Lie Pseudo Groups. Peter J. Olver University of Minnesota olver. Paris, December, 2009

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1 Lie Pseudo Groups Peter J. Olver University of Minnesota olver Paris, December, 2009

2 Sur la théorie, 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 mémoires de Cartan, premiere exploration à travers une jungle presque impénétrable; mais cell-ci menace de se refermer sur les sentiers déjà tracés, si l on ne procède bientôt à un indispensable travail de défrichement. André Weil, 1947

3 What s the Difficulty with Infinite Dimensional Groups? Lie invented Lie groups to study symmetry and solution of differential equations. In Lie s time, there were no abstract Lie groups. All groups were realized by their action on a space. Therefore, Lie saw no essential distinction between finitedimensional and infinite-dimensional group actions. However, with the advent of abstract Lie groups, the two subjects have gone in radically different directions. The general theory of finite-dimensional Lie groups has been rigorously formalized and applied. But there is still no generally accepted abstract object that represents an infinite-dimensional Lie pseudo-group!

4 Ehresmann s Trinity 1953: Lie Pseudo-groups Jets Groupoids

5 Ehresmann s Trinity 1953: Lie pseudo-groups Jets Groupoids

6 Ehresmann s Trinity 1953: Lie pseudo-groups Jets Groupoids

7 Ehresmann s Trinity 1953: Lie pseudo-groups Jets Groupoids

8 Lie Pseudo-groups History Lie, Medolaghi, Vessiot É. Cartan Ehresmann, Libermann Kuranishi, Spencer, Singer, Sternberg, Guillemin, Kumpera,...

9 Relativity Lie Pseudo-groups Applications 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 dimensions): K P, Davey-Stewartson,... Kac Moody Morphology and shape recognition Control theory Geometric numerical integration Lie groups!

10 Lie Pseudo-groups Moving Frames Motivation: To develop an algorithmic invariant calculus for Lie group and pseudo-group actions. Classify and construct differential invariants includingtheirgeneratorsandsyzygies invariantdifferential forms, invariant differential operators, invariant differential equations, invariant variational problems, etc. Tools: The equivariant approach to moving frames which can be implemented for arbitrary Lie group and most Lie pseudo-group actions along with the induced invariant variational bicomplex. Additional benefits: A new, elementary approach to the structure theory for Lie pseudo-groups, including explicit construction of Maurer Cartan forms and direct, elementary determination of structure equations from the infinitesimal generators. = PJO, Fels, Pohjanpelto, Cheh, Itskov, Valiquette

11 Lie Pseudo-groups Further applications Symmetry groups of differential equations Vessiot group splitting; explicit solutions Gauge theories Calculus of variations Invariant geometric flows Computer vision and mathematical morphology Geometric numerical integration

12 Pseudo-groups M analytic (smooth) manifold Definition. A pseudo-group is a collection of local analytic diffeomorphisms φ :domφ M M such that Identity: 1 M G Inverses: φ 1 G Restriction: U dom φ = φ U G Continuation: domφ = U κ and φ U κ G = φ G Composition: imφ dom ψ = ψ φ G

13 The Diffeomorphism Pseudo-group M m-dimensional manifold D = D(M) pseudo-group of all local analytic diffeomorphisms Z = φ(z) z =(z 1,...,z m ) source coordinates Z =(Z 1,...,Z m ) target coordinates L ψ (φ) =ψ φ R ψ (φ) =φ ψ 1 left action right action

14 Jets For 0 n : Given a smooth map φ : M M, writteninlocalcoordinatesas Z = φ(z), let j n φ z denote its n jet at z M, i.e.,itsn th order Taylor polynomial or series based at z. J n (M,M) isthen th order jet bundle, whosepointsarethejets. Local coordinates on J n (M,M): (z,z (n) )=(... z a... Z b... ZA b... ), Zb A = k Z b z a 1 z a k

15 Diffeomorphism Jets The n th order diffeomorphism jet bundle is the subbundle D (n) = D (n) (M) J n (M,M) consisting of n th order jets of local diffeomorphisms φ : M M. The Inverse Function Theorem tells us that D (n) is defined by the non-vanishing of the Jacobian determinant: det( Z a b )=det( Za / z b ) 0 D (n) forms a groupoid under composition of Taylor polynomials/series.

16 Double fibration: Groupoid Structure σ (n) M D (n) τ (n) M σ (n) (z,z (n) )=z source map τ (n) (z,z (n) )=Z target map You are only allowed to multiply h (n) g (n) if σ (n) (h (n) )=τ (n) (g (n) ) Composition of Taylor polynomials/series is well-defined only when the source of the second matches the target of the first.

17 One-dimensional case: M = R Source coordinate: x Target coordinate: X Local coordinates on D (n) (R) g (n) =(x, X, X x,x xx,x xxx,...,x n ) Diffeomorphism jet: X [ h ] = X + X x h X xx h X xxx h3 + = Taylor polynomial/series at a source point x

18 Groupoid multiplication of diffeomorphism jets: (X, X, X X, X XX,... ) (x, X,X x,x xx,... ) =(x, X, X X X x, X X X xx + X XX X 2 x,...) = Composition of Taylor polynomials/series The groupoid multiplication (or Taylor composition) is only defined when the source coordinate X of the first multiplicand matches the target coordinate X of the second. The higher order terms are expressed in terms of Bell polynomials according to the general Fàa di Bruno formula.

19 Pseudo-group Jets Any pseudo-group G Ddefines Definition. a Lie sub-groupoid G (n) D (n). G is regular if, for n 0, its jets σ : G (n) M form an embedded subbundle of σ : D (n) projection π n+1 n Definition. : G (n+1) G (n) is a fibration. M and the A regular, analytic pseudo-group G is called a Lie pseudo-group of order n 1ifevery local diffeomorphism φ Dsatisfying j n φ G (n ) belongs it: φ G.

20 In local coordinates, G (n ) differential equations D (n ) forms a system of F (n ) (z,z (n ) )=0 called the determining system of the pseudo-group. The Lie condition requires that every local solution to the determining system belongs to the pseudo-group. What about integrability/involutivity? Lemma. In the analytic category, for sufficiently large n 0 the determining system G (n) D (n) of a regular pseudogroup is an involutive system of partial differential equations. Proof : Cartan Kuranishi + local solvability.

21 In local coordinates, G (n ) differential equations D (n ) forms a system of F (n ) (z,z (n ) )=0 called the determining system of the pseudo-group. The Lie condition requires that every local solution to the determining system belongs to the pseudo-group. What about integrability/involutivity? Lemma. In the analytic category, for sufficiently large n 0 the determining system G (n) D (n) of a regular pseudogroup is an involutive system of partial differential equations. Proof : Cartan Kuranishi + local solvability.

22 In local coordinates, G (n ) differential equations D (n ) forms a system of F (n ) (z,z (n ) )=0 called the determining system of the pseudo-group. The Lie condition requires that every local solution to the determining system belongs to the pseudo-group. What about integrability/involutivity? Lemma. In the analytic category, for sufficiently large n 0 the determining system G (n) D (n) of a regular pseudogroup is an involutive system of partial differential equations. Proof : regularity + Cartan Kuranishi + local solvability.

23 Lie Completion of a Pseudo-group Definition. The Lie completion G G of a regular pseudogroup is defined as the space of all analytic diffeomorphisms φ that solve the determining system G (n ). Theorem. G and G have the same differential invariants, the same invariant differential forms, etc. Thus, for local geometry, there is no loss in generality assuming all (regular) pseudo-groups are Lie pseudo-groups!

24 Lie Completion of a Pseudo-group Definition. The Lie completion G G of a regular pseudogroup is defined as the space of all analytic diffeomorphisms φ that solve the determining system G (n ). Theorem. G and G have the same differential invariants, the same invariant differential forms, etc. Thus, for local geometry, there is no loss in generality assuming all (regular) pseudo-groups are Lie pseudo-groups!

25 ANon-LiePseudo-group X = φ(x) Y = φ(y) where φ D(R) On the off-diagonal set M = { (x, y) x y }, thepseudogroup G is regular of order 1, and G (1) D (1) is defined by the first order determining system X y = Y x =0 X x,y y 0 The general solution to the determining system G (1) forms the Lie completion G: X = φ(x) Y = ψ(y) where φ, ψ D(R)

26 Recall: Structure of Lie Pseudo-groups The structure of a finite-dimensional Lie group G is specified by its Maurer Cartan forms a basis µ 1,...,µ r for the right-invariant one-forms: dµ k = i<j C k ij µi µ j

27 What should be the Maurer Cartan forms of a Lie pseudo-group? Cartan: Use exterior differential systems and prolongation to determine the structure equations. I propose a direct approach based on the following observation: The Maurer Cartan forms for a Lie group and hence Lie pseudo-group can be identified with the rightinvariant one-forms on the jet groupoid G ( ). The structure equations can be determined immediately from the infinitesimal determining equations.

28 What should be the Maurer Cartan forms of a Lie pseudo-group? Cartan: Use exterior differential systems and prolongation to determine the structure equations. I propose a direct approach based on the following observation: The Maurer Cartan forms for a Lie group and hence Lie pseudo-group can be identified with the rightinvariant one-forms on the jet groupoid G ( ). The structure equations can be determined immediately from the infinitesimal determining equations.

29 What should be the Maurer Cartan forms of a Lie pseudo-group? Cartan: Use exterior differential systems and prolongation to determine the structure equations. I propose a direct approach based on the following observation: The Maurer Cartan forms for a Lie group and hence Lie pseudo-group can be identified with the rightinvariant one-forms on the jet groupoid G ( ). The structure equations can be determined immediately from the infinitesimal determining equations.

30 What should be the Maurer Cartan forms of a Lie pseudo-group? Cartan: Use exterior differential systems and prolongation to determine the structure equations. I propose a direct approach based on the following observation: The Maurer Cartan forms for a Lie group and hence Lie pseudo-group can be identified with the rightinvariant one-forms on the jet groupoid G ( ). The structure equations can be determined immediately from the infinitesimal determining equations.

31 What should be the Maurer Cartan forms of a Lie pseudo-group? Cartan: Use exterior differential systems and prolongation to determine the structure equations. I propose a direct approach based on the following observation: The Maurer Cartan forms for a Lie group and hence Lie pseudo-group can be identified with the rightinvariant one-forms on the jet groupoid G ( ). The structure equations can be determined immediately from the infinitesimal determining equations.

32 The Variational Bicomplex The differential one-forms on an infinite jet bundle split into two types: horizontal forms contact forms Consequently, the exterior derivative on D ( ) splits d = d M + d G into horizontal (manifold) and contact (group) components, leading to the variational bicomplex structure on the algebra of differential forms on D ( ).

33 For the diffeomorphism jet bundle Local coordinates: D ( ) J (M,M) z 1,...z m, Z 1,... Z m,... ZA b,... }{{}}{{} }{{} source target jet Horizontal forms: dz 1,...,dz m Basis contact forms: Θ b A = d G Zb A = dz b A m a =1 Z a A,a dza

34 One-dimensional case: M = R Local coordinates on D ( ) (R) (x, X, X x,x xx,x xxx,...,x n,... ) Horizontal form: Contact forms: dx Θ=dX X x dx Θ x = dx x X xx dx Θ xx = dx xx X xxx dx.

35 Maurer Cartan Forms Definition. The Maurer Cartan forms for the diffeomorphism pseudo-group are the right-invariant one-forms on the diffeomorphism jet groupoid D ( ). Key observation: Since the right action only affects source coordinates, the target coordinate functions Z a are right-invariant. Thus, when we decompose dz a = σ a + µ a horizontal contact both components σ a,µ a are right-invariant one forms.

36 Maurer Cartan Forms Definition. The Maurer Cartan forms for the diffeomorphism pseudo-group are the right-invariant one-forms on the diffeomorphism jet groupoid D ( ). Key observation: Since the right action only affects source coordinates, the target coordinate functions Z a are right-invariant. Thus, when we decompose dz a = σ a + µ a horizontal contact both components σ a,µ a are right-invariant one forms.

37 Maurer Cartan Forms Definition. The Maurer Cartan forms for the diffeomorphism pseudo-group are the right-invariant one-forms on the diffeomorphism jet groupoid D ( ). Key observation: Since the right action only affects source coordinates, the target coordinate functions Z a are right-invariant. Thus, when we decompose dz a = σ a + µ a horizontal contact both components σ a,µ a are right-invariant one forms.

38 Invariant horizontal forms: σ a = d M Z a = m b =1 Z a b dzb Dual invariant total differentiation operators: D Z a = m b =1 ( Z a b ) 1 D z b Thus, the invariant contact forms µ b A are obtained by invariant differentiation of the order zero contact forms: µ b = d G Z b =Θ b = dz b m a =1 Z b a dza µ b A = DA Z µb = D Z a 1 D Z anµ b b =1,...,m, #A 0

39 One-dimensional case: M = R Contact forms: Θ=dX X x dx Θ x = D x Θ=dX x X xx dx Θ xx = D 2 x Θ=dX xx X xxx dx Right-invariant horizontal form: σ = d M X = X x dx Invariant differentiation: D X = 1 X x D x

40 Invariant contact forms: µ =Θ=dX X x dx µ X = D X µ = Θ x X x = dx x X xx dx X x µ XX = D 2 X µ = X x Θ xx X xx Θ x X 3 x µ n = D n X µ = X x dx xx X xx dx x +(X2 xx X x X xxx ) dx X 3 x.

41 The Structure Equations for the Diffeomorphism Pseudo group dµ b A = C b,b,c A,c,d Formal Maurer Cartan series: µ b [ H ] = A µc B µd C 1 A! µb A HA H =(H 1,...,H m ) formal parameters dµ[ H ] = µ[ H ] ( µ[ H ] dz ) dσ = dµ[ 0 ] = µ[ 0 ] σ

42 The Structure Equations for the Diffeomorphism Pseudo group dµ b A = C b,b,c A,c,d Formal Maurer Cartan series: µ b [ H ] = A µc B µd C 1 A! µb A HA H =(H 1,...,H m ) formal parameters dµ[ H ] = µ[ H ] ( µ[ H ] dz ) dσ = dµ[ 0 ] = µ[ 0 ] σ

43 Structure equations: One-dimensional case: M = R dσ = µ X σ dµ[ H ] = dµ dh [ H ] (µ[ H ] dz) where σ = X x dx = dx µ µ[ H ] = µ + µ X H µ XX H2 + µ[ H ] dz = σ + µ X H µ XX H2 + dµ dh [ H ] = µ X + µ XX H µ XXX H2 +

44 In components: dσ = µ 1 σ dµ n = µ n+1 σ + = σ µ n+1 n 1 i =0 [ n+1 2 ] j =1 ( ) n i n 2j +1 n +1 µ i+1 µ n i ( n +1 j ) µ j µ n+1 j. = Cartan

45 The Maurer Cartan Forms for a Lie Pseudo-group The Maurer Cartan forms for a pseudo-group G D are obtained by restricting the diffeomorphism Maurer Cartan forms σ a,µ b A to G( ) D ( ). The resulting one-forms are no longer linearly independent, but the dependencies can be determined directly from the infinitesimal generators of G.

46 The Maurer Cartan Forms for a Lie Pseudo-group The Maurer Cartan forms for a pseudo-group G D are obtained by restricting the diffeomorphism Maurer Cartan forms σ a,µ b A to G( ) D ( ). The resulting one-forms are no longer linearly independent, but the dependencies can be determined directly from the infinitesimal generators of G.

47 Infinitesimal Generators g Liealgebraofinfinitesimalgeneratorsof the pseudo-group G z =(x, u) localcoordinatesonm Vector field: v = m a =1 ζ a (z) z a = p i=1 ξ i x i + q ϕ α α=1 u α Vector field jet: j n v ζ (n) =(... ζ b A... ) ζ b A = #A ζ b z A = k ζ b z a 1 z a k

48 The infinitesimal generators of G are the solutions to the infinitesimal determining equations L(z,ζ (n) )=0 ( ) obtained by linearizing the nonlinear determining equations at the identity. If G is the symmetry group of a system of differential equations, then ( ) isthe(involutivecompletionof)theusual Lie determining equations for the symmetry group.

49 The infinitesimal generators of G are the solutions to the infinitesimal determining equations L(z,ζ (n) )=0 ( ) obtained by linearizing the nonlinear determining equations at the identity. If G is the symmetry group of a system of differential equations, then ( ) isthe(involutivecompletionof)theusual Lie determining equations for the symmetry group.

50 Theorem. The Maurer Cartan forms on G ( ) satisfy the invariantized infinitesimal determining equations L(... Z a... µ b A... )=0 ( ) obtained from the infinitesimal determining equations L(... z a... ζ b A... )=0 ( ) by replacing source variables z a by target variables Z a derivatives of vector field coefficients ζ b A by right-invariant Maurer Cartan forms µ b A

51 The Structure Equations for a Lie Pseudo-group Theorem. The structure equations for the pseudo-group G are obtained by restricting the universal diffeomorphism structure equations dµ[ H ] = µ[ H ] ( µ[ H ] dz ) to the solution space of the linear algebraic system L(... Z a,... µ b A,...)=0.

52 Comparison of Structure Equations If the action is transitive, then our structure equations are isomorphic to 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 an adapted coordinate chart, which requires identification of the invariants on M. Ourscanbefoundinanysystemoflocalcoordinates. Cartan s procedure for identifying the invariant forms is recursive, and not easy to implement. Ours follow immediately from the structure equations for the diffeomorphism pseudo-group using merely linear algebra. For finite-dimensional intransitive Lie group actions, Cartan s pseudogroup 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 canbenonisomorphic. Ours are always isomorphic.

53 Comparison of Structure Equations If the action is transitive, then our structure equations are isomorphic to 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 an adapted coordinate chart, which requires identification of the invariants on M. Ourscanbefoundinanysystemoflocalcoordinates. Cartan s procedure for identifying the invariant forms is recursive, and not easy to implement. Ours follow immediately from the structure equations for the diffeomorphism pseudo-group using merely linear algebra. For finite-dimensional intransitive Lie group actions, Cartan s pseudogroup 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 canbenonisomorphic. Ours are always isomorphic.

54 Comparison of Structure Equations If the action is transitive, then our structure equations are isomorphic to 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 an adapted coordinate chart, which requires identification of the invariants on M. Ourscanbefoundinanysystemoflocalcoordinates. Cartan s procedure for identifying the invariant forms is recursive, and not easy to implement. Ours follow immediately from the structure equations for the diffeomorphism pseudo-group using merely linear algebra. For finite-dimensional intransitive Lie group actions, Cartan s pseudogroup 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 canbenonisomorphic. Ours are always isomorphic.

55 Comparison of Structure Equations If the action is transitive, then our structure equations are isomorphic to 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 an adapted coordinate chart, which requires identification of the invariants on M. Ourscanbefoundinanysystemoflocalcoordinates. Cartan s procedure for identifying the invariant forms is recursive, and not easy to implement. Ours follow immediately from the structure equations for the diffeomorphism pseudo-group using merely linear algebra. For finite-dimensional intransitive Lie group actions, Cartan s pseudogroup 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 canbenonisomorphic. Ours are always isomorphic.

56 Comparison of Structure Equations If the action is transitive, then our structure equations are isomorphic to 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 an adapted coordinate chart, which requires identification of the invariants on M. Ourscanbefoundinanysystemoflocalcoordinates. Cartan s procedure for identifying the invariant forms is recursive, and not easy to implement. Ours follow immediately from the structure equations for the diffeomorphism pseudo-group using merely linear algebra. For finite-dimensional intransitive Lie group actions, Cartan s pseudogroup 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 canbenonisomorphic. Ours are always isomorphic.

57 Lie Kumpera Example Infinitesimal generators: X = f(x) U = u f (x) v = ξ x + ϕ u = ξ(x) x ξ (x) u u Linearized determining system ξ x = ϕ u ξ u =0 ϕ u = ϕ u

58 Maurer Cartan forms: σ = u U dx = f x dx, τ = U x dx + U u du = u f xx dx + f x du f x 2 µ = dx U u dx = df f x dx, ν = du U x dx U u du = u f x 2 ( df x f xx dx ) µ X = du u du U x dx U = df x f xx dx f x, µ U =0 ν X = U u (du x U xx dx) U x u (du U x dx) ν U = du u + du U x dx U = u f x 3 ( df xx f xxx dx )+u f xx f x 4 ( df x f xx dx ) = df x f xx dx f x

59 First order linearized determining system: ξ x = ϕ u ξ u =0 ϕ u = ϕ u First order Maurer Cartan determining system: µ X = ν U µ U =0 ν U = ν U Substituting into the full diffeomorphism structure equations yields the (first order) structure equations: dµ = dσ = ν σ U, dν = ν X σ ν τ U dν X = ν XX σ ν X (τ +2ν) U

60 Symmetry Groups Review System of differential equations: ν (x, u (n) )=0, ν =1, 2,...,k By a symmetry, wemeanatransformationthatmapssolutions to solutions. Lie: To find the symmetry group of the differential equations, work infinitesimally. The vector field v = p i=1 ξ i (x, u) x i + q α=1 ϕ α (x, u) u α is an infinitesimal symmetry if its flow exp(t v) isaoneparameter symmetry group of the differential equation.

61 We prolong v to the jet space whose coordinates are the derivatives appearing in the differential equation: p v (n) = ξ i q n x i + ϕ J α where ϕ J α = D J i=1 ( ϕ α p i=1 Infinitesimal invariance criterion: α=1 #J=0 u α i ξi ) + u α J p u α J,i ξi i=1 D J v (n) ( ν )=0 whenever =0. Infinitesimal determining equations: L(x, u; ξ (n),ϕ (n) )=0 totalderivatives We can determine the structure of the symmetry group without solving the determining equations!

62 Symmetry generator: Prolongation: where The Korteweg devries equation v = τ(t, x, u) t v (3) = v + ϕ t u t + u xxx + uu x =0 + ξ(t, x, u) x u t + ϕ x + ϕ(t, x, u) u u x + + ϕ xxx u xxx ϕ t = ϕ t + u t ϕ u u t τ t u 2 t τ u u x ξ t u t u x ξ u ϕ x = ϕ x + u x ϕ u u t τ x u t u x τ u u x ξ x u 2 x ξ u ϕ xxx = ϕ xxx +3u x ϕ u +

63 Infinitesimal invariance: v (3) (u t + u xxx + uu x )=ϕ t + ϕ xxx + uϕ x + u x ϕ =0 on solutions Infinitesimal determining equations: τ x = τ u = ξ u = ϕ t = ϕ x =0 ϕ = ξ t 2 3 uτ t ϕ u = 2 3 τ t = 2 ξ x τ tt = τ tx = τ xx = = ϕ uu =0 General solution: τ = c 1 +3c 4 t, ξ = c 2 + c 3 t + c 4 x, ϕ = c 3 2c 4 u.

64 Basis for symmetry algebra g KdV : v 1 = t, v 2 = x, v 3 = t x + u, v 4 =3t t + x x 2u u. The symmetry group G KdV is four-dimensional (x, t, u) (λ 3 t + a, λ x + ct+ b, λ 2 u + c)

65 v 1 = t, v 2 = x, v 3 = t x + u, v 4 =3t t + x x 2u u. Commutator table: v 1 v 2 v 3 v 4 v v 1 v v 1 3 v 2 v 3 0 v v 3 v 4 v 1 3 v 2 2 v 3 0 Entries: [v i, v j ]= k C k ij v k. Ck ij structureconstantsofg

66 Diffeomorphism Maurer Cartan forms: µ t, µ x, µ u,µ t T, µt X,µt U,µx T,...,µu U,µt TT, µt TX,... Infinitesimal determining equations: τ x = τ u = ξ u = ϕ t = ϕ x =0 ϕ = ξ t 2 3 uτ t ϕ u = 2 3 τ t = 2 ξ x τ tt = τ tx = τ xx = = ϕ uu =0 Maurer Cartan determining equations: µ t X = µt U = µx U = µu T = µu X =0, µ u = µ x T 2 3 Uµt T, µu U = 2 3 µt T = 2 µx X, µ t TT = µt TX = µt XX = = µu UU =... =0.

67 Basis (dim G KdV =4): µ 1 = µ t, µ 2 = µ x, µ 3 = µ u, µ 4 = µ t T. Substituting into the full diffeomorphism structure equations yields the structure equations for g KdV : dµ 1 = µ 1 µ 4, dµ 2 = µ 1 µ Uµ1 µ µ2 µ 4, dµ 3 = 2 3 µ3 µ 4, dµ 4 =0. dµ i = C i jk µj µ k

68 Basis (dim G KdV =4): µ 1 = µ t, µ 2 = µ x, µ 3 = µ u, µ 4 = µ t T. Substituting into the full diffeomorphism structure equations yields the structure equations for g KdV : dµ 1 = µ 1 µ 4, dµ 2 = µ 1 µ U µ1 µ µ2 µ 4, dµ 3 = 2 3 µ3 µ 4, dµ 4 =0. dµ i = C i jk (Z) µj µ k

69 Essential Invariants The pseudo-group structure equations live on the bundle τ : G ( ) M, andthestructurecoefficientscjk i constructed above may vary from point to point. In the case of a finite-dimensional Lie group action, G ( ) G M, andthismeansthebasisofmaurer Cartan forms on each fiber of G ( ) is varying with the target point Z M. However,wecanalwaysmakeaZ dependent change of basis to make the structure coefficients constant. However, for infinite-dimensional pseudo-groups, it may not be possible to find such a change of Maurer Cartan basis, leading to the concept of essential invariants.

70 Kadomtsev Petviashvili (KP) Equation Symmetry generators: ( u t uu x u xxx ) x ± 3 4 u yy =0 v f = f(t) t yf (t) y + ( 1 3 xf (t) 2 9 y2 f (t) ) x + ( 2 3 uf (t)+ 2 9 xf (t) 4 27 y2 f (t) ) u, w g = g(t) y 2 3 yg (t) x 4 9 yg (t) u, z h = h(t) x h (t) u. = Kac Moody loop algebra A (1) 4

71 Navier Stokes Equations u t + u u = p + ν u, u =0. Symmetry generators: v α = α(t) x + α (t) u α (t) x p v 0 = t s = x x +2t t u u 2 p p r = x x + u u w h = h(t) p

72 Action of Pseudo-groups on Submanifolds a.k.a. Solutions of Differential Equations G Lie pseudo-group acting on p-dimensional submanifolds: N = {u = f(x)} M For example, G may be the symmetry group of a system of differential equations (x, u (n) )=0 and the submanifolds are the graphs of solutions u = f(x). Goal: Understand G invariant objects (moduli spaces)

73 Prolongation J n =J n (M,p) n th order submanifold jet bundle Local coordinates : z (n) =(x, u (n) )=(... x i... u α J... ) Prolonged action of G (n) on submanifolds: (x, u (n) ) (X, Û (n) ) Coordinate formulae: Û α J = F J α (x, u(n),g (n) ) = Implicit differentiation.

74 Differential Invariants A differential invariant is an invariant function I :J n R for the prolonged pseudo-group action Invariant differential operators: I(g (n) (x, u (n) )) = I(x, u (n) ) = curvature, torsion,... D 1,...,D p = arc length derivative If I is a differential invariant, so is D j I. I(G) thealgebraofdifferentialinvariants

75 The Basis Theorem Theorem. The differential invariant algebra I(G) islocally generated by a finite number of differential invariants I 1,...,I l and p = dims invariant differential operators D 1,...,D p meaning that every differential invariant can be locally expressed as a function of the generating invariants and their invariant derivatives: D J I κ = D j1 D j2 D jn I κ. = Lie groups: Lie, Ovsiannikov = Lie pseudo-groups: Tresse, Kumpera, Kruglikov Lychagin, Muñoz Muriel Rodríguez, Pohjanpelto O

76 Key Issues Minimal basis of generating invariants: I 1,...,I l Commutation formulae for the invariant differential operators: [ D j, D k ]= p i=1 Syzygies (functional relations) among Y i jk D i = Non-commutative differential algebra the differentiated invariants: Φ(... D J I κ... ) 0 = Codazzi relations

77 Computing Differential Invariants The infinitesimal method: v(i) =0 foreveryinfinitesimalgenerator v g Moving frames. Completely algebraic. = Requires solving differential equations. Can be adapted to arbitrary group and pseudo-group actions. Describes the complete structure of the differential invariant algebra I(G) using only linear algebra & differentiation! Prescribes differential invariant signatures for equivalence and symmetry detection.

78 Moving Frames In the finite-dimensional Lie group case, a moving frame is defined as an equivariant map ρ (n) :J n G

79 However, we do not have an appropriate abstract object to represent our pseudo-group G. Consequently, the moving frame will be an equivariant section ρ (n) :J n H (n) of the pulled-back pseudo-group jet groupoid: G (n) H (n) M J n.

80 Moving Frames for Pseudo Groups Definition. A(right)moving frame of order n is a rightequivariant section ρ (n) : V n H (n) defined on an open subset V n J n. = Groupoid action. Proposition. Amovingframeofordern exists if and only if G (n) acts freely and regularly.

81 Moving Frames for Pseudo Groups Definition. A(right)moving frame of order n is a rightequivariant section ρ (n) : V n H (n) defined on an open subset V n J n. = Groupoid action. Proposition. Amovingframeofordern exists if and only if G (n) acts freely and regularly.

82 Freeness For Lie group actions, freeness means trivial isotropy: G z = { g G g z = z } = {e}. For infinite-dimensional pseudo-groups, this definition cannot work, and one must restrict to the transformation jets of order n, usingthen th order isotropy subgroup: Definition. G (n) = { g (n) G (n) z (n) z g (n) z (n) = z (n) } At a jet z (n) J n,thepseudo-groupg acts freely if G (n) = { 1 (n) z (n) z } locally freely if G (n) z (n) is a discrete subgroup of G (n) z the orbits have dimension r n =dimg (n) z = Kumpera s growth bounds on Spencer cohomology.

83 Persistence of Freeness Theorem. If n 1andG (n) acts (locally) freely at z (n) J n, then it acts (locally) freely at any z (k) J k with π n k(z(k) )=z (n) for all k>n.

84 The Normalization Algorithm To construct a moving frame : I. Compute the prolonged pseudo-group action u α K U α K = F α K (x, u(n),g (n) ) by implicit differentiation. II. Choose a cross-section to the pseudo-group orbits: u α κ J κ = c κ, κ =1,...,r n =fiberdimg (n)

85 III. Solve the normalization equations U α κ J κ = F α κ J κ (x, u (n),g (n) )=c κ for the n th order pseudo-group parameters g (n) = ρ (n) (x, u (n) ) IV. Substitute the moving frame formulas into the unnormalized jet coordinates u α K = F K α(x, u(n),g (n) ). The resulting functions form a complete system of n th order differential invariants IK α (x, u(n) )=FK α (x, u(n),ρ (n) (x, u (n) ))

86 Invariantization Amovingframeinducesaninvariantizationprocess,denoted ι, that projects functions to invariants, differential operators to invariant differential operators; differential forms to invariant differential 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.

87 Invariantization In particular, invariantization of the jet coordinates leads to a complete system of functionally independent differential invariants: ι(x i )=H i ι(u α J )=Iα J Phantom differential invariants: I α κ J κ = c κ The non-constant invariants form a complete system of functionally independent differential invariants Replacement Theorem I(... x i... u α J... )=ι( I(... xi... u α J... )) = I(... H i... I α J... )

88 The Invariant Variational Bicomplex Differential functions = ι(x i )=H i differential invariants ι(u α J )=Iα J Differential forms = invariant differential forms ι( dx i )=ϖ i ι(θ α K )=ϑα K Differential operators = invariant differential operators ι (D x i )=D i

89 Recurrence Formulae Invariantization and differentiation do not commute The recurrence formulae connect the differentiated invariants with their invariantized counterparts: D i I α J = Iα J,i + M α J,i = M α J,i correctionterms

90 Once established, the recurrence formulae completely prescribe the structure of the differential invariant algebra I(G) thankstothefunctionalindependence of the non-phantom normalized differential invariants. The recurrence formulae can be explicitly determined using only the infinitesimal generators and linear differential algebra!

91 Horizontal coframe Lie Tresse Kumpera Example X = f(x), Y = y, U = u f (x) d H X = f x dx, d H Y = dy, Implicit differentiations D X = 1 f x D x, D Y =D y.

92 Prolonged pseudo-group transformations on surfaces S R 3 : X = f Y = y U = u f x U X = u x f 2 x U XX = u xx f 3 x u f xx f 3 x 3u x f xx f 4 x U Y = u y f x u f xxx f 4 x + 3u f 2 xx f 5 x U XY = u xy f 2 x u y f xx f 3 x U YY = u yy f x f, f x, f xx, f xxx,... pseudo-group parameters = action is free at every order.

93 Coordinate cross-section X = f =0, U = u f x =1, U X = u x f 2 x Moving frame u f xx f 3 x f =0, f x = u, f xx = u x, f xxx = u xx. Differential invariants U Y = u y J = ι(u f y )= u y x u =0, U XX = =0. U XY = J 1 = ι(u xy )= uu xy u x u y u 3 U YY = J 2 = ι(u xy )= u yy u U XXY J 3 = ι(u xxy ) U XY Y J 4 = ι(u xyy ) U YYY J 5 = ι(u yyy )

94 Invariant horizontal forms d H X = f x dx udx, Invariant differentiations D 1 = 1 u D x d H Y = dy dy, D 2 =D y Higher order differential invariants: D m 1 Dn 2 J J,1 = D 1 J = uu xy u x u y u 3 = J 1, J,2 = D 2 J = uu yy u2 y u 2 = J 2 J 2. Recurrence formulae: D 1 J = J 1, D 2 J = J 2 J 2, D 1 J 1 = J 3, D 2 J 1 = J 4 3 JJ 1, D 1 J 2 = J 4, D 2 J 2 = J 5 JJ 2,

95 Korteweg devries Equation Prolonged Symmetry Group Action: T = e 3λ 4 (t + λ 1 ) X = e λ 4 (λ 3 t + x + λ 1 λ 3 + λ 2 ) U = e 2λ 4 (u + λ 3 ) U T = e 5λ 4 (u t λ 3 u x ) U X = e 3λ 4 u x U TT = e 8λ 4 (u tt 2λ 3 u tx + λ 3 2 u xx ) U TX = D X D T U = e 6λ 4 (u tx λ 3 u xx ) U XX = e 4λ 4 u xx.

96 Cross Section: T = e 3λ 4 (t + λ 1 )=0 X = e λ 4 (λ 3 t + x + λ 1 λ 3 + λ 2 )=0 U = e 2λ 4 (u + λ 3 )=0 U T = e 5λ 4 (u t λ 3 u x )=1 Moving Frame: λ 1 = t, λ 2 = x, λ 3 = u, λ 4 = 1 5 log(u t + uu x )

97 Normalized differential invariants: u I 01 = ι(u x )= x (u t + uu x ) 3/5 I 20 = ι(u tt )= u tt +2uu tx + u2 u xx (u t + uu x ) 8/5 I 11 = ι(u tx )= I 02 = ι(u xx )= Invariant differential operators:. u tx + uu xx (u t + uu x ) 6/5 u xx (u t + uu x ) 4/5 D 1 = ι(d t )=(u t + uu x ) 3/5 D t + u(u t + uu x ) 3/5 D x, D 2 = ι(d x )=(u t + uu x ) 1/5 D x.

98 Commutation formula: [ D 1, D 2 ]=I 01 D 1 Recurrence formulae: D 1 I 01 = I I I 01 I 20, D 2 I 01 = I I I 01 I 11, D 1 I 20 = I 30 +2I I 01 I I2 20, D 2 I 20 = I 21 +2I 01 I I2 01 I I 11 I 20, D 1 I 11 = I 21 + I I 01 I I 11 I 20, D 2 I 11 = I 12 + I 01 I I2 01 I I2 11, D 1 I 02 = I I 01 I I 02 I 20, D 2 I 02 = I I2 01 I I 02 I 11,..

99 Generating differential invariants: I 01 = ι(u x )= Fundamental syzygy: u x (u t + uu x ) 3/5, I 20 = ι(u tt )=u tt +2uu tx + u2 u xx (u t + uu x ) 8/5. D 2 1 I I 01 D 1 I 20 D 2 I 20 + ( 1 5 I I 01) D1 I 01 D 2 I I 01 I I2 01 I I3 01 =0.

100 The Master Recurrence Formula p d H IJ α = ( D i IJ α ) ωi = i=1 p i=1 I α J,i ωi + ψ α J where ψ α J = ι( ϕ α J )=Φα J (... Hi... IJ α... ;... γb A... ) are the invariantized prolonged vector field coefficients, which are particular linear combinations of γa b = ι(ζb A ) invariantized Maurer Cartan forms prescribed by the invariantized prolongation map. The invariantized Maurer Cartan forms are subject to the invariantized determining equations: L(H 1,...,H p,i 1,...,I q,...,γ b A,...)=0

101 d H I α J = p i=1 I α J,i ωi + ψ α J (... γb A... ) Step 1: Solve the phantom recurrence formulas 0= d H I α J = p i=1 I α J,i ωi + ψ α J (... γb A... ) for the invariantized Maurer Cartan forms: γ b A = p i=1 J b A,i ωi ( ) Step 2: Substitute ( ) into the non-phantom recurrence formulae to obtain the explicit correction terms.

102 Only uses linear differential algebra based on the specification of cross-section. Does not require explicit formulas for the moving frame, the differential invariants, the invariant differential operators, or even the Maurer Cartan forms!

103 Lie Tresse Kumpera Example (continued) Phantom recurrence formulae: X = f(x), Y = y, U = u f (x) 0=dH = ϖ 1 + γ, 0=dI 10 = J 1 ϖ 2 + ϑ 1 γ 2, 0=dI 00 = Jϖ 2 + ϑ γ 1, 0=dI 20 = J 3 ϖ 2 + ϑ 3 γ 3, Solve for pulled-back Maurer Cartan forms: γ = ϖ 1, γ 2 = J 1 ϖ 2 + ϑ 1, γ 1 = Jϖ 2 + ϑ, γ 3 = J 3 ϖ 2 + ϑ 3,

104 Recurrence formulae: dy = ϖ 2 dj = J 1 ϖ 1 +(J 2 J 2 ) ϖ 2 + ϑ 2 Jϑ, dj 1 = J 3 ϖ 1 +(J 4 3 JJ 1 ) ϖ 2 + ϑ 4 Jϑ 1 J 1 ϑ, dj 2 = J 4 ϖ 1 +(J 5 JJ 2 ) ϖ 2 + ϑ 5 J 2 ϑ,

105 The Korteweg devries Equation (continued) Recurrence formula: di jk = I j+1,k ω 1 + I j,k+1 ω 2 + ι(ϕ jk ) Invariantized Maurer Cartan forms: ι(τ) =λ, ι(ξ) =µ, ι(ϕ) =ψ = ν, ι(τ t )=ψ t = λ t,... Invariantized determining equations: λ x = λ u = µ u = ν t = ν x =0 ν = µ t ν u = 2 µ x = 2 3 λ t λ tt = λ tx = λ xx = = ν uu = =0 Invariantizations of prolonged vector field coefficients: ι(τ) =λ, ι(ξ) =µ, ι(ϕ) =ν, ι(ϕ t )= I 01 ν 5 3 λ t, ι(ϕ x )= I 01 λ t, ι(ϕ tt )= 2I 11 ν 8 3 I 20 λ t,...

106 Phantom recurrence formulae: 0= d H H 1 = ω 1 + λ, 0= d H H 2 = ω 2 + µ, 0= d H I 00 = I 10 ω 1 + I 01 ω 2 + ψ = ω 1 + I 01 ω 2 + ν, 0= d H I 10 = I 20 ω 1 + I 11 ω 2 + ψ t = I 20 ω 1 + I 11 ω 2 I 01 ν 5 3 λ t, = Solve for λ = ω 1, µ = ω 2, ν = ω 1 I 01 ω 2, λ t = 3 5 (I 20 + I 01 )ω (I 11 + I2 01 )ω2. Non-phantom recurrence formulae: d H I 01 = I 11 ω 1 + I 02 ω 2 I 01 λ t, d H I 20 = I 30 ω 1 + I 21 ω 2 2I 11 ν 8 3 I 20 λ t, d H I 11 = I 21 ω 1 + I 12 ω 2 I 02 ν 2I 11 λ t, d H I 02 = I 12 ω 1 + I 03 ω I 02 λ t,.

107 D 1 I 01 = I I I 01 I 20, D 2 I 01 = I I I 01 I 11, D 1 I 20 = I 30 +2I I 01 I I2 20, D 2 I 20 = I 21 +2I 01 I I2 01 I I 11 I 20, D 1 I 11 = I 21 + I I 01 I I 11 I 20, D 2 I 11 = I 12 + I 01 I I2 01 I I2 11, D 1 I 02 = I I 01 I I 02 I 20, D 2 I 02 = I I2 01 I I 02 I 11,..

108 Gröbner Basis Approach Suppose G acts freely at order n. The differential invariants of order > n are naturally identified with polynomials belonging to a certain algebraic module J, called the invariantized prolonged symbol module which is defined as the invariantized pull-back of the symbol module for the infinitesimal determining equations under a certain explicit linear map.

109 Constructive Basis Theorem Theorem. A system of generating differential invariants in one-to-one correspondence with the Gröbner basis elements of the invariantized prolonged symbol module J >n plus, possibly, a finite number of differential invariants of order n.

110 Syzygy Theorem Theorem. Every differential syzygy among the generating differential invariants is either a syzygy among those of order n, or arises from an algebraic syzygy among the Gröbner basis polynomials in J >n,orcomesfromacommutatorsyzygy among the invariant differential operators.

111 The Symbol Module Linearized determining equations L(z,ζ (n) )=0 t =(t 1,...,t m ), T =(T 1,...,T m ) { T = P (t, T )= Symbol module: m a =1 P a (t) T a } R[t] R m R[t, T ] I T

112 The Prolonged Symbol Module s =(s 1,...,s p ), S =(S 1,...,S q ), Ŝ = T (s, S) = Define the linear map { s i = β i (t) =t i + q α =1 q α =1 T α (s) S α } R[s] R q R[s, S] u α i t p+α, i =1,...,p, S α = B α (T )=T p+α p i=1 u α i T i, α =1,...,q. Prolonged symbol module: = symbol of the vector field prolongation operation J =(β ) 1 (I)