The Structure of Dierential Invariant Algebras of Lie Groups and Lie Pseudo–Groups

Peter J. Olver University of Minnesota http://www.math.umn.edu/ olver

=⇒ Juha Pohjanpelto Jeongoo Cheh

0 Sur la theorie, si importante sans doute, mais pour nous si obscure, des groupes de Lie innis, nous ne savons rien que ce qui trouve dans les memoires de Cartan, premiere exploration a travers une jungle presque impenetrable; mais celle-ci menace de se refermer sur les sentiers deja traces, si l’on ne procede bientˆot a un indispensable travail de defrichement.

— Andre Weil, 1947

1 Pseudo-groups in Action

Lie — Medolaghi — Vessiot Cartan . . . Guillemin, Sternberg Kuranishi, Spencer, Goldschmidt, Kumpera, . . . Relativity Noether’s Second Theorem Gauge theory and eld theories Maxwell, Yang–Mills, conformal, string, . . . Fluid Mechanics, Metereology Euler, Navier–Stokes, boundary layer, quasi-geostropic , . . . Linear and linearizable PDEs Solitons (in 2 + 1 dimensions) K–P, Davey-Stewartson, . . . Image processing Numerical methods — geometric integration Kac–Moody symmetry algebras Lie groups!

2 What’s New?

Direct constructive algorithms for:

Invariant Maurer–Cartan forms Structure equations Moving frames Dierential invariants Invariant dierential operators Constructive Basis Theorem Syzygies and recurrence formulae Grobner basis constructions Further applications

=⇒ Symmetry groups of dierential equations =⇒ Vessiot group splitting =⇒ Gauge theories =⇒ Calculus of variations

3 Dierential Invariants

G — transformation group acting on p-dimensional submani- folds N = {u = f(x)} M

G(n) — prolonged action on the submanifold jet space Jn = Jn(M, p)

Dierential invariant I : Jn → R I(g(n) (x, u(n))) = I(x, u(n)) =⇒ curvature, torsion, . . . Invariant dierential operators:

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

? ? I(G) — the algebra of dierential invariants ? ?

4 The Basis Theorem

Theorem. The dierential invariant algebra I(G) is gener-

ated by a nite number of dierential invariants I1, . . . , I`, meaning that every dierential invariant can be locally expressed as a function of the generating invariants and their invariant derivatives:

DJ I = Dj1Dj2 DjnI.

=⇒ Tresse, Kumpera

♦ ♦ functional independence ♦ ♦

? ? Constructive Version ? ?

=⇒ Computational algebra & Grobner bases

5 The Algebra of Dierential Invariants

Key Issues:

Minimal basis of generating invariants: I1, . . . , I` Commutation formulae for the invariant dierential operators: p i [ Dj, Dk ] = Aj,k Di iX=1 =⇒ Non-commutative dierential algebra Syzygies (functional relations) among the dierentiated invariants:

( . . . DJ I . . . ) 0 =⇒ Gauss–Codazzi relations

Applications: Equivalence and signatures of submanifolds and characteri- zation of moduli spaces Computation of invariant variational problems:

L( . . . DJ I . . . ) dω Z Group splitting of PDEs

6 Basic Themes

The structure of a (connected) pseudo-group is xed by its Lie algebra of innitesimal generators: g The innitesimal generators satisfy an overdetermined system of linear partial dierential equations — the determining equations: F = 0 The basic structure of an overdetermined system of PDEs is xed by the algebraic structure of its symbol module: I

The structure of the dierential invariant algebra I(G) is xed by the prolonged innitesimal generators: g(∞) These satisfy an overdetermined system of partial dier- ential equations — the prolonged determining equa- tions: H = 0 The basic structure of the prolonged determining equa- tions is xed by the algebraic structure of its prolonged symbol module: J

7 Jets and Prolongation Jet = Taylor polynomial/series (Ehresmann) th jnv — n order jet of a vector eld Jn = Jn(M, p) — jets of p-dimensional submanifolds N M v(n) — prolonged vector eld on Jn

The prolongation map takes (jets of) vector elds on M to vector elds on the jet space Jn.

(∞) p : j∞v 7→ v There is an induced dual prolongation map p : J → I on the symbol modules, which is algebraic (at suciently high order).

With this “algebraic prolongation” in hand, the structure of the prolonged symbol module, and hence the dierential invariants, is algorithmically determined by the structure of the pseudo-group’s symbol module.

8 Pseudo-groups

Denition. A pseudo-group is a collection of local dieomorphisms ϕ : M → M such that

Identity: 1M ∈ G, Inverses: ϕ1 ∈ G, Restriction: U dom ϕ =⇒ ϕ | U ∈ G,

Composition: im ϕ dom =⇒ ϕ ∈ G.

Denition. A Lie pseudo-group G is a pseudo-group whose transformations are the solutions to an involutive system of partial dierential equations:

F (z, ϕ(n)) = 0.

Nonlinear determining equations =⇒ analytic (Cartan–Kahler)

? ? Key complication: 6 ∃ Abstract object G ? ?

9 A Non-Lie Pseudo-group

Acting on M = R2:

X = ϕ(x), Y = ϕ(y)

where ϕ ∈ D(R). Cannot be characterized by a system of partial dierential equations (x, y, X(n), Y (n)) = 0

Theorem. (Johnson, Itskov) Any non-Lie pseudo-group can be completed to a Lie pseudo-group with the same dierential invariants.

Completion of previous example:

X = ϕ(x), Y = (y)

where ϕ, ∈ D(R).

10 Innitesimal Generators

g — Lie algebra of innitesimal generators of the pseudo-group G z = (x, u) — local coordinates on M

Vector eld: m ∂ p ∂ q ∂ v = a(z) = i + ϕ ∂za ∂xi ∂u aX=1 iX=1 X=1 Vector eld jet: (n) b jnv 7→ = ( . . . A . . . ) #A b k b b ∂ ∂ A = = ∂zA ∂za1 ∂zak Innitesimal (Linearized) Determining Equations

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

Remark: If G is the symmetry group of a system of dierential equations (x, u(n)) = 0, then () is the (involutive completion of) the usual Lie determining equations for the symmetry group.

11 Symmetry Groups — Review

System of dierential equations:

(n) (x, u ) = 0, = 1, 2, . . . , k Prolonged vector eld: p ∂ q n ∂ v(n) = i + ϕ ∂xi J ∂u iX=1 X=1 #XJ=0 J where b p p i i ϕ J = DJ ϕ ui + uJ,i à ! iX=1 iX=1 b (n) (n) (n) J (x, u ; , ϕ ) Innitesimal invariance: (n) v () = 0 whenever = 0. Innitesimal determining equations:

L(x, u; (n), ϕ(n)) = 0

i i L( . . . , x , . . . , u , . . . , A, . . . , ϕA, . . . ) = 0 =⇒ involutive completion

12 The Korteweg–deVries equation

ut + uxxx + uux = 0 Symmetry generator: ∂ ∂ ∂ v = (t, x, u) + (t, x, u) + ϕ(t, x, u) ∂t ∂x ∂u Prolongation: ∂ ∂ ∂ v(3) = v + ϕt + ϕx + + ϕxxx ∂ut ∂ux ∂uxxx where t 2 ϕ = ϕt + utϕu utt ut u uxt utuxu x 2 ϕ = ϕx + uxϕu utx utuxu uxx uxu xxx ϕ = ϕxxx + 3uxϕu +

Innitesimal invariance: (3) t xxx x v (ut + uxxx + uux) = ϕ + ϕ + u ϕ + ux ϕ = 0 on solutions

13 ut + uxxx + uux = 0

Innitesimal determining equations:

x = u = u = ϕt = ϕx = 0 2 2 ϕ = t 3 ut ϕu = 3 t = 2 x

tt = tx = xx = = ϕuu = 0

General solution:

= c1 + 3c4t, = c2 + c3t + c4x, ϕ = c3 2c4u.

Basis for symmetry algebra:

∂t, ∂x, t ∂x + ∂u, 3t ∂t + x ∂x 2u ∂u.

The symmetry group GKdV is four-dimensional (x, t, u) 7→ (3t + a, x + ct + b, 2u + c)

14 Moving Frames

For a nite-dimensional Lie group action, a moving frame is dened to be a (right) equivariant map

(n) : Jn → G where (n)(g(n) z(n)) = (n)(z(n)) g1

Moving frames are explicitly constructed by choosing a cross-section to the group orbits, and solving the normaliza- tion equations for the group parameters.

The existence of a moving frame requires that the group action be free, i.e., that there is no isotropy, or (locally) that the group orbits have the same dimension as the group itself.

15 Moving Frames for Pseudo–Groups

The role of the group parameters is played by the pseudo- group jet coordinates, while the group is replaced by a certain principal bundle H(n) ' Jn G(n) → Jn. There are an innite number of pseudo-group parameters to normalize, which are to be done order by order. (Or else use the Taylor series approach.) However, at each nite order, the algebraic manipulations in the moving frame normalization procedure are identical to the nite- dimensional calculus. Since the pseudo-group G is innite-dimensional, classical freeness is impossible! Rather, a pseudo-group action is said to act freely at a jet z(n) ∈ Jn if the only pseudo- group elements which x z(n) are those whose n-jet coincides with the identity n-jet.

Theorem. If G acts freely on Jn for n 1, then it acts freely on all Jk for k n.

n? — order of freeness.

16 Normalization To construct a moving frame :

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

(n) uJ = c, = 1, . . . , rn = ber dim G

II. Solve the normalization equations

(n) (n) FJ (x, u , g ) = c for the pseudo-group parameters g(n) = (n)(x, u(n))

III. Invariantization maps dierential functions to dierential invariants: : F (x, u(n)) 7→ I(x, u(n)) = F ( (n)(x, u(n)) (x, u(n)) )

=⇒ an algebra morphism and a projection: = I(x, u(n)) = ( I(x, u(n)) )

17 Invariantization

A moving frame induces an invariantization process, denoted , that projects functions to invariants, dierential operators to invariant dierential operators; dierential forms to invari- ant dierential 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.

18 Invariantization

In particular, invariantization of the jet coordinates leads to a complete system of functionally independent dierential invariants: i i (x ) = H (uJ ) = IJ

Phantom dierential invariants: IJ = c The non-constant invariants form a functionally inde- pendent generating set for the dierential invariant algebra I(G) Replacement Theorem i i I( . . . x . . . uJ . . . ) = ( I( . . . x . . . uJ . . . ) ) i = I( . . . H . . . IJ . . . )

♦ Dierential forms =⇒ invariant dierential forms ( dxi) = ωi i = 1, . . . , p

♦ Dierential operators =⇒ invariant dierential operators

( Dxi ) = Di i = 1, . . . , p

19 Recurrence Formulae

? ? Invariantization and dierentiation ? ? do not commute

The recurrence formulae connect the dierentiated invariants with their invariantized counterparts:

DiIJ = IJ,i + MJ,i

=⇒ MJ,i — correction terms

♥ Once established, they completely prescribe the structure of the dierential invariant algebra I(G) — thanks to the functional independence of the non-phantom normalized dierential invariants.

? ? The recurrence formulae can be explicitly determined using only the innitesimal generators and linear dierential algebra!

20 The Key Formula

p p i i dIJ = ( DiIJ ) ω = IJ,i ω + J iX=1 iX=1 b

where i b J = ( ϕ J ) = J ( . . . H . . . IJ . . . ; . . . A . . . )

are theb invarianb tized prolonged vector eld coecients, which are particular linear combinations of b b A = (A) — invariantized Maurer–Cartan forms prescribed by the invariantized prolongation map.

Proposition. The invariantized Maurer–Cartan forms are subject to the invariantized determining equations: 1 p 1 q b L(H , . . . , H , I , . . . , I , . . . , A, . . . ) = 0

21 p i b dH IJ = IJ,i ω + J ( . . . A . . . ) iX=1 b

Step 1: Solve the phantom recurrence formulas p i b 0 = dH IJ = IJ,i ω + J ( . . . A . . . ) iX=1 b for the invariantized Maurer–Cartan forms: p b b i A = JA,i ω () iX=1 Step 2: Substitute () into the non-phantom recurrence formulae to obtain the explicit correction terms. ♦ Only uses linear dierential algebra based on the specica- tion of cross-section. ♥ Does not require explicit formulas for the moving frame, the dierential invariants, the invariant dierential opera- tors, or even the Maurer–Cartan forms!

22 Korteweg–deVries equation

Symmetry Group Action:

34 T = e (t + 1) = 0

4 X = e (3t + x + 13 + 2) = 0

24 U = e (u + 3) = 0

Prolonged Action:

54 UT = e (ut 3ux),

34 UX = e ux,

84 2 UT T = e (utt 23utx + 3uxx),

64 UT X = DXDT U = e (utx 3uxx),

44 UXX = e uxx, . .

Cross Section: 34 T = e (t + 1) = 0

4 X = e (3t + x + 13 + 2) = 0

24 U = e (u + 3) = 0

54 UT = e (ut 3ux) = 1 Moving Frame:

1 1 = t, 2 = x, 3 = u, 4 = 5 log(ut + uux)

23 Phantom Invariants: 1 H = (t) = 0, I00 = (u) = 0, 2 H = (x) = 0, I10 = (ut) = 1.

Normalized dierential invariants:

ux I01 = (ux) = 3/5 (ut + uux) 2 utt + 2uutx + u uxx I20 = (utt) = 8/5 (ut + uux)

utx + uuxx I11 = (utx) = 6/5 (ut + uux)

uxx I02 = (uxx) = 4/5 (ut + uux) u I = (u ) = xxx 03 xxx u + uu . t x . Replacement Theorem:

ut + uux + uxxx 0 = (ut + uux + uxxx) = 1 + I03 = . ut + uux

Invariant horizontal one-forms: 1 3/5 ω = (dt) = (ut + uux) dt,

2 1/5 1/5 ω = (dx) = u(ut + uux) dt + (ut + uux) dx.

Invariant dierential operators:

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

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

24 Recurrence formula:

1 2 jk dIjk = Ij+1,kω + Ij,k+1ω + (ϕ )

Invariantized Maurer–Cartan forms:

t () = , () = , (ϕ) = = , (t) = = t, . . .

Invariantized determining equations:

x = u = u = t = x = 0 2 = t u = 2 x = 3 t

tt = tx = xx = = uu = = 0

Invariantizations of prolonged vector eld coecients:

t 5 () = , () = , (ϕ) = , (ϕ ) = I01 3 t, x tt 8 (ϕ ) = I01t, (ϕ ) = 2I11 3I20t, . . .

25 Phantom recurrence formulae: 1 1 0 = dH H = ω + , 2 2 0 = dH H = ω + , 1 2 1 2 0 = dH I00 = I10ω + I01ω + = ω + I01ω + , 1 2 t 1 2 5 0 = dH I10 = I20ω + I11ω + = I20ω + I11ω I01 3 t,

1 2 1 2 =⇒ Solve for = ω , = ω , = ω I01ω , 3 1 3 2 2 t = 5 (I20 + I01)ω + 5 (I11 + I01)ω . Non-phantom recurrence formulae:

1 2 dH I01 = I11ω + I02ω I01t, 1 2 8 dH I20 = I30ω + I21ω 2I11 3I20t, 1 2 dH I11 = I21ω + I12ω I02 2I11t, 1 2 4 dH I02 = I12ω + I03ω 3I02t, . .

3 2 3 3 3 3 D1I01 = I11 5 I01 5 I01I20, D2I01 = I02 5 I01 5 I01I11,

8 8 2 8 2 8 D1I20 = I30 + 2I11 5 I01I20 5 I20, D2I20 = I21 + 2I01I11 5 I01I20 5 I11I20,

6 6 6 2 6 2 D1I11 = I21 + I02 5 I01I11 5 I11I20, D2I11 = I12 + I01I02 5 I01I11 5 I11,

4 4 4 2 4 D1I02 = I12 5 I01I02 5 I02I20, D2I02 = I03 5 I01I02 5 I02I11, . . . .

26 Generating dierential invariants:

2 ux utt + 2uutx + u uxx I01 = (ux) = 3/5 , I20 = (utt) = 8/5 . (ut + uux) (ut + uux) Invariant dierential operators:

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

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

Commutation formula:

[ D1, D2 ] = I01 D1 Fundamental syzygy:

2 3 1 19 D1I01 + 5 I01D1I20 D2I20 + 5 I20 + 5 I01 D1I01 ³ ´ 6 2 7 2 24 3 D2I01 25 I01I20 25 I01I20 + 25 I01 = 0.

27 The Symbol Module

Vector eld: m ∂ v = b(z) ∂zb aX=1 Vector eld jet:

(∞) b j∞v ⇐⇒ = ( . . . A . . . ) #A b k b b ∂ ∂ A = = ∂zA ∂za1 ∂zak

Determining Equations for v ∈ g

b L(z; . . . A . . . ) = 0 ()

28 Duality

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

Polynomial module: m R Rm R T = P (t, T ) = Pa(t) Ta ' [t] [t, T ] ( ) aX=1 ∞ T ' (J T M|z) Dual pairing: b b j∞v ; tAT = A. D E Each polynomial m A b (z; t, T ) = hb (z) tAT ∈ T bX=1 #XAn induces a linear partial dierential equation

(n) L(z, ) = j∞v ; (z; t, T ) D E m A b = hb (z) A = 0 bX=1 #XAn

29 The Linear Determining Equations

Annihilator: L = (J∞g)⊥

Determining Equations

j∞v ; = 0 for all ∈ L ⇐⇒ v ∈ g D E

Symbol = highest degree terms: m (n) A b [L(z, )] = H[(z; t, T )] = hb (z) tAT . bX=1 #XA=n

Symbol submodule:

I = H(L)

=⇒ Formal integrability (involutivity)

30 Prolonged Duality

Prolonged vector eld: p ∂ ∂ v(∞) = i(x, u) + ϕ (x, u(k)) ∂xi J ∂u iX=1 ,JX J b

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

e e e “Prolonged” polynomial module:

p q Rp R Rq S = (s, S, s) = cisi + (s) S ' ( [s] ) ( ) iX=1 X=1 b e e b ∞ S ' T J |z(∞) b Dual pairing:

(∞) i v ; si = D E p (∞) e i v ; S = Q = ϕ ui D E iX=1 (∞) (n) (n) v ; sJ S = ϕ J = J (u ; ) D E b

31 Algebraic Prolongation

Prolongation of vector elds: p : J∞g 7→ g(∞) (∞) j∞v 7→ v

Dual prolongation map: p : S → T

(∞) j∞v ; p () = p(j∞v) ; = v ; D E D E D E

? ? On the symbol level, p is algebraic ? ?

32 Prolongation Symbols

Dene the linear map : R2m → Rm q si = i(t) = ti + ui tp+, i = 1, . . . , p, X=1 p S = B(T ) = Tp+ ui Ti, = 1, . . . , q. iX=1

Pull-back map [ (s1, . . . , sp, S1, . . . , Sq) ]

= ( 1(t), . . . , p(t), B1(T ) . . . , Bq(T ) )

Lemma. The symbols of the prolonged vector eld coe- cients are (i) = T i (ϕ ) = T +p (Q ) = (S ) = Bb(T ) (ϕ J ) = (sJ S ) = (sj1 sjnS )

b = j1(t) jn(t) B(T )

33 Prolonged annihilator:

Z = (p)1L = (g(∞))⊥

h v(∞) ; i = 0 for all v ∈ g ⇐⇒ ∈ Z

Prolonged symbol subbundle:

U = H(Z) J∞(M, p) S

Prolonged symbol module:

J = ()1(I)

Warning: U J

But U n = J n when n > n? n? — order of freeness.

34 Algebraic Recurrence

Polynomial:

(k) J (k) (I ; s, S) = ha (I ) sJ S ∈ S ,JX e b

Dierential invariant:

J (k) I˜ = ha (I ) IJ ,JX

Recurrence:

Di I˜ = IDi˜ Isi˜ + Ri,˜

order I˜ = n

n ? ∈ J , n > n =⇒ order IDi˜ = n + 1

f e order Ri,˜ n

35 Algebra =⇒ Invariants

I — symbol module

determining equations for g

b M ' T / I — complementary monomials tAT pseudo-group parameters

Maurer–Cartan forms

N — leading monomials sJ S

normalized dierential invariants IJ

K = S / N — complementary monomials sKS

cross-section coordinates uK = cK

phantom dierential invariants IK

J = ()1(I)

Freeness: : K → M

g

36 Generating Dierential Invariants

Theorem. The dierential invariant algebra is generated by dierential invariants that are in one-to-one correspon- dence with the Grobner basis elements of the prolonged symbol module plus, possibly, a nite number of dieren- tial invariants of order n?.

37 Syzygies

Theorem. Every dierential syzygy among the generating dierential invariants is either a syzygy among those of order n?, or arises from an algebraic syzygy among the Grobner basis polynomials in J .

f

38