Valentin Lychagin - NTNU · Quotients Valentin Lychagin Institute of Mathematics and Statistics,...

Quotients

Valentin Lychagin

Institute of Mathematics and Statistics,University of Tromsø, Norway

Geometry and Lie theory.Dedicated to Eldar Straume on his 70th birthday.

Trondheim, 03.11.16

Valentin Lychagin (University of Tromso) QuotientsGeometry and Lie theory. Dedicated to Eldar Straume on his 70th birthday. Trondheim, 03.11.16 1

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Plan of the talk

Some observations and quasi historical remarks

Lie-Tresse theorem (joint with Boris Kruglikov)

what is a differential invariant?finiteness of singularitiesfiniteness of differential invariant algebras and fields

Applications

Arnold conjecture (joint with Boris Kruglikov)Riemannian and Einstein manifolds (joint with Valeriy Yumaguzhin)differential contra algebraic invariants in the theory of algebraic Liegroup actions (joint with Pavel Bibikov)

binary forms and n-ary formsinvariants of irreducible representations of semisimple Lie groups


/ 26

Plan of the talk



what is a differential invariant?

finiteness of singularitiesfiniteness of differential invariant algebras and fields

Applications




/ 26

Plan of the talk



what is a differential invariant?finiteness of singularities

finiteness of differential invariant algebras and fields

Applications




/ 26

Plan of the talk




Applications




/ 26

Plan of the talk




Applications

Arnold conjecture (joint with Boris Kruglikov)

Riemannian and Einstein manifolds (joint with Valeriy Yumaguzhin)differential contra algebraic invariants in the theory of algebraic Liegroup actions (joint with Pavel Bibikov)



/ 26

Plan of the talk




Applications

Arnold conjecture (joint with Boris Kruglikov)Riemannian and Einstein manifolds (joint with Valeriy Yumaguzhin)

differential contra algebraic invariants in the theory of algebraic Liegroup actions (joint with Pavel Bibikov)



/ 26

Plan of the talk




Applications




/ 26

Plan of the talk




Applications


binary forms and n-ary forms

invariants of irreducible representations of semisimple Lie groups


/ 26

Plan of the talk




Applications




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The General Moduli Problem

"Describe" the orbit space Ω/G of a group G -action on a space Ω.

Main collection:

Ω -is a smooth manifold, G - is a Lie group, G ×Ω→ Ω proper andfree action =⇒ ΩG smooth manifold and Ω→ ΩG is aprincipal G - bundle (J.L. Koszul and R. Palais).Orbits are separated by smooth invariants.

Ω- is an affi ne manifold,G -is a semi-simple Lie group, G ×Ω→ Ω-algebraic action =⇒ ΩG affi ne manifold (D. Hilbert).Regular orbits are separated by polynomial invariants.Ω- is an algebraic manifold,G -is an algebraic Lie group, G ×Ω→ Ω-algebraic action =⇒ regular orbits are separated by rationalinvariants (M. Rosenlicht).There is no Hilbert’s 14th problem!


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Main collection:

Ω -is a smooth manifold, G - is a Lie group, G ×Ω→ Ω proper andfree action =⇒ ΩG smooth manifold and Ω→ ΩG is aprincipal G - bundle (J.L. Koszul and R. Palais).Orbits are separated by smooth invariants.Ω- is an affi ne manifold,G -is a semi-simple Lie group, G ×Ω→ Ω-algebraic action =⇒ ΩG affi ne manifold (D. Hilbert).Regular orbits are separated by polynomial invariants.

Ω- is an algebraic manifold,G -is an algebraic Lie group, G ×Ω→ Ω-algebraic action =⇒ regular orbits are separated by rationalinvariants (M. Rosenlicht).There is no Hilbert’s 14th problem!


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Main collection:

Ω -is a smooth manifold, G - is a Lie group, G ×Ω→ Ω proper andfree action =⇒ ΩG smooth manifold and Ω→ ΩG is aprincipal G - bundle (J.L. Koszul and R. Palais).Orbits are separated by smooth invariants.Ω- is an affi ne manifold,G -is a semi-simple Lie group, G ×Ω→ Ω-algebraic action =⇒ ΩG affi ne manifold (D. Hilbert).Regular orbits are separated by polynomial invariants.Ω- is an algebraic manifold,G -is an algebraic Lie group, G ×Ω→ Ω-algebraic action =⇒ regular orbits are separated by rationalinvariants (M. Rosenlicht).There is no Hilbert’s 14th problem!


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The Differential Moduli Problem

"Describe" the orbit space Ω/G , where Ω is a solution space of

a differential equation and G is a symmetry pseudogroup.

Jet level:

PDEs system E · ⊂ J∞, G -symmetry Lie pseudogroup.

E = J∞.Lie-Tresse Theorem: Microlocally (i.e. in a neighborhood of J∞)algebra differential G -invariants is generated by a number of basicdifferential invariants and G -invariant total derivations.(S. Lie, A.Tresse, A. Kumpera for Lie pseudogroups and L. Ovsyannikov and P.Olver for Lie groups).

Lie-Tresse Theorem for PDEs: E ⊂ J∞is a formally integrable PDEssystem , G -Lie pseudogroup of symmetries. (B.Kruglikov & VL).


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The Differential Moduli Problem

"Describe" the orbit space Ω/G , where Ω is a solution space of

a differential equation and G is a symmetry pseudogroup.

Jet level:

PDEs system E · ⊂ J∞, G -symmetry Lie pseudogroup.

E = J∞.Lie-Tresse Theorem: Microlocally (i.e. in a neighborhood of J∞)algebra differential G -invariants is generated by a number of basicdifferential invariants and G -invariant total derivations.(S. Lie, A.Tresse, A. Kumpera for Lie pseudogroups and L. Ovsyannikov and P.Olver for Lie groups).Lie-Tresse Theorem for PDEs: E ⊂ J∞is a formally integrable PDEssystem , G -Lie pseudogroup of symmetries. (B.Kruglikov & VL).


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Basic Setup

M is a smooth manifold, Jk0 (M,M) - the manifold of k -jets ofdiffeomorphisms of M, Jk (M, n) - the manifold of k-jets of ofsubmanifolds in M, having dimension n.Affi ne and algebraic structures:

· · · −→ J3 (M, n) S3τ∗⊗ν−→ J2 (M, n) S

2τ∗⊗ν−→ J1 (M, n)Grn(TM )−→ M

and

· · · −→ J30 (M,M)S3T∗⊗T−→ J20 (M,M)

S2T∗⊗T−→ J10 (M,M)Gl(TM )−→ M×M → M

G is a Lie pseudogroup, acting on M, G k ⊂ Jk0 (M,M) , k = 1, 2, ...is the corresponding Lie equation.E · ⊂ J∞ (M, n) , Ek ⊂ Jk (M, n), is a PDEs system onsubmanifolds (of dimension n ) of M.Given point a ∈ M, by Jka and Eka we denote the fibres of projectionsJk (M, n)→ M and Ek → M at the point a, G ka are stabilizers G

k ofthe point.


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Basic Setup


· · · −→ J3 (M, n) S3τ∗⊗ν−→ J2 (M, n) S

2τ∗⊗ν−→ J1 (M, n)Grn(TM )−→ M

and

· · · −→ J30 (M,M)S3T∗⊗T−→ J20 (M,M)

S2T∗⊗T−→ J10 (M,M)Gl(TM )−→ M×M → M

G is a Lie pseudogroup, acting on M, G k ⊂ Jk0 (M,M) , k = 1, 2, ...is the corresponding Lie equation.

E · ⊂ J∞ (M, n) , Ek ⊂ Jk (M, n), is a PDEs system onsubmanifolds (of dimension n ) of M.Given point a ∈ M, by Jka and Eka we denote the fibres of projectionsJk (M, n)→ M and Ek → M at the point a, G ka are stabilizers G

k ofthe point.


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Basic Setup


· · · −→ J3 (M, n) S3τ∗⊗ν−→ J2 (M, n) S

2τ∗⊗ν−→ J1 (M, n)Grn(TM )−→ M

and

· · · −→ J30 (M,M)S3T∗⊗T−→ J20 (M,M)

S2T∗⊗T−→ J10 (M,M)Gl(TM )−→ M×M → M

G is a Lie pseudogroup, acting on M, G k ⊂ Jk0 (M,M) , k = 1, 2, ...is the corresponding Lie equation.E · ⊂ J∞ (M, n) , Ek ⊂ Jk (M, n), is a PDEs system onsubmanifolds (of dimension n ) of M.

Given point a ∈ M, by Jka and Eka we denote the fibres of projectionsJk (M, n)→ M and Ek → M at the point a, G ka are stabilizers G

k ofthe point.


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Basic Setup


· · · −→ J3 (M, n) S3τ∗⊗ν−→ J2 (M, n) S

2τ∗⊗ν−→ J1 (M, n)Grn(TM )−→ M

and

· · · −→ J30 (M,M)S3T∗⊗T−→ J20 (M,M)

S2T∗⊗T−→ J10 (M,M)Gl(TM )−→ M×M → M

G is a Lie pseudogroup, acting on M, G k ⊂ Jk0 (M,M) , k = 1, 2, ...is the corresponding Lie equation.E · ⊂ J∞ (M, n) , Ek ⊂ Jk (M, n), is a PDEs system onsubmanifolds (of dimension n ) of M.Given point a ∈ M, by Jka and Eka we denote the fibres of projectionsJk (M, n)→ M and Ek → M at the point a, G ka are stabilizers G

k ofthe point.


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Assumptions

The action of G on M is transitive.

The prolongated actions of G on Jk (M, n) , k = 1, 2, ... are algebraic.That is, G ka are algebraic groups acting algebraically on algebraicmanifolds Jka .E · is G -invariant formally integrable PDE system and Eka ⊂ Jka areirreducible algebraic submanifolds.


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Assumptions


The prolongated actions of G on Jk (M, n) , k = 1, 2, ... are algebraic.That is, G ka are algebraic groups acting algebraically on algebraicmanifolds Jka .

E · is G -invariant formally integrable PDE system and Eka ⊂ Jka areirreducible algebraic submanifolds.


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Assumptions


The prolongated actions of G on Jk (M, n) , k = 1, 2, ... are algebraic.That is, G ka are algebraic groups acting algebraically on algebraicmanifolds Jka .E · is G -invariant formally integrable PDE system and Eka ⊂ Jka areirreducible algebraic submanifolds.


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Differential Invariants

"Common sense": A smooth function I densely defined on equationE l ⊂ Jl (M, n) and invariant with respect to the prolongated G -action is called a differential G -invariant of order ≤ l .

"Narrow sense": By a rational differential G -invariant of order ≤ lwe mean a differential G -invariant which is rational along fibresπl ,0 : E l → M.

"Good sense": By a differential G -invariant of order ≤ l we mean adifferential G -invariant which is rational along fibres πs ,0 : E s → M,for some s ≤ l , and polynomial along fibres πl ,s : E l → E s .


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Differential Invariants

"Common sense": A smooth function I densely defined on equationE l ⊂ Jl (M, n) and invariant with respect to the prolongated G -action is called a differential G -invariant of order ≤ l ."Narrow sense": By a rational differential G -invariant of order ≤ lwe mean a differential G -invariant which is rational along fibresπl ,0 : E l → M.

"Good sense": By a differential G -invariant of order ≤ l we mean adifferential G -invariant which is rational along fibres πs ,0 : E s → M,for some s ≤ l , and polynomial along fibres πl ,s : E l → E s .


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Invariant derivations

Total vector field densely defined on differential equation E · andinvariant under the prolonged G -action is called an G -invariantdifferentiation.

Similar to the case of differential invariants we’ll play with coeffi cientsof the total vector fields in order to get invariant derivations in "goodsense".

In reality, one needs derivations but not only total vector fields!


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Cartan-Kuranishi type theorem for singularities

We say that a closed subset S ⊂ Ek is Zariski closed if all its intersectionsSa = S ∩ Jka are Zariski closed.

TheoremThere exists a number l and a Zariski closed invariant proper subset Σl ⊂E l such that the action is regular in E · r π−1∞,l (Σl ) i.e. for any k ≥ l , theorbits of G on Ek r π−1k ,l (Σl ) are closed, have the same dimension andseperated by "good" differential invariants.


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Lie-Tresse theorem

Our second result gives finiteness for differential invariants.

TheoremThere exists a number l and a Zariski closed invariant proper subset Σl ⊂E l such that the algebra of good differential invariants separates theregular orbits and is finitely generated in the following sense.There exists a finite number of good differential invariants I1, ...., In and ofgood invariant derivations ∇1, ..,∇s such that any good differentialinvariant is a polynomial of ∇J (Ii ) , where ∇J (Ii ) = ∇j1 · · · ∇jr , forsome multi-indices J, with coeffi cients being rational functions of I .


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Comments

For smooth functions the global Lie-Tresse theorem fails, while themicro-local one follows via the implicit functions theorem. Thepolynomial version of the theorem also fails.

Provided E la are Stein manifolds the obtained finite generationproperty holds for the bigger algebra of meromorphic differentialG -invariants.

The theorem holds if G acts in a transitive way on some manifold Ek .Then all algebraic properties should be required for bundlesπr ,k : E r → Ek , where r > k.An important issue (not appearing micro-locally) is that some of thederivations ∇j may not be represented by total vector fields.Finiteness theorem valid invariant differential forms, tensors and othernatural geometric objects.


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Comments



The theorem holds if G acts in a transitive way on some manifold Ek .Then all algebraic properties should be required for bundlesπr ,k : E r → Ek , where r > k.

An important issue (not appearing micro-locally) is that some of thederivations ∇j may not be represented by total vector fields.Finiteness theorem valid invariant differential forms, tensors and othernatural geometric objects.


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Comments



The theorem holds if G acts in a transitive way on some manifold Ek .Then all algebraic properties should be required for bundlesπr ,k : E r → Ek , where r > k.An important issue (not appearing micro-locally) is that some of thederivations ∇j may not be represented by total vector fields.

Finiteness theorem valid invariant differential forms, tensors and othernatural geometric objects.


/ 26

Comments



The theorem holds if G acts in a transitive way on some manifold Ek .Then all algebraic properties should be required for bundlesπr ,k : E r → Ek , where r > k.An important issue (not appearing micro-locally) is that some of thederivations ∇j may not be represented by total vector fields.Finiteness theorem valid invariant differential forms, tensors and othernatural geometric objects.


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Arnold conjecture on Poincarè function

V. Arnold (1994) made a conjecture that Poincarè functions in differentialmoduli problems are rational.In our case:

Rk the field of rational differential G -invariants of order ≤ k;rk = trdeg(Rk ) - the trancedence degree.

dk = rk − rk−1 -dimension of differential invariants of pure order k .HG ,E : k 7−→ dk - the Hilbert function of the pseudogroup action.PG ,E (z) = ∑k dk z

k - the Poincaré function of the pseudogroupaction.Then:

1 The Hilbert function HG is a polynomial for large k.2 The Poincaré function equals

PG (z) =p (z)

(1− z)d,

for some polynomial p(z) and integer d > 0.


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Rk the field of rational differential G -invariants of order ≤ k;rk = trdeg(Rk ) - the trancedence degree.dk = rk − rk−1 -dimension of differential invariants of pure order k .

HG ,E : k 7−→ dk - the Hilbert function of the pseudogroup action.PG ,E (z) = ∑k dk z



PG (z) =p (z)

(1− z)d,



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Rk the field of rational differential G -invariants of order ≤ k;rk = trdeg(Rk ) - the trancedence degree.dk = rk − rk−1 -dimension of differential invariants of pure order k .HG ,E : k 7−→ dk - the Hilbert function of the pseudogroup action.

PG ,E (z) = ∑k dk zk - the Poincaré function of the pseudogroup

action.Then:


PG (z) =p (z)

(1− z)d,



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Rk the field of rational differential G -invariants of order ≤ k;rk = trdeg(Rk ) - the trancedence degree.dk = rk − rk−1 -dimension of differential invariants of pure order k .HG ,E : k 7−→ dk - the Hilbert function of the pseudogroup action.PG ,E (z) = ∑k dk z

k - the Poincaré function of the pseudogroupaction.

Then:


PG (z) =p (z)

(1− z)d,



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PG (z) =p (z)

(1− z)d,



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1 The Hilbert function HG is a polynomial for large k.

2 The Poincaré function equals

PG (z) =p (z)

(1− z)d,



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PG (z) =p (z)

(1− z)d,



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Riemannian structures

Let (M, g) be a fixed n -dimensional Riemannian manifold.

Ω = g, G = Diffeo (M) ,

Ricg ∈ S2T∗ is the Ricci tensor of the metric g , and Rg : T→ T thecorresponding operator.

J1 = Tr Rg , J2 = Tr R2g , ..., Jn = Tr Rngare rational differential invariants of order 2.We say that metric g is Ricci regular at a point a ∈ M if totaldifferentials dJ1, dJ2, ..., dJn are linear independent at the pointa3 = [g ]3a ∈ J3

(S2T∗

).

Here [g ]3a is the 3 - jet of g at the point, J3(S2T∗

)the manifold of 3

-jets of the bundle S2T∗M → M.Denote by

Σ3 = dJ1 ∧ dJ2 ∧ · · · ∧ dJn = 0 ⊂ J3(S2T∗

)be the set of Ricci singular points.


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Ω = g, G = Diffeo (M) ,Ricg ∈ S2T∗ is the Ricci tensor of the metric g , and Rg : T→ T thecorresponding operator.


(S2T∗

).


)the manifold of 3


Σ3 = dJ1 ∧ dJ2 ∧ · · · ∧ dJn = 0 ⊂ J3(S2T∗



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J1 = Tr Rg , J2 = Tr R2g , ..., Jn = Tr Rngare rational differential invariants of order 2.

We say that metric g is Ricci regular at a point a ∈ M if totaldifferentials dJ1, dJ2, ..., dJn are linear independent at the pointa3 = [g ]3a ∈ J3

(S2T∗

).


)the manifold of 3


Σ3 = dJ1 ∧ dJ2 ∧ · · · ∧ dJn = 0 ⊂ J3(S2T∗



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(S2T∗

).


)the manifold of 3

-jets of the bundle S2T∗M → M.

Denote by

Σ3 = dJ1 ∧ dJ2 ∧ · · · ∧ dJn = 0 ⊂ J3(S2T∗



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(S2T∗

).


)the manifold of 3


Σ3 = dJ1 ∧ dJ2 ∧ · · · ∧ dJn = 0 ⊂ J3(S2T∗



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Differential invariant algebraBasic invariants

At Ricci regular points we represent the metric in the following form

g = ∑i≤jGij dJi · dJj .

Then Gij are rational differential invariants of order 3 defined at Ricciregular points J3

(S2T∗

)\ Σ3.


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Differential invariant algebraBasic derivations

The Tresse derivations

∇1 =DDJj

, ....,∇n =DDJn

are rational, defined and linear independent at Ricci regular points.The basic frame formed by gradients of basic invariants:

E1 = gradg J1, ....,En = gradg Jn.


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Differential invariant algebra

By differential invariant of order k we mean an invariant which is rationalalong fibres Jk

(S2T∗

)→ M and polynomial along fibres

Jk(S2T∗

)→ J3

(S2T∗

).

TheoremAlgebra metric differential invariants generated by basic invariants Gij andTresse derivatives ∇1, ...,∇n.


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Metric differential invariantsFactor equation

Space Rn with fixed coordinates (x1, ..., xn) .

Covering

J : M r Σg → Dg ⊂ Rn,

J = (J1 (g) , ..., Jn (g)) ,

where Ji (g) are the values of Ji at metric g , Σg - Ricci singularpoints, Dg = Im (J) .Factor equation Emetric for metrics h on Rn :

TrRh = x1, TrR2h = x2, · · · , TrRnh = xn.

For any metric g , having Ricci regular points, metricgJ = ∑i ,j Gij (g) dJi (g) · dJj (g) is a solution of Emetric overDg ⊂ Rn.

Equivalence classes of metrics ⇐⇒ Solutions of PDEs system Emetric


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Space Rn with fixed coordinates (x1, ..., xn) .Covering


J = (J1 (g) , ..., Jn (g)) ,

where Ji (g) are the values of Ji at metric g , Σg - Ricci singularpoints, Dg = Im (J) .

Factor equation Emetric for metrics h on Rn :





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J = (J1 (g) , ..., Jn (g)) ,






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J = (J1 (g) , ..., Jn (g)) ,




Equivalence classes of metrics ⇐⇒ Solutions of PDEs system EmetricValentin Lychagin (University of Tromso) Quotients

Geometry and Lie theory. Dedicated to Eldar Straume on his 70th birthday. Trondheim, 03.11.16 17/ 26

Einstein Manifolds

By Einstein manifold (M, g) we mean an oriented 4 -dimensionalLorentzian manifold (signature g equals (1, 3)) satisfying Einstein equationRicg = λ g .

Denote byWg : Λ2T∗ → Λ2T∗

the Weyl tensor. Then the Hodge operator ∗ : Λ2T∗ → Λ2T∗ definesa complex structure in the bundle Λ2T∗ and Wg is C -linear operatorwith TrCWg = 0.

LetTrCW

2g = J1 +

√−1 J2, TrCW

3g = J3 +

√−1 J4.

Then J1, J2, J3, J4 are rational differential invariants of order 2.We say that Einstein metric g is Weyl regular at a point a ∈ M iftotal differentials dJ1, dJ2, ..., dJ4 are linear independent at the pointa3 = [g ]3a ∈ J3

(S2T∗

).

Denote by Σ3 in J3(S2T∗

)the set of Weyl singular points.


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Einstein Manifolds



the Weyl tensor. Then the Hodge operator ∗ : Λ2T∗ → Λ2T∗ definesa complex structure in the bundle Λ2T∗ and Wg is C -linear operatorwith TrCWg = 0.Let

TrCW2g = J1 +

√−1 J2, TrCW

3g = J3 +

√−1 J4.

Then J1, J2, J3, J4 are rational differential invariants of order 2.

We say that Einstein metric g is Weyl regular at a point a ∈ M iftotal differentials dJ1, dJ2, ..., dJ4 are linear independent at the pointa3 = [g ]3a ∈ J3

(S2T∗

).




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Einstein Manifolds



the Weyl tensor. Then the Hodge operator ∗ : Λ2T∗ → Λ2T∗ definesa complex structure in the bundle Λ2T∗ and Wg is C -linear operatorwith TrCWg = 0.Let

TrCW2g = J1 +

√−1 J2, TrCW

3g = J3 +

√−1 J4.

Then J1, J2, J3, J4 are rational differential invariants of order 2.We say that Einstein metric g is Weyl regular at a point a ∈ M iftotal differentials dJ1, dJ2, ..., dJ4 are linear independent at the pointa3 = [g ]3a ∈ J3

(S2T∗

).




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Differential invariant algebra

Basic invariants: Ifg = ∑

i≤jGij dJi · dJj ,

then Gij are rational differential invariants of order 3, defined at Weylregular points.

Basic derivations: The Tresse derivations

∇1 =DDJj

, ....,∇4 =DDJ4

are rational, defined and linear independent at Weyl regular points.

TheoremAlgebra differential invariants for Einstein metrics is generated by basicinvariants Gij and Tresse derivatives ∇1, ...,∇4.


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Factor equation

Space C2 with fixed coordinates(z1 = x1 +

√−1 y1, z2 = x2 +

√−1 y2

).

Covering

J : M r Weyl Sing → Dg ⊂ C2,

J =(J1 (g) +

√−1 J2 (g) , J3 (g) +

√−1 J4 (g)

),

where Ji (g) are the values of Ji on metric g , Dg = Im (J) .Factor equation EEinstein metric for (1, 3) metrics h on C2 ×C2 :

Rich = λ h, TrCW2h = z1, TrCW

3h = z2.

For any Einstein metric g , having Weyl regular points, metricgJ = ∑i ,j Gij (g) dJi (g) · dJj (g) is a solution of EEinstein metric overDg ⊂ C2 ×C2.

Equivalence classes of Einstein metrics⇐⇒ Solutions of PDEssystem EEinstein metric


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Factor equation


√−1 y1, z2 = x2 +

√−1 y2

).

Covering


J =(J1 (g) +

√−1 J2 (g) , J3 (g) +

√−1 J4 (g)

),

where Ji (g) are the values of Ji on metric g , Dg = Im (J) .

Factor equation EEinstein metric for (1, 3) metrics h on C2 ×C2 :


3h = z2.




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Factor equation


√−1 y1, z2 = x2 +

√−1 y2

).

Covering


J =(J1 (g) +

√−1 J2 (g) , J3 (g) +

√−1 J4 (g)

),

where Ji (g) are the values of Ji on metric g , Dg = Im (J) .Factor equation EEinstein metric for (1, 3) metrics h on C2 ×C2 :


3h = z2.




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Binary Forms

Binary form over C degree n:

P = ∑iaixk−iy i .

Bk the space of binary forms of degree n.

Gl (2,C) - action on Bk : Sl (2,C) - change of coordinates,λ ∈ C∗ : P 7−→ λP.

nk minimal number of generators of polynomial Sl (2,C) -invariantson Bk .


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Remarks: What is known?

n3 = 1, I4; n4 = 2, I2, I3 - The debut of theory of invariants (Boole,Cayley,Einsenstein,1840-1850)

n5 = 4, I4, I8, I12,I18, (Caley and Hermit (I18), 1854), I18 has degree18 and contains 800 monomials.

n6 = 5, (Gordan, 1885)

n7 ≤ 33 (Gall, 1888), n7 = 30 (Dixmier and Lazard, 1986).Bedratyuk (2007)- programm for finding minimal set of generators.

n8 = 9 (Gall, 1888 and Shioda, 1967)

Kac, V.G. (1982)- number of generators grows exponentially with k


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Differential contra algebraic invariants

Binary forms are analytic solutions of the Euler equation:

xux + yuy − ku = 0 ⊂ J1.

Gl (2,C) is a symmetry groupdifferential invariants are polynomials in ux , uy , .... and rational in u.

Basic invariant

H =uxxuyy − u2xy

u2

Invariant derivation

∇ = uyuddx− uxuddy

Regular orbits: H 6= 0, u 6= 0.Algebra differential invariants generated by H and ∇. It separatesregular orbits.


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xux + yuy − ku = 0 ⊂ J1.

Gl (2,C) is a symmetry group

differential invariants are polynomials in ux , uy , .... and rational in u.

Basic invariant

H =uxxuyy − u2xy

u2





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xux + yuy − ku = 0 ⊂ J1.


Basic invariant

H =uxxuyy − u2xy

u2





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xux + yuy − ku = 0 ⊂ J1.


Basic invariant

H =uxxuyy − u2xy

u2



Regular orbits: H 6= 0, u 6= 0.

Algebra differential invariants generated by H and ∇. It separatesregular orbits.


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xux + yuy − ku = 0 ⊂ J1.


Basic invariant

H =uxxuyy − u2xy

u2





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On Classification of Binary Forms

LetJ1 = H, J2 = ∇ (H) , J3 = ∇2 (H)

andji = Ji (P) ,

i = 1, 2, 3, their values on binary form P.

j1, j2, j3 are rational functions on C2 =⇒ ∃ irreducible polynomialFP (z1, z2, z3) such that

FP (j1, j2, j3) = 0.

We call Fp invariant of binary form P.We say that a binary form P is regular if j1j3 − 3

2 j22 6= 0.

TheoremTwo regular binary forms are Gl (2,C) -equivalent iff their invariantscoincide (up to multiplier).


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LetJ1 = H, J2 = ∇ (H) , J3 = ∇2 (H)

andji = Ji (P) ,

i = 1, 2, 3, their values on binary form P.j1, j2, j3 are rational functions on C2 =⇒ ∃ irreducible polynomialFP (z1, z2, z3) such that

FP (j1, j2, j3) = 0.

We call Fp invariant of binary form P.

We say that a binary form P is regular if j1j3 − 32 j22 6= 0.



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LetJ1 = H, J2 = ∇ (H) , J3 = ∇2 (H)

andji = Ji (P) ,

i = 1, 2, 3, their values on binary form P.j1, j2, j3 are rational functions on C2 =⇒ ∃ irreducible polynomialFP (z1, z2, z3) such that

FP (j1, j2, j3) = 0.

We call Fp invariant of binary form P.We say that a binary form P is regular if j1j3 − 3

2 j22 6= 0.



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Remarks and generalizations

Orbits and automorphic DEs:

xux + yuy − ku = 0

FP (H,∇H,∇2H) = 0

is an Gl (2,C) -automorphic PDEs system of 4th order.

Example: ForP = x100 + y100,

one hasFP = 9702 J1J3 − 9602 J22 + 19208 J31 .

Rational binary forms

n -ary forms

Invariants of irreducible representations of semi simple Lie groups


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Thank you for your attention


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Valentin Lychagin - NTNU · Quotients Valentin Lychagin Institute of Mathematics and Statistics,...

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