Global Lie-Tresse theorem - School of Mathematics · Structure of the Algebra of Diﬀerential...

Structure of the Algebra of Differential InvariantsDiscussion, development and applications

Global Lie-Tresse theorem

Boris Kruglikov

joint work with Valentin Lychagin

University of Tromsø, Norway

Peter Olver LX conference (Minnesota 2012) Boris Kruglikov ⋄ Global Lie-Tresse theorem


Introduction and HistoryActions of pseudogroups: Basic conceptsMain results

Group actions

The concepts of group and invariant take each day a more

preponderant place in mathematics and tend to dominate

this entire science of our epoch. Sophus Lie

The classical problem in invariant theory is to describe the action ofa group G on a space M . This can be seen as a problem ofdescription of the quotient space M/G.

In the case, when G is a Lie group and M a manifold, the resultingorbifold M/G can have complicated singularities. Invariantfunctions describe regular obits and in some cases separate them.

Finite generation property for the algebra of all invariants was thetopic of D.Hilbert’s XIV problem. For them the Notherian propertywas proved in the case of reductive groups actions.




Geometry vs. Algebra

According to Erlangen program of F.Klein a geometry ischaracterized by the invariants of a transitive transformation group.An important ingredient is the notion of a continuous group in bothfinite and infinite-dimensional cases. This was introduced by S.Lie,who also considered the actions of these groups ininfinite-dimensional spaces and studied their differential invariants.

While homogeneous geometry was central in Klein’s program,E.Cartan re-shaped it by developing the theory of structures, whichare homogeneous up to certain order. This was the base for hisapproach to the solution of equivalence problem.

However for infinite groups of Lie and Cartan the Notherianproperty generally does not hold for differential invariants. Insteadfiniteness is guaranteed by the Lie-Tresse theorem, which usesinvariant functions and invariant derivations as generators.




Lie-Tresse theorem

This theorem is a phenomenological statement motivated by Lieand Tresse. It was rigorously proved for un-constrained actions ofpseudogroups (on regular strata in the space of jets) byA.Kumpera, and further elaborated by L.Ovsyannikov, P.Olver andothers. It was extended for pseudogroup actions on differentialequations (which can be, for instance, singular strata ofun-constrained actions) by BK and V.Lychagin.

In these approaches the generating property of the algebra A ofdifferential invariants holds micro-locally – on open (not alwaysZariski open or G-invariant) domains in jet-space J∞.

In our approach we overcome this difficulty by consideringalgebraic actions (essentially all known examples are such) andrestricting to differential invariants that are rational functions byjets of certain order ≤ l and polynomial by higher jets.




Rational differential invariants are natural in the classificationproblems and all known examples are such, even though one canencounter many classical expressions with roots.

Example

The proper motion group E(2)+ = SO(2) ⋉R2 acts on the curvesin Euclidean R2(x, y). The classical curvature

K =y′′(x)

√

(1 + y′(x)2)3

is not an invariant of E(2)+ (reflection (x, y) 7→ (−x,−y) preservesthe circle x2 + y2 = R2 but transforms K 7→ −K; K = ±R−1),but its square K2 is a bona fide rational differential invariant.Notice however that the Lie group E(2)+ is connected, and that Kis invariant under the action of its Lie algebra.




A pseudogroup G ⊂ Diff loc(M) acting on a manifold M consists ofa collection of local diffeomorphisms ϕ, each bearing own domainof definition dom(ϕ) and range im(ϕ), with properties:

idM ∈ G and dom(idM ) = im(idM ) =M ,

If ϕ,ψ ∈ G, then ϕ ◦ ψ ∈ G whenever dom(ϕ) ⊂ im(ψ),

If ϕ ∈ G, then ϕ−1 ∈ G and dom(ϕ−1) = im(ϕ),

ϕ ∈ G iff for every open subset U ∈ dom(ϕ) the restrictionϕ|U ∈ G,

The pseudogroup is of order k if this is the minimal numbersuch that ϕ ∈ G ⇔ ∀a ∈ dom(ϕ) : [ϕ]ka ∈ Gk.

Here Gk ⊂ Jkreg(M,M) is the space of k-jets of the elements from

the pseudogroup in the space of k-jets.We will assume that action of G on M is transitive, i.e. any twopoints on M can be superposed by a pseudogroup element.




Algebraic pseudogroup

Let Jk(M,n) be the space of k-jets of submanifolds N ⊂M ofdimension n. If (x, y) are local coordinates in M such thatN : y = y(x), then the local coordinates in Jk(M,n) are (x, yσ)with |σ| ≤ k. The space of infinte jets is J∞ = ∪kJ

k(M,n).

We will consider the lifted action of G on the space of jets

Gk ∋ ϕk : Jk(M,n) → Jk(M,n).

For a point a ∈M the stabilizer Gka = {ϕ ∈ Gk : ϕ(a) = a} acts

on the space Jka (M,n) of k-jets at a.

Definition

The G action on M is called algebraic if Gka is an algebraic group

acting algebraically on the algebraic manifold Jka .




G-action on an algebraic equation

More generally, consider the pseudogroup action on a differentialequation E considered as a submanifold in jets, so thatEk ⊂ Jk(M,n) is G-invariant. We will assume that Ek

a as amanifold is algebraic and irreducible.Notice that this property concerns the behavior only with respect tothe derivatives, so for instance sin-Gordon uxy = sinu or Liouvilleuxy = eu are algebraic differential equations from this perspective.

Alternatively instead of the pseudogroup G we can consider its Liealgebra sheaf g. The flow of vector fields from gk preserves Ek.

Now we increase the order k for both pseudogroup and theequation, using the prolongations, and calculate the invariants ofthe prolonged action.




Differential Invariants and Invariant Differentiations

A function I ∈ C∞loc(E

k) constant on the orbits of the above actionis called a differential invariant: ϕ∗

k(I) = I ∀ϕk ∈ Gk. For the Liesheaf g the defining equation is LX(I) = 0. Uniting the invariantsby k we get the subspace of G-invariant functions in C∞

loc(E∞).

We will not consider the general smooth/analytic functions, butrestrict to the algebra of invariant functions Al = ∪kA

lk that are

smooth by the base variables, rational by the fibers of πl : Jl →M

and polynomial by higher jets of order k > l.

Vector field v ∈ C∞(E∞)⊗C∞(M) D(M) invariant under theaction of pseudogroup G is called an invariant differentiation. Wewould like to have it rational as well. Then it acts as follows:

v : Alk−1 → Al

k.

Examples are Levi-Civita derivative in Riemannian geometry,Study derivative in projective geometry etc ... Tresse derivatives.Peter Olver LX conference (Minnesota 2012) Boris Kruglikov ⋄ Global Lie-Tresse theorem



Cartan-Kuranishi type theorem for pseudogroup actions

A closed subset S ⊂ Ek is called Zariski closed if its intersection Sawith every fiber Ek

a , a ∈M , is Zariski closed. Our first main resultsays that regularity is guaranteed by a finite number of conditions.

Theorem

Consider an algebraic action of a pseudogroup G on a formally

integrable irreducible differential equation E over M such that Gacts transitively on M . There exists a number l and a Zariski

closed invariant proper subset Sl ⊂ E l such that the action is

regular in π−1∞,l(E

l \ Sl) ⊂ E∞, i.e. for any k ≥ l the orbits of Gk

on Ek \ π−1k,l (Sl) are closed, have the same dimension and

algebraically fiber the space. In other words, there exists a rational

geometric quotient

(Ek \ π−1

k,l (Sl))/Gk ≃ Yk.




Global Lie-Tresse theorem

Our second main result gives finiteness for differential invariants.We keep the assumptions of the previous theorem.

Theorem

There exists a number l and a Zariski closed invariant proper subset

Sl ⊂ E l such that the algebra Al of differential invariants on E∞

separates the regular orbits from E∞ \ π−1∞,l(Sl) and is finitely

generated in the following sense.

There exists a finite number of functions I1, . . . , It ∈ Al and a finite

number of rational invariant derivations ∇1, . . . ,∇s : Al → Al such

that any function from Al is a polynomial of differential invariants

∇JIi, where ∇J = ∇j1 · · · ∇jr for some multi-indices

J = (j1, . . . , jr), with coefficients being rational functions of Ii.

An important issue (not appearing micro-locally) is that some of thederivations ∇j may not be represented by horizontal vector fields.




Provided Ek is a Stein space the obtained finite generation propertyholds for the bigger algebra M(E∞)G of meromorphic G-invariantfunctions. For smooth functions the global Lie-Tresse theorem fails,while the micro-local one follows via the implicit functions theorem.The polynomial version of the theorem also fails.

We can also calculate and bound the asymptotic for the number rkof independent differential invariants of pure order k in aneighborhood of a regular point. By virtue of our results rk is apolynomial (the Hilbert function) for large k ≫ 1. This impliesrationality of the Poincaré function, answering the question ofV.Arnold (1994):

Theorem

For transitive actions the Poincaré series∑

∞

k=0 rk · zk equals

P (z) =R(z)

(1− z)dfor some polynomial R(z) and integer d ∈ Z+.




Further results

Some other results proved in the same manner are: finitenesstheorem for invariant derivations, differential forms and othernatural geometric objects (tensors, differential operators,connections etc).We also obtain finiteness for differential syzygies and highersyzygies, and interpret this as G-equivariant Cartan-Kuranishitheorem. We can informally summarize the main results as follows:

Remark

Hilbert and Rosenlicht theorems allow to treat the quotient of analgebraic variety by an algebraic group action as an algebraicvariety. Our global version of the Lie-Tresse theorem allows to treatthe quotient of a differential equation by an algebraic pseudogroupaction as a differential equation.



Some classical problemsExamples and Counter-ExamplesAn application in Mathematical Physics

GL2(C)-classification of binary forms (P.Bibikov-V.Lychagin)

A binary form is a homogeneous polynomial from Sn(C2)∗.SL2(C) acts on it by coordinate change, the center – by re-scaling.The algebras of classical invariants are known in degrees n ≤ 8.

n = 3 solved by Bool in 1841,

n = 4 by Bool, Cayley and Eisenstein in 1841-1850 (cross ratioof the four points on P1 or j-invariant of elliptic curve),

n = 5 by Cayley (the invariant algebra is generated by 4homogeneous polynomials of deg = 4, 8, 12, 18, satisfying ahomogeneous relation of degree 36; the invariant of degree 18consists of more than 800 monomials),

n = 6, 7, 8 solved by Gordan, Shioda, Dixmier and Lazard(explicit invariants for n = 7 found by Bedratyuk in 2007).

The new idea is to identify Sn(C2)∗ with the solution space of theEuler equation E : xux + yuy = nu and to study differentialinvariants of GL2(C) on E∞.Peter Olver LX conference (Minnesota 2012) Boris Kruglikov ⋄ Global Lie-Tresse theorem



The prolongations of E are given by the equations (uij = DixD

jyu)

xui,m−i + yui−1,m−i+1 = (n−m+ 1)ui−1,m−i.

So E i+1 → E(i) is an affine 1D bundle.

Theorem

The algebra of diff.invariants equals A = 〈H,∇H,∇2H, . . . 〉,

H =uxxuyy − u2xy

u2, ∇ =

uyuDx −

uxuDy.

Let f ∈ Sn(C2)∗ be a binary form and j4(f) ⊂ J4(C2). Denote

I1(f) = H|j4(f), I2(f) = ∇H|j4(f), I3(f) = ∇2H|j4(f) the

homogeneous functions on C2, and let F (I1, I2, I3) = 0 be the

basic syzygy between them. Then the correspondence f 7→ F is the

fundamental invariant of the binary form that separates the orbits.

In a similar manner ternary and higher forms are classified.Peter Olver LX conference (Minnesota 2012) Boris Kruglikov ⋄ Global Lie-Tresse theorem



Liouville-Darboux problem (BK)

Consider the classification of Riemannian metrics on M2.Here G = Diff loc(M) acts on the sections of the bundle S2

+T∗M .

A basis in differential invariants of order i = 2 + l is obtained viathe form d⊗l

∇K (order of arguments matters!):

Iij = d⊗l∇K(gradK, . . . , gradK

︸︷︷︸

l−j

, sgradK, . . . , sgradK︸︷︷︸

j

).

The first invariants are: I2 = K and I3 = |∇K|2. Starting fromi = 4 there are l+ 1 = i− 1 invariants Iij and we re-enumerate theindex j by letters (I4a instead of I40, I5d instead of I53 etc). Forinstance I4b = d⊗2

∇K(gradK, sgradK).




There are two invariant Lie derivatives ∇1 = LgradK and∇2 = LsgradK . They generate the whole algebra of differentialinvariants A via the following infinite sequence of identities:

∇1 I2 = I3, ∇2 I2 = 0, ∇1 I3 = 2I4a, ∇2 I3 = 2I4b,

∇1 I4a = I5a +2(I24a + I24b)

I3, ∇2 I4a = I5b +

2I4b(I4a + I4c)

I3,

∇1 I4b = I5b +I4b(I4a + I4c)

I3, ∇2 I4b = I5c +

I24c − I4aI4c + 2I24bI3

+ I2I23 ,

∇1 I4c = I5c +2(I4aI4c − I24b)

I3, ∇2 I4c = I5d, . . .

They can be obtained successively from the commutation relations:

[gradK, sgradK] = −2I4bI3

gradK +I4a − I4c

I3sgradK.




We apply this to the classical problem of characterizing surfaceswith polynomial in momenta integrals of the geodesic flow (andcounting the integrals Jk of degree k).

dimJ1 =

3 if K = const (I3 = 0),1 if I4b = 0, I5d = 0,0 else.

dimJ2 =

6 if K = const,4 if I4b = 0, J5 = 0, J4 = 03 if J6a = J6b = J6c = J6d = 0,

2 if J̃6 = 0, J7a = J7b = J7c = J7d = 0,1 else.

Here

J4 = 3(I4a − I4c)(I4a + 4I4c)I4c − 15I2I33 (I4a + 4I4c) + 25I53 ,

J5 = 5I3(I5a − I5c) + (I4a − I4c)(I4c − 6I4a)− 25I2I33 etc.




There are examples of pseudogroup actions, when some of invariantderivatives act trivially, and also when the number of invariantderivations is less than n. Here is an example, when not allderivations can be represented by G-invariant vector fields.

Example. Consider the pseudogroup G on R3 = R2(x, y)× R1(u)with the transitive Lie algebra sheaf

g = 〈f(x, y) ∂x, ∂y, ∂u〉.

This algebra lifted to J∞(R2) acts transitively in the complementto the equation E = {ux = 0}, so there are no differentialinvariants. The invariant horizontal fields are

1

uxDx, Dy −

uyux

Dx.

Restriction to E yields non-trivial algebra of differential invariantsA = 〈uy, uyy , uyyy, . . . 〉, generated by uy and the invariantderivation Dy mod 〈Dx〉. However there are no invarianthorizontal vector fields at all!Peter Olver LX conference (Minnesota 2012) Boris Kruglikov ⋄ Global Lie-Tresse theorem



Non-algebraic situation

Let g = 〈∂x1 , . . . , ∂x4 , (x2∂x1 − x1∂x2)− λ(x4∂x3 − x3∂x4)〉 actupon M = R4(x1, x2, x3, x4), λ ∈ R \Q. Extend the action triviallyto J0(M) = R5(x1, x2, x3, x4, u) and prolong it to J∞(M).The action on J1 exhibits irrational winding on T 2, so micro-localand global invariants differ. The latter have the following basis (incomplex coordinates z = x1 + ix2, w = x3 + ix4):

u, I ′1 = |uz|2, I ′′1 = |uw|

2 and Ipqrs =1

upzuqz̄u

rwu

sw̄

∂p+q+r+su

∂zp∂z̄q∂wr∂w̄s.

The algebra A is finitely generated by the differential invariants u,I ′1, I

′′1 , I1100, I1010, I0011 and the invariant derivatives

∇z =1

uzDz, ∇z̄ =

1

uz̄Dz̄, ∇w =

1

uwDw, ∇w̄ =

1

uw̄Dw̄.

Thus the Lie-Tresse finiteness holds, but A does not separate theG-orbits.Peter Olver LX conference (Minnesota 2012) Boris Kruglikov ⋄ Global Lie-Tresse theorem



Singular systems

Consider the equivalence problem for Hamiltonian systems near anon-degenerate linearly stable equilibrium point 0. Here thepseudogroup G consists of germs of symplectic diffeomorphisms of(R2n, ω) preserving 0. Its subgroup acts on the space E of thegerms of functions H vanishing at 0 to order 2 such that theoperator ω−1d20H has purely imaginary spectrum and noresonances.

Differential invariants in this problem occur only in even orders, andthe Poincaré function is equal to

P (z) =1

(1− z2)n.

Consequently the algebra of invariants A is not finitely generated(even in a generalized sense), though it separates the orbits.




Symmetries of the Plebañski equation (BK & O.Morozov)

Consider the group SDiff(2) of volume-preserving transformationsof M = R2. Its Lie algebra g consists of divergence-free vectorfields, and it is isomorphic to the Poisson algebra P of functions on(R2,̟), ̟ = dt ∧ dz.This algebra naturally lifts its action to TM = R4(t, z, x, y), wherethe symplectic form identified from T ∗M by raising the indices isω = dx ∧ dz + dt ∧ dy. This lift is related to the total derivative∇ : C∞(M) → C∞(TM), and it is the unique 1st order operatorpreserving the Poisson brackets, but it is defined only up to adjointaction of P:

A 7→ ∇A+ {q,A}, ∇ = x∂t + y∂z.

This leads to extension of the Poisson algebra to the graded Liealgebra P0 ⊕ P1 (Pi ≃ P): (A0, A1) 7→ ∇A0 +A1.




Extending the action to J0(TM) = TM × R1(u) is equivalent tocalculating the cohomology group H1(g, C∞(TM)) = R2

(non-trivial element ∇3 representing this cohomology isSDiff(2)-analog of the Gelfand-Fuks cocycle).

Trying to extend this to the algebra H1 = P0 ⊕ P1 violates theJacobi identity because of the cut tails, and we shall allow moregradings. The process naturally stops withH3 = P0 ⊕P1 ⊕ P2 ⊕ P3, representation of this algebra inD(J0(TM)) being (unique modulo some singular cases)

V0(A0) = A0

z∂t −A0t ∂z + (A0

tzx+A0zzy)∂x − (A0

ttx+A0tzy)∂y

−1

6∇3(A0)∂u,

V1(A1) = A1

z∂x −A1t ∂y −

1

2∇2(A1)∂u,

V2(A2) = ∇(A2)∂u,

V3(A3) = A3∂u.




Let G be the Lie pseudogroup corresponding to the graded Liealgebra H3, whose action naturally extends to J∞(TM).

Theorem

The lowest order differential invariant of the above action is

I = uty − uxz + uxxuyy − u2xy.

The whole algebra A of differential invariants is generated by I,

J = E2E4 − E1E3 − uxxE23 + 2uxyE2E3 − uyyE

23 ,

where E1 = Dt(I1), E2 = Dx(I1), E3 = Dy(I1), E4 = Dz(I1), andthe invariant derivations

D1 = E3Dx − E2Dy,

D2 = E3Dt + E4Dx − E1Dy −E2Dz,

D3 = E6Dx − E5Dy,

D4 = E6Dt − E7Dx + E8Dy −E5Dz.

withPeter Olver LX conference (Minnesota 2012) Boris Kruglikov ⋄ Global Lie-Tresse theorem



The invariant I is fundamental in the sense that its vanishingI = 0 leads to degeneration of the whole algebra – all otherinvariants vanish (or go to ∞).

The equation I = 0 is known as the second Plebañski equation, andit is the basic equation of self-dual gravity. It is a rare integrableequation possessing a huge local symmetry algebra. This symmetryalgebra H3 is resposnisble for many exact solutions of Plebañski II.

Remark

The whole symmetry algebra of the 2nd Plebañski equations is by 3dimensions bigger (the symmetry algebra of I – by 2 dimensions).It is obtained from our infinite-dimensional algebraP0 ⊕ P1 ⊕ P2 ⊕ P3 by two simple successive right extensions.


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