the non-phantom normalized di!erential invariants.

$ $ The recurrence formulae can be explicitly determined

using only the infinitesimal generators and linear

di!erential algebra!

Lie–Tresse–Kumpera Example

X = f(x), Y = y, U =u

f $(x)

Horizontal coframe

dH X = fx dx, dH Y = dy,

Implicit di!erentiations

DX =1

fxDx, DY = Dy.

Prolonged pseudo-group transformations on surfaces S ) R3:

X = f Y = y U =u

fx

UX =ux

f 2x

2u fxxf 3x

UY =uy

fx

UXX =uxx

f 3x

23ux fxxf 4x

2u fxxxf 4x

+3u f 2

xx

f 5x

UXY =uxy

f 2x

2uy fxxf 3x

UY Y =uyy

fx

f, fx, fxx, fxxx, . . . — pseudo-group parameters

=( action is free at every order.

Coordinate cross-section

X = f = 0, U =u

fx= 1, UX =

ux

f 2x

2u fxxf 3x

= 0, UXX = · · · = 0.

Moving frame

f = 0, fx = u, fxx = ux, fxxx = uxx.

Di!erential invariants

UY =uy

fx42* J = /(uy) =

uy

u

UXY = · · · 42* J1 = /(uxy) =uuxy 2 uxuy

u3

UY Y = · · · 42* J2 = /(uxy) =uyy

uUXXY 42* J3 = /(uxxy) UXY Y 42* J4 = /(uxyy) UY Y Y 42* J5 = /(uyyy)

Invariant horizontal forms

dH X = fx dx 42* u dx, dH Y = dy 42* dy,

Invariant di!erentiations

D1 =1

uDx D2 = Dy

Higher order di!erential invariants: Dm1 Dn

2 J

J,1 = D1J =uuxy 2 uxuy

u3= J1,

J,2 = D2J =uuyy 2 u2

y

u2= J2 2 J2.

Recurrence formulae:D1J = J1, D2J = J2 2 J2,

D1J1 = J3, D2J1 = J4 2 3J J1,

D1J2 = J4, D2J2 = J5 2 J J2,

Korteweg–deVries Equation

Prolonged Symmetry Group Action:

T = e3'4(t+ ,1)

X = e'4(,3t+ x+ ,1,3 + ,2)

U = e!2'4(u+ ,3)

UT = e!5'4(ut 2 ,3ux)

UX = e!3'4ux

UTT = e!8'4(utt 2 2,3utx + ,32uxx)

UTX = DXDTU = e!6'4(utx 2 ,3uxx)

UXX = e!4'4uxx

...

Cross Section:

T = e3'4(t+ ,1) = 0

X = e'4(,3t+ x+ ,1,3 + ,2) = 0

U = e!2'4(u+ ,3) = 0

UT = e!5'4(ut 2 ,3ux) = 1

Moving Frame:

,1 = 2t, ,2 = 2x, ,3 = 2u, ,4 =15 log(ut + uux)

Normalized di!erential invariants:

I01 = /(ux) =ux

(ut + uux)3/5

I20 = /(utt) =utt + 2uutx + u2uxx

(ut + uux)8/5

I11 = /(utx) =utx + uuxx

(ut + uux)6/5

I02 = /(uxx) =uxx

(ut + uux)4/5

...

Invariant di!erential operators:

D1 = /(Dt) = (ut + uux)!3/5Dt + u(ut + uux)

!3/5Dx,

D2 = /(Dx) = (ut + uux)!1/5Dx.

Commutation formula:

[D1,D2 ] = I01 D1

Recurrence formulae:

D1I01 = I11 235 I

201 2

35 I01I20, D2I01 = I02 2

35 I

301 2

35 I01I11,

D1I20 = I30 + 2I11 285 I01I20 2

85 I

220, D2I20 = I21 + 2I01I11 2

85 I

201I20 2

85 I11I20,

D1I11 = I21 + I02 265 I01I11 2

65 I11I20, D2I11 = I12 + I01I02 2

65 I

201I11 2

65 I

211,

D1I02 = I12 245 I01I02 2

45 I02I20, D2I02 = I03 2

45 I

201I02 2

45 I02I11,

......

Generating di!erential invariants:

I01 = /(ux) =ux

(ut + uux)3/5

, I20 = /(utt) =utt + 2uutx + u2uxx

(ut + uux)8/5

.

Fundamental syzygy:

D21I01 +

35 I01D1I20 2D2I20 +

,15 I20 +

195 I01

-D1I01

2D2I01 2625 I01I

220 2

725 I

201I20 +

2425 I

301 = 0.

The Master Recurrence Formula

dH I$J =p$

i=1

(DiI$J )3i =

p$

i=1

I$J,i 3i + ." $

J

where." $

J = /( .) $J ) = %$J( . . . Hi . . . I$J . . . ; . . . 4bA . . . )

are the invariantized prolonged vector field coe"cients, whichare particular linear combinations of

4bA = /('bA) — invariantized Maurer–Cartan formsprescribed by the invariantized prolongation map.

• The invariantized Maurer–Cartan forms are subject to theinvariantized determining equations :

L(H1, . . . , Hp, I1, . . . , Iq, . . . , 4bA, . . . ) = 0

dH I$J =p$

i=1

I$J,i 3i + ." $

J( . . . 4bA . . . )

Step 1: Solve the phantom recurrence formulas

0 = dH I$J =p$

i=1

I$J,i 3i + ." $

J( . . . 4bA . . . )

for the invariantized Maurer–Cartan forms:

4bA =p$

i=1

JbA,i 3

i (5)

Step 2: Substitute (5) into the non-phantom recurrenceformulae to obtain the explicit correction terms.

$ Only uses linear di!erential algebra based on the specifica-tion of cross-section.

& Does not require explicit formulas for the moving frame, thedi!erential invariants, the invariant di!erential operators, oreven the Maurer–Cartan forms!

Lie–Tresse–Kumpera Example (continued)

X = f(x), Y = y, U =u

f $(x)

Phantom recurrence formulae:

0 = dH = 01 + 4, 0 = dI10 = J102 + 21 2 42,

0 = dI00 = J 02 + 22 41, 0 = dI20 = J302 + 23 2 43,

Solve for pulled-back Maurer–Cartan forms:

4 = 201, 42 = J102 + 21,

41 = J 02 + 2, 43 = J302 + 23,

Recurrence formulae: dy = 02

dJ = J101 + (J2 2 J2)02 + 22 2 J 2,

dJ1 = J301 + (J4 2 3J J1)0

2 + 24 2 J 21 2 J1 2,

dJ2 = J401 + (J5 2 J J2)0

2 + 25 2 J2 2,

The Korteweg–deVries Equation (continued)

Recurrence formula:

dIjk = Ij+1,k31 + Ij,k+13

2 + /()jk)

Invariantized Maurer–Cartan forms:

/(*) = ,, /(() = µ, /()) = " = +, /(*t) = "t = ,t, . . .

Invariantized determining equations:

,x = ,u = µu = +t = +x = 0

+ = µt +u = 2 2µx = 2 23 ,t

,tt = ,tx = ,xx = · · · = +uu = · · · = 0

Invariantizations of prolonged vector field coe"cients:

/(*) = ,, /(() = µ, /()) = +, /()t) = 2I01+ 253 ,t,

/()x) = 2I01,t, /()tt) = 22I11+ 283I20,t, . . .

Phantom recurrence formulae:0 = dH H1 = 31 + ,,

0 = dH H2 = 32 + µ,

0 = dH I00 = I1031 + I013

2 + " = 31 + I0132 + +,

0 = dH I10 = I2031 + I113

2 + "t = I2031 + I113

2 2 I01+ 253 ,t,

=( Solve for , = 231, µ = 232, + = 231 2 I0132,

,t =35 (I20 + I01)3

1 + 35 (I11 + I201)3

2.

Non-phantom recurrence formulae:

dH I01 = I1131 + I023

2 2 I01,t,

dH I20 = I3031 + I213

2 2 2I11+ 283I20,t,

dH I11 = I2131 + I123

2 2 I02+ 2 2I11,t,

dH I02 = I1231 + I033

2 2 43I02,t,

...

D1I01 = I11 235I

201 2

35I01I20, D2I01 = I02 2

35 I

301 2

35 I01I11,

D1I20 = I30 + 2I11 285I01I20 2

85I

220, D2I20 = I21 + 2I01I11 2

85I

201I20 2

85I11I20,

D1I11 = I21 + I02 265I01I11 2

65I11I20, D2I11 = I12 + I01I02 2

65I

201I11 2

65I

211,

D1I02 = I12 245I01I02 2

45I02I20, D2I02 = I03 2

45I

201I02 2

45I02I11,

......

GrobnerBasisApproach

Suppose G acts freely at order n#.

The di!erential invariants of order > n# arenaturally identified with polynomials belonging to acertain algebraic module J , called the invariantizedprolonged symbol module which is defined as theinvariantized pull-back of the symbol module for theinfinitesimal determining equations under a certainexplicit linear map.

Constructive Basis Theorem

Theorem. A system of generating di!erential in-variants in one-to-one correspondence with theGrobner basis elements of the invariantized pro-longed symbol module J>n!

plus, possibly, afinite number of di!erential invariants of order, n#.

Syzygy Theorem

Theorem. Every di!erential syzygy among the gen-erating di!erential invariants is either a syzygyamong those of order , n#, or arises from an alge-braic syzygy among the Grobner basis polynomi-als in J>n!

, or comes from a commutator syzygyamong the invariant di!erential operators.

The Symbol Module

Linearized determining equations

L(z, '(n)) = 0

t = (t1, . . . , tm), T = (T1, . . . , Tm)

T =

3

P (t, T ) =m$

a=1

Pa(t)Ta

4

6 R[ t ]!Rm ) R[ t, T ]

Symbol module:I ) T

The Prolonged Symbol Module

s = (s1, . . . , sp), S = (S1, . . . , Sq),

.S =

3

T (s, S) =q$

$=1

T$(s)S$

4

6 R[s]!Rq ) R[s, S]

Define the linear map

si = 5i(t) = ti +q$

$=1

u$i tp+$, i = 1, . . . , p,

S$ = B$(T ) = Tp+$ 2p$

i=1

u$i Ti, 6 = 1, . . . , q.

=( “symbol”” of the vector field prolongation operation

Prolonged symbol module:

J = ($5)!1(I)

Documents

LiePseudo–Groups olver/t_/psgparis.pdf · une jungle presque impénétrable; mais cell-ci menace de se refermer sur les sentiers déja tracés, si l’on ne procède bientot