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)
Quotients
Geometry and Lie theory. Dedicated to Eldar
/ 26
Plan of the talk
Some observations and quasi historical remarks
Valentin Lychagin (University of Tromso)
Quotients
Geometry and Lie theory. Dedicated to Eldar
/ 26
Plan of the talk
Some observations and quasi historical remarks
Lie-Tresse theorem (joint with Boris Kruglikov)
Valentin Lychagin (University of Tromso)
Quotients
Geometry and Lie theory. Dedicated to Eldar
/ 26
Plan of the talk
Some observations and quasi historical remarks
Lie-Tresse theorem (joint with Boris Kruglikov)
what is a di¤erential invariant?
Valentin Lychagin (University of Tromso)
Quotients
Geometry and Lie theory. Dedicated to Eldar
/ 26
Plan of the talk
Some observations and quasi historical remarks
Lie-Tresse theorem (joint with Boris Kruglikov)
what is a di¤erential invariant?
…niteness of singularities
Valentin Lychagin (University of Tromso)
Quotients
Geometry and Lie theory. Dedicated to Eldar
/ 26
Plan of the talk
Some observations and quasi historical remarks
Lie-Tresse theorem (joint with Boris Kruglikov)
what is a di¤erential invariant?
…niteness of singularities
…niteness of di¤erential invariant algebras and …elds
Valentin Lychagin (University of Tromso)
Quotients
Geometry and Lie theory. Dedicated to Eldar
/ 26
Plan of the talk
Some observations and quasi historical remarks
Lie-Tresse theorem (joint with Boris Kruglikov)
what is a di¤erential invariant?
…niteness of singularities
…niteness of di¤erential invariant algebras and …elds
Applications
Valentin Lychagin (University of Tromso)
Quotients
Geometry and Lie theory. Dedicated to Eldar
/ 26
Plan of the talk
Some observations and quasi historical remarks
Lie-Tresse theorem (joint with Boris Kruglikov)
what is a di¤erential invariant?
…niteness of singularities
…niteness of di¤erential invariant algebras and …elds
Applications
Arnold conjecture (joint with Boris Kruglikov)
Valentin Lychagin (University of Tromso)
Quotients
Geometry and Lie theory. Dedicated to Eldar
/ 26
Plan of the talk
Some observations and quasi historical remarks
Lie-Tresse theorem (joint with Boris Kruglikov)
what is a di¤erential invariant?
…niteness of singularities
…niteness of di¤erential invariant algebras and …elds
Applications
Arnold conjecture (joint with Boris Kruglikov)
Riemannian and Einstein manifolds (joint with Valeriy Yumaguzhin)
Valentin Lychagin (University of Tromso)
Quotients
Geometry and Lie theory. Dedicated to Eldar
/ 26
Plan of the talk
Some observations and quasi historical remarks
Lie-Tresse theorem (joint with Boris Kruglikov)
what is a di¤erential invariant?
…niteness of singularities
…niteness of di¤erential invariant algebras and …elds
Applications
Arnold conjecture (joint with Boris Kruglikov)
Riemannian and Einstein manifolds (joint with Valeriy Yumaguzhin)
di¤erential contra algebraic invariants in the theory of algebraic Lie
group actions (joint with Pavel Bibikov)
Valentin Lychagin (University of Tromso)
Quotients
Geometry and Lie theory. Dedicated to Eldar
/ 26
Plan of the talk
Some observations and quasi historical remarks
Lie-Tresse theorem (joint with Boris Kruglikov)
what is a di¤erential invariant?
…niteness of singularities
…niteness of di¤erential invariant algebras and …elds
Applications
Arnold conjecture (joint with Boris Kruglikov)
Riemannian and Einstein manifolds (joint with Valeriy Yumaguzhin)
di¤erential contra algebraic invariants in the theory of algebraic Lie
group actions (joint with Pavel Bibikov)
binary forms and n-ary forms
Valentin Lychagin (University of Tromso)
Quotients
Geometry and Lie theory. Dedicated to Eldar
/ 26
Plan of the talk
Some observations and quasi historical remarks
Lie-Tresse theorem (joint with Boris Kruglikov)
what is a di¤erential invariant?
…niteness of singularities
…niteness of di¤erential invariant algebras and …elds
Applications
Arnold conjecture (joint with Boris Kruglikov)
Riemannian and Einstein manifolds (joint with Valeriy Yumaguzhin)
di¤erential contra algebraic invariants in the theory of algebraic Lie
group actions (joint with Pavel Bibikov)
binary forms and n-ary forms
invariants of irreducible representations of semisimple Lie groups
Valentin Lychagin (University of Tromso)
Quotients
Geometry and Lie theory. Dedicated to Eldar
/ 26
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 and
free action =) Ω G smooth manifold and Ω ! Ω G is a
principal G - bundle (J.L. Koszul and R. Palais).
Orbits are separated by smooth invariants.
Valentin Lychagin (University of Tromso)
Quotients
Geometry and Lie theory. Dedicated to Eldar
/ 26
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 and
free action =) Ω G smooth manifold and Ω ! Ω G is a
principal G - bundle (J.L. Koszul and R. Palais).
Orbits are separated by smooth invariants.
Ω- is an a¢ ne manifold,G -is a semi-simple Lie group, G Ω ! Ω
-algebraic action =) Ω G a¢ ne manifold (D. Hilbert).
Regular orbits are separated by polynomial invariants.
Valentin Lychagin (University of Tromso)
Quotients
Geometry and Lie theory. Dedicated to Eldar
/ 26
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 and
free action =) Ω G smooth manifold and Ω ! Ω G is a
principal G - bundle (J.L. Koszul and R. Palais).
Orbits are separated by smooth invariants.
Ω- is an a¢ ne manifold,G -is a semi-simple Lie group, G Ω ! Ω
-algebraic action =) Ω G a¢ 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 rational
invariants (M. Rosenlicht).
There is no Hilbert’s 14th problem!
Valentin Lychagin (University of Tromso)
Quotients
Geometry and Lie theory. Dedicated to Eldar
/ 26
The Di¤erential Moduli Problem
"Describe" the orbit space Ω/G , where Ω is a solution space of
a di¤erential 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 di¤erential G -invariants is generated by a number of basic
di¤erential invariants and G -invariant total derivations.(S. Lie, A.
Tresse, A. Kumpera for Lie pseudogroups and L. Ovsyannikov and P.
Olver for Lie groups).
Valentin Lychagin (University of Tromso)
Quotients
Geometry and Lie theory. Dedicated to Eldar
/ 26
The Di¤erential Moduli Problem
"Describe" the orbit space Ω/G , where Ω is a solution space of
a di¤erential 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 di¤erential G -invariants is generated by a number of basic
di¤erential 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 PDEs
system , G -Lie pseudogroup of symmetries. (B.Kruglikov & VL).
Valentin Lychagin (University of Tromso)
Quotients
Geometry and Lie theory. Dedicated to Eldar
/ 26
Basic Setup
M is a smooth manifold, Jk0 (M, M ) - the manifold of k -jets of
di¤eomorphisms of M, Jk (M, n ) - the manifold of k-jets of of
submanifolds in M, having dimension n.
A¢ ne and algebraic structures:
! J3 (M, n)
S3 τ
! J2 (M, n)
ν
S2 τ
! J1 (M, n)
ν
Grn (TM )
! M
and
! J30 (M, M )
Valentin Lychagin (University of Tromso)
S3 T
T
! J20 (M, M )
Quotients
S2 T
T
! J10 (M, M )
Gl(TM )
! M
M!
Geometry and Lie theory. Dedicated to Eldar
/ 26
Basic Setup
M is a smooth manifold, Jk0 (M, M ) - the manifold of k -jets of
di¤eomorphisms of M, Jk (M, n ) - the manifold of k-jets of of
submanifolds in M, having dimension n.
A¢ ne and algebraic structures:
! J3 (M, n)
S3 τ
! J2 (M, n)
ν
S2 τ
! J1 (M, n)
ν
Grn (TM )
! M
and
! J30 (M, M )
S3 T
T
! J20 (M, M )
S2 T
G is a Lie pseudogroup, acting on M, G k
is the corresponding Lie equation.
Valentin Lychagin (University of Tromso)
Quotients
T
! J10 (M, M )
Gl(TM )
! M
M!
Jk0 (M, M ) , k = 1, 2, ...
Geometry and Lie theory. Dedicated to Eldar
/ 26
Basic Setup
M is a smooth manifold, Jk0 (M, M ) - the manifold of k -jets of
di¤eomorphisms of M, Jk (M, n ) - the manifold of k-jets of of
submanifolds in M, having dimension n.
A¢ ne and algebraic structures:
! J3 (M, n)
S3 τ
! J2 (M, n)
ν
S2 τ
! J1 (M, n)
ν
Grn (TM )
! M
and
! J30 (M, M )
S3 T
T
! J20 (M, M )
S2 T
T
! J10 (M, M )
Gl(TM )
! 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 ) , fE k
Jk (M, n )g, is a PDEs system on
submanifolds (of dimension n ) of M.
Valentin Lychagin (University of Tromso)
Quotients
Geometry and Lie theory. Dedicated to Eldar
/ 26
Basic Setup
M is a smooth manifold, Jk0 (M, M ) - the manifold of k -jets of
di¤eomorphisms of M, Jk (M, n ) - the manifold of k-jets of of
submanifolds in M, having dimension n.
A¢ ne and algebraic structures:
! J3 (M, n)
S3 τ
! J2 (M, n)
ν
S2 τ
! J1 (M, n)
ν
Grn (TM )
! M
and
! J30 (M, M )
S3 T
T
! J20 (M, M )
S2 T
T
! J10 (M, M )
Gl(TM )
! 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 ) , fE k
Jk (M, n )g, is a PDEs system on
submanifolds (of dimension n ) of M.
Given point a 2 M, by Jka and Eak we denote the …bres of projections
Jk (M, n ) ! M and E k ! M at the point a, Gak are stabilizers G k of
the point.
Valentin Lychagin (University of Tromso)
Quotients
Geometry and Lie theory. Dedicated to Eldar
/ 26
Assumptions
The action of G on M is transitive.
Valentin Lychagin (University of Tromso)
Quotients
Geometry and Lie theory. Dedicated to Eldar
/ 26
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, Gak are algebraic groups acting algebraically on algebraic
manifolds Jka .
Valentin Lychagin (University of Tromso)
Quotients
Geometry and Lie theory. Dedicated to Eldar
/ 26
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, Gak are algebraic groups acting algebraically on algebraic
manifolds Jka .
E is G -invariant formally integrable PDE system and Eak
irreducible algebraic submanifolds.
Valentin Lychagin (University of Tromso)
Quotients
Jka are
Geometry and Lie theory. Dedicated to Eldar
/ 26
Di¤erential Invariants
"Common sense": A smooth function I densely de…ned on equation
E l Jl (M, n) and invariant with respect to the prolongated G action is called a di¤erential G -invariant of order l.
Valentin Lychagin (University of Tromso)
Quotients
Geometry and Lie theory. Dedicated to Eldar
/ 26
Di¤erential Invariants
"Common sense": A smooth function I densely de…ned on equation
E l Jl (M, n) and invariant with respect to the prolongated G action is called a di¤erential G -invariant of order l.
"Narrow sense": By a rational di¤erential G -invariant of order l
we mean a di¤erential G -invariant which is rational along …bres
π l ,0 : E l ! M.
Valentin Lychagin (University of Tromso)
Quotients
Geometry and Lie theory. Dedicated to Eldar
/ 26
Di¤erential Invariants
"Common sense": A smooth function I densely de…ned on equation
E l Jl (M, n) and invariant with respect to the prolongated G action is called a di¤erential G -invariant of order l.
"Narrow sense": By a rational di¤erential G -invariant of order l
we mean a di¤erential G -invariant which is rational along …bres
π l ,0 : E l ! M.
"Good sense": By a di¤erential G -invariant of order l we mean a
di¤erential G -invariant which is rational along …bres π s ,0 : E s ! M,
for some s l, and polynomial along …bres π l ,s : E l ! E s .
Valentin Lychagin (University of Tromso)
Quotients
Geometry and Lie theory. Dedicated to Eldar
/ 26
Invariant derivations
Total vector …eld densely de…ned on di¤erential equation E and
invariant under the prolonged G -action is called an G -invariant
di¤erentiation.
Valentin Lychagin (University of Tromso)
Quotients
Geometry and Lie theory. Dedicated to Eldar
/ 26
Invariant derivations
Total vector …eld densely de…ned on di¤erential equation E and
invariant under the prolonged G -action is called an G -invariant
di¤erentiation.
Similar to the case of di¤erential invariants we’ll play with coe¢ cients
of the total vector …elds in order to get invariant derivations in "good
sense".
Valentin Lychagin (University of Tromso)
Quotients
Geometry and Lie theory. Dedicated to Eldar
/ 26
Invariant derivations
Total vector …eld densely de…ned on di¤erential equation E and
invariant under the prolonged G -action is called an G -invariant
di¤erentiation.
Similar to the case of di¤erential invariants we’ll play with coe¢ cients
of the total vector …elds in order to get invariant derivations in "good
sense".
In reality, one needs derivations but not only total vector …elds!
Valentin Lychagin (University of Tromso)
Quotients
Geometry and Lie theory. Dedicated to Eldar
/ 26
Cartan-Kuranishi type theorem for singularities
We say that a closed subset S E k is Zariski closed if all its intersections
Sa = S \ Jka are Zariski closed.
Theorem
There exists a number l and a Zariski closed invariant proper subset Σl
E l such that the action is regular in E r π ∞,l1 (Σl ) i.e. for any k l, the
orbits of G on E k r π k ,l1 (Σl ) are closed, have the same dimension and
seperated by "good" di¤erential invariants.
Valentin Lychagin (University of Tromso)
Quotients
Geometry and Lie theory. Dedicated to Eldar
/ 26
Lie-Tresse theorem
Our second result gives …niteness for di¤erential invariants.
Theorem
There exists a number l and a Zariski closed invariant proper subset Σl
E l such that the algebra of good di¤erential invariants separates the
regular orbits and is …nitely generated in the following sense.
There exists a …nite number of good di¤erential invariants I1 , ...., In and of
good invariant derivations r1 , .., rs such that any good di¤erential
invariant is a polynomial of rJ (Ii ) , where rJ (Ii ) = rj1
rjr , for
some multi-indices J, with coe¢ cients being rational functions of I .
Valentin Lychagin (University of Tromso)
Quotients
Geometry and Lie theory. Dedicated to Eldar
/ 26
Comments
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.
Valentin Lychagin (University of Tromso)
Quotients
Geometry and Lie theory. Dedicated to Eldar
/ 26
Comments
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.
Provided Eal are Stein manifolds the obtained …nite generation
property holds for the bigger algebra of meromorphic di¤erential
G -invariants.
Valentin Lychagin (University of Tromso)
Quotients
Geometry and Lie theory. Dedicated to Eldar
/ 26
Comments
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.
Provided Eal are Stein manifolds the obtained …nite generation
property holds for the bigger algebra of meromorphic di¤erential
G -invariants.
The theorem holds if G acts in a transitive way on some manifold E k .
Then all algebraic properties should be required for bundles
π r ,k : E r ! E k , where r > k.
Valentin Lychagin (University of Tromso)
Quotients
Geometry and Lie theory. Dedicated to Eldar
/ 26
Comments
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.
Provided Eal are Stein manifolds the obtained …nite generation
property holds for the bigger algebra of meromorphic di¤erential
G -invariants.
The theorem holds if G acts in a transitive way on some manifold E k .
Then all algebraic properties should be required for bundles
π r ,k : E r ! E k , where r > k.
An important issue (not appearing micro-locally) is that some of the
derivations rj may not be represented by total vector …elds.
Valentin Lychagin (University of Tromso)
Quotients
Geometry and Lie theory. Dedicated to Eldar
/ 26
Comments
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.
Provided Eal are Stein manifolds the obtained …nite generation
property holds for the bigger algebra of meromorphic di¤erential
G -invariants.
The theorem holds if G acts in a transitive way on some manifold E k .
Then all algebraic properties should be required for bundles
π r ,k : E r ! E k , where r > k.
An important issue (not appearing micro-locally) is that some of the
derivations rj may not be represented by total vector …elds.
Finiteness theorem valid invariant di¤erential forms, tensors and other
natural geometric objects.
Valentin Lychagin (University of Tromso)
Quotients
Geometry and Lie theory. Dedicated to Eldar
/ 26
Arnold conjecture on Poincarè function
V. Arnold (1994) made a conjecture that Poincarè functions in di¤erential
moduli problems are rational.
In our case:
Rk the …eld of rational di¤erential G -invariants of order
rk = trdeg(Rk ) - the trancedence degree.
Valentin Lychagin (University of Tromso)
Quotients
k;
Geometry and Lie theory. Dedicated to Eldar
/ 26
Arnold conjecture on Poincarè function
V. Arnold (1994) made a conjecture that Poincarè functions in di¤erential
moduli problems are rational.
In our case:
Rk the …eld of rational di¤erential G -invariants of order k;
rk = trdeg(Rk ) - the trancedence degree.
dk = rk rk 1 -dimension of di¤erential invariants of pure order k.
Valentin Lychagin (University of Tromso)
Quotients
Geometry and Lie theory. Dedicated to Eldar
/ 26
Arnold conjecture on Poincarè function
V. Arnold (1994) made a conjecture that Poincarè functions in di¤erential
moduli problems are rational.
In our case:
Rk the …eld of rational di¤erential G -invariants of order k;
rk = trdeg(Rk ) - the trancedence degree.
dk = rk rk 1 -dimension of di¤erential invariants of pure order k.
HG ,E : k 7 ! dk - the Hilbert function of the pseudogroup action.
Valentin Lychagin (University of Tromso)
Quotients
Geometry and Lie theory. Dedicated to Eldar
/ 26
Arnold conjecture on Poincarè function
V. Arnold (1994) made a conjecture that Poincarè functions in di¤erential
moduli problems are rational.
In our case:
Rk the …eld of rational di¤erential G -invariants of order k;
rk = trdeg(Rk ) - the trancedence degree.
dk = rk rk 1 -dimension of di¤erential 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 pseudogroup
action.
Valentin Lychagin (University of Tromso)
Quotients
Geometry and Lie theory. Dedicated to Eldar
/ 26
Arnold conjecture on Poincarè function
V. Arnold (1994) made a conjecture that Poincarè functions in di¤erential
moduli problems are rational.
In our case:
Rk the …eld of rational di¤erential G -invariants of order k;
rk = trdeg(Rk ) - the trancedence degree.
dk = rk rk 1 -dimension of di¤erential 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 pseudogroup
action.
Then:
Valentin Lychagin (University of Tromso)
Quotients
Geometry and Lie theory. Dedicated to Eldar
/ 26
Arnold conjecture on Poincarè function
V. Arnold (1994) made a conjecture that Poincarè functions in di¤erential
moduli problems are rational.
In our case:
Rk the …eld of rational di¤erential G -invariants of order k;
rk = trdeg(Rk ) - the trancedence degree.
dk = rk rk 1 -dimension of di¤erential 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 pseudogroup
action.
Then:
1
The Hilbert function HG is a polynomial for large k.
Valentin Lychagin (University of Tromso)
Quotients
Geometry and Lie theory. Dedicated to Eldar
/ 26
Arnold conjecture on Poincarè function
V. Arnold (1994) made a conjecture that Poincarè functions in di¤erential
moduli problems are rational.
In our case:
Rk the …eld of rational di¤erential G -invariants of order k;
rk = trdeg(Rk ) - the trancedence degree.
dk = rk rk 1 -dimension of di¤erential 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 pseudogroup
action.
Then:
1
2
The Hilbert function HG is a polynomial for large k.
The Poincaré function equals
PG ( z ) =
p (z )
(1
,
z )d
for some polynomial p (z ) and integer d > 0.
Valentin Lychagin (University of Tromso)
Quotients
Geometry and Lie theory. Dedicated to Eldar
/ 26
Riemannian structures
Let (M, g ) be a …xed n -dimensional Riemannian manifold.
Ω = fg g, G = Diffeo (M ) ,
Valentin Lychagin (University of Tromso)
Quotients
Geometry and Lie theory. Dedicated to Eldar
/ 26
Riemannian structures
Let (M, g ) be a …xed n -dimensional Riemannian manifold.
Ω = fg g, G = Diffeo (M ) ,
bg : T ! T the
Ricg 2 S2 T is the Ricci tensor of the metric g , and R
corresponding operator.
Valentin Lychagin (University of Tromso)
Quotients
Geometry and Lie theory. Dedicated to Eldar
/ 26
Riemannian structures
Let (M, g ) be a …xed n -dimensional Riemannian manifold.
Ω = fg g, G = Diffeo (M ) ,
bg : T ! T the
Ricg 2 S2 T is the Ricci tensor of the metric g , and R
corresponding operator.
bg , J2 = Tr R
bg2 , ..., Jn = Tr R
bgn
J1 = Tr R
are rational di¤erential invariants of order 2.
Valentin Lychagin (University of Tromso)
Quotients
Geometry and Lie theory. Dedicated to Eldar
/ 26
Riemannian structures
Let (M, g ) be a …xed n -dimensional Riemannian manifold.
Ω = fg g, G = Diffeo (M ) ,
bg : T ! T the
Ricg 2 S2 T is the Ricci tensor of the metric g , and R
corresponding operator.
bg , J2 = Tr R
bg2 , ..., Jn = Tr R
bgn
J1 = Tr R
are rational di¤erential invariants of order 2.
We say that metric g is Ricci regular at a point a 2 M if total
b 1 , dJ
b 2 , ..., dJ
b n are linear independent at the point
di¤erentials dJ
3
3
2
a3 = [ g ] a 2 J S T .
Here [g ]3a is the 3 - jet of g at the point, J3 S2 T the manifold of 3
-jets of the bundle S2 T M ! M.
Valentin Lychagin (University of Tromso)
Quotients
Geometry and Lie theory. Dedicated to Eldar
/ 26
Riemannian structures
Let (M, g ) be a …xed n -dimensional Riemannian manifold.
Ω = fg g, G = Diffeo (M ) ,
bg : T ! T the
Ricg 2 S2 T is the Ricci tensor of the metric g , and R
corresponding operator.
bg , J2 = Tr R
bg2 , ..., Jn = Tr R
bgn
J1 = Tr R
are rational di¤erential invariants of order 2.
We say that metric g is Ricci regular at a point a 2 M if total
b 1 , dJ
b 2 , ..., dJ
b n are linear independent at the point
di¤erentials dJ
3
3
2
a3 = [ g ] a 2 J S T .
Here [g ]3a is the 3 - jet of g at the point, J3 S2 T the manifold of 3
-jets of the bundle S2 T M ! M.
Denote by
b 1 ^ dJ
b 2^
Σ3 = fdJ
b n = 0g
^ dJ
J3 S2 T
be the set of Ricci singular points.
Valentin Lychagin (University of Tromso)
Quotients
Geometry and Lie theory. Dedicated to Eldar
/ 26
Di¤erential invariant algebra
Basic invariants
At Ricci regular points we represent the metric in the following form
g=
∑ Gij
i j
b i
dJ
b j.
dJ
Then Gij are rational di¤erential invariants of order 3 de…ned at Ricci
regular points J3 S2 T n Σ3 .
Valentin Lychagin (University of Tromso)
Quotients
Geometry and Lie theory. Dedicated to Eldar
/ 26
Di¤erential invariant algebra
Basic derivations
The Tresse derivations
r1 =
D
D
, ...., rn =
DJj
DJn
are rational, de…ned and linear independent at Ricci regular points.
The basic frame formed by gradients of basic invariants:
E1 = gradg J1 , ...., En = gradg Jn .
Valentin Lychagin (University of Tromso)
Quotients
Geometry and Lie theory. Dedicated to Eldar
/ 26
Di¤erential invariant algebra
By di¤erential invariant of order k we mean an invariant which is rational
along …bres Jk S2 T ! M and polynomial along …bres
Jk S2 T ! J3 S2 T .
Theorem
Algebra metric di¤erential invariants generated by basic invariants Gij and
Tresse derivatives r1 , ..., rn .
Valentin Lychagin (University of Tromso)
Quotients
Geometry and Lie theory. Dedicated to Eldar
/ 26
Metric di¤erential invariants
Factor equation
Space Rn with …xed coordinates (x1 , ..., xn ) .
Valentin Lychagin (University of Tromso)
Quotients
Geometry and Lie theory. Dedicated to Eldar
/ 26
Metric di¤erential invariants
Factor equation
Space Rn with …xed coordinates (x1 , ..., xn ) .
Covering
M r Σg ! Dg
Rn ,
J
:
J
= (J1 (g ) , ..., Jn (g )) ,
where Ji (g ) are the values of Ji at metric g , Σg - Ricci singular
points, Dg = Im (J ) .
Valentin Lychagin (University of Tromso)
Quotients
Geometry and Lie theory. Dedicated to Eldar
/ 26
Metric di¤erential invariants
Factor equation
Space Rn with …xed coordinates (x1 , ..., xn ) .
Covering
M r Σg ! Dg
Rn ,
J
:
J
= (J1 (g ) , ..., Jn (g )) ,
where Ji (g ) are the values of Ji at metric g , Σg - Ricci singular
points, Dg = Im (J ) .
Factor equation Emetric for metrics h on Rn :
Tr Rh = x1 , Tr Rh2 = x2 ,
Valentin Lychagin (University of Tromso)
Quotients
, Tr Rhn = xn .
Geometry and Lie theory. Dedicated to Eldar
/ 26
Metric di¤erential invariants
Factor equation
Space Rn with …xed coordinates (x1 , ..., xn ) .
Covering
M r Σg ! Dg
Rn ,
J
:
J
= (J1 (g ) , ..., Jn (g )) ,
where Ji (g ) are the values of Ji at metric g , Σg - Ricci singular
points, Dg = Im (J ) .
Factor equation Emetric for metrics h on Rn :
Tr Rh = x1 , Tr Rh2 = x2 ,
, Tr Rhn = xn .
For any metric g , having Ricci regular points, metric
gJ = ∑i ,j Gij (g ) dJi (g ) dJj (g ) is a solution of Emetric over
Dg
Rn .
Valentin Lychagin (University of Tromso)
Quotients
Geometry and Lie theory. Dedicated to Eldar
/ 26
Metric di¤erential invariants
Factor equation
Space Rn with …xed coordinates (x1 , ..., xn ) .
Covering
M r Σg ! Dg
Rn ,
J
:
J
= (J1 (g ) , ..., Jn (g )) ,
where Ji (g ) are the values of Ji at metric g , Σg - Ricci singular
points, Dg = Im (J ) .
Factor equation Emetric for metrics h on Rn :
Tr Rh = x1 , Tr Rh2 = x2 ,
, Tr Rhn = xn .
For any metric g , having Ricci regular points, metric
gJ = ∑i ,j Gij (g ) dJi (g ) dJj (g ) is a solution of Emetric over
Dg
Rn .
Equivalence classes of metrics () Solutions of PDEs system Emetric
Valentin Lychagin (University of Tromso)
Quotients
Geometry and Lie theory. Dedicated to Eldar
/ 26
Einstein Manifolds
By Einstein manifold (M, g ) we mean an oriented 4 -dimensional
Lorentzian manifold (signature g equals (1, 3)) satisfying Einstein equation
Ricg = λ g .
Denote by
Wg : Λ 2 T ! Λ 2 T
the Weyl tensor. Then the Hodge operator : Λ2 T ! Λ2 T de…nes
a complex structure in the bundle Λ2 T and Wg is C -linear operator
with TrC Wg = 0.
Valentin Lychagin (University of Tromso)
Quotients
Geometry and Lie theory. Dedicated to Eldar
/ 26
Einstein Manifolds
By Einstein manifold (M, g ) we mean an oriented 4 -dimensional
Lorentzian manifold (signature g equals (1, 3)) satisfying Einstein equation
Ricg = λ g .
Denote by
Wg : Λ 2 T ! Λ 2 T
the Weyl tensor. Then the Hodge operator : Λ2 T ! Λ2 T de…nes
a complex structure in the bundle Λ2 T and Wg is C -linear operator
with TrC Wg = 0.
Let
p
p
TrC Wg2 = J1 +
1 J2 , TrC Wg3 = J3 +
1 J4 .
Then J1 , J2 , J3 , J4 are rational di¤erential invariants of order 2.
Valentin Lychagin (University of Tromso)
Quotients
Geometry and Lie theory. Dedicated to Eldar
/ 26
Einstein Manifolds
By Einstein manifold (M, g ) we mean an oriented 4 -dimensional
Lorentzian manifold (signature g equals (1, 3)) satisfying Einstein equation
Ricg = λ g .
Denote by
Wg : Λ 2 T ! Λ 2 T
the Weyl tensor. Then the Hodge operator : Λ2 T ! Λ2 T de…nes
a complex structure in the bundle Λ2 T and Wg is C -linear operator
with TrC Wg = 0.
Let
p
p
TrC Wg2 = J1 +
1 J2 , TrC Wg3 = J3 +
1 J4 .
Then J1 , J2 , J3 , J4 are rational di¤erential invariants of order 2.
We say that Einstein metric g is Weyl regular at a point a 2 M if
b 1 , dJ
b 2 , ..., dJ
b 4 are linear independent at the point
total di¤erentials dJ
3
3
2
a3 = [ g ] a 2 J S T .
Valentin Lychagin (University of Tromso)
Quotients
Geometry and Lie theory. Dedicated to Eldar
/ 26
Einstein Manifolds
By Einstein manifold (M, g ) we mean an oriented 4 -dimensional
Lorentzian manifold (signature g equals (1, 3)) satisfying Einstein equation
Ricg = λ g .
Denote by
Wg : Λ 2 T ! Λ 2 T
the Weyl tensor. Then the Hodge operator : Λ2 T ! Λ2 T de…nes
a complex structure in the bundle Λ2 T and Wg is C -linear operator
with TrC Wg = 0.
Let
p
p
TrC Wg2 = J1 +
1 J2 , TrC Wg3 = J3 +
1 J4 .
Then J1 , J2 , J3 , J4 are rational di¤erential invariants of order 2.
We say that Einstein metric g is Weyl regular at a point a 2 M if
b 1 , dJ
b 2 , ..., dJ
b 4 are linear independent at the point
total di¤erentials dJ
3
3
2
a3 = [ g ] a 2 J S T .
Denote by Σ3 in J3 S2 T the set of Weyl singular points.
Valentin Lychagin (University of Tromso)
Quotients
Geometry and Lie theory. Dedicated to Eldar
/ 26
Di¤erential invariant algebra
Basic invariants: If
g=
∑ Gij
i j
b i
dJ
b j,
dJ
then Gij are rational di¤erential invariants of order 3, de…ned at Weyl
regular points.
Valentin Lychagin (University of Tromso)
Quotients
Geometry and Lie theory. Dedicated to Eldar
/ 26
Di¤erential invariant algebra
Basic invariants: If
g=
∑ Gij
i j
b i
dJ
b j,
dJ
then Gij are rational di¤erential invariants of order 3, de…ned at Weyl
regular points.
Basic derivations: The Tresse derivations
r1 =
D
D
, ...., r4 =
DJj
DJ4
are rational, de…ned and linear independent at Weyl regular points.
Valentin Lychagin (University of Tromso)
Quotients
Geometry and Lie theory. Dedicated to Eldar
/ 26
Di¤erential invariant algebra
Basic invariants: If
g=
∑ Gij
i j
b i
dJ
b j,
dJ
then Gij are rational di¤erential invariants of order 3, de…ned at Weyl
regular points.
Basic derivations: The Tresse derivations
r1 =
D
D
, ...., r4 =
DJj
DJ4
are rational, de…ned and linear independent at Weyl regular points.
Theorem
Algebra di¤erential invariants for Einstein metrics is generated by basic
invariants Gij and Tresse derivatives r1 , ..., r4 .
Valentin Lychagin (University of Tromso)
Quotients
Geometry and Lie theory. Dedicated to Eldar
/ 26
Factor equation
Space C2 with
p …xed coordinatesp
z1 = x1 +
1 y1 , z2 = x2 +
1 y2 .
Valentin Lychagin (University of Tromso)
Quotients
Geometry and Lie theory. Dedicated to Eldar
/ 26
Factor equation
Space C2 with
p …xed coordinatesp
z1 = x1 +
1 y1 , z2 = x2 +
1 y2 .
Covering
J
J
M r fWeyl Singg ! Dg
C2 ,
p
p
= J1 (g ) +
1 J2 ( g ) , J3 ( g ) +
1 J4 (g ) ,
:
where Ji (g ) are the values of Ji on metric g , Dg = Im (J ) .
Valentin Lychagin (University of Tromso)
Quotients
Geometry and Lie theory. Dedicated to Eldar
/ 26
Factor equation
Space C2 with
p …xed coordinatesp
z1 = x1 +
1 y1 , z2 = x2 +
1 y2 .
Covering
J
J
M r fWeyl Singg ! Dg
C2 ,
p
p
= 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, TrC Wh2 = z1 , TrC Wh3 = z2 .
Valentin Lychagin (University of Tromso)
Quotients
Geometry and Lie theory. Dedicated to Eldar
/ 26
Factor equation
Space C2 with
p …xed coordinatesp
z1 = x1 +
1 y1 , z2 = x2 +
1 y2 .
Covering
J
J
M r fWeyl Singg ! Dg
C2 ,
p
p
= 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, TrC Wh2 = z1 , TrC Wh3 = z2 .
For any Einstein metric g , having Weyl regular points, metric
gJ = ∑i ,j Gij (g ) dJi (g ) dJj (g ) is a solution of EEinstein metric over
Dg
C2 C2 .
Valentin Lychagin (University of Tromso)
Quotients
Geometry and Lie theory. Dedicated to Eldar
/ 26
Factor equation
Space C2 with
p …xed coordinatesp
z1 = x1 +
1 y1 , z2 = x2 +
1 y2 .
Covering
J
J
M r fWeyl Singg ! Dg
C2 ,
p
p
= 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, TrC Wh2 = z1 , TrC Wh3 = z2 .
For any Einstein metric g , having Weyl regular points, metric
gJ = ∑i ,j Gij (g ) dJi (g ) dJj (g ) is a solution of EEinstein metric over
Dg
C2 C2 .
Equivalence classes of Einstein metrics () Solutions of PDEs
system EEinstein metric
Valentin Lychagin (University of Tromso)
Quotients
Geometry and Lie theory. Dedicated to Eldar
/ 26
Binary Forms
Binary form over C degree n:
P=
∑ ai x k
i i
y .
i
Bk the space of binary forms of degree n.
Valentin Lychagin (University of Tromso)
Quotients
Geometry and Lie theory. Dedicated to Eldar
/ 26
Binary Forms
Binary form over C degree n:
P=
∑ ai x k
i i
y .
i
Bk the space of binary forms of degree n.
Gl (2, C) - action on Bk : Sl (2, C) - change of coordinates,
λ 2 C : P 7 ! λP.
Valentin Lychagin (University of Tromso)
Quotients
Geometry and Lie theory. Dedicated to Eldar
/ 26
Binary Forms
Binary form over C degree n:
P=
∑ ai x k
i i
y .
i
Bk the space of binary forms of degree n.
Gl (2, C) - action on Bk : Sl (2, C) - change of coordinates,
λ 2 C : P 7 ! λP.
nk minimal number of generators of polynomial Sl (2, C) -invariants
on Bk .
Valentin Lychagin (University of Tromso)
Quotients
Geometry and Lie theory. Dedicated to Eldar
/ 26
Remarks: What is known?
n3 = 1, I4 ; n4 = 2, I2 , I3 - The debut of theory of invariants (Boole,
Cayley,Einsenstein,1840-1850)
Valentin Lychagin (University of Tromso)
Quotients
Geometry and Lie theory. Dedicated to Eldar
/ 26
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 degree
18 and contains 800 monomials.
Valentin Lychagin (University of Tromso)
Quotients
Geometry and Lie theory. Dedicated to Eldar
/ 26
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 degree
18 and contains 800 monomials.
n6 = 5, (Gordan, 1885)
Valentin Lychagin (University of Tromso)
Quotients
Geometry and Lie theory. Dedicated to Eldar
/ 26
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 degree
18 and contains 800 monomials.
n6 = 5, (Gordan, 1885)
n7 33 (Gall, 1888), n7 = 30 (Dixmier and Lazard, 1986).
Bedratyuk (2007)- programm for …nding minimal set of generators.
Valentin Lychagin (University of Tromso)
Quotients
Geometry and Lie theory. Dedicated to Eldar
/ 26
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 degree
18 and contains 800 monomials.
n6 = 5, (Gordan, 1885)
n7 33 (Gall, 1888), n7 = 30 (Dixmier and Lazard, 1986).
Bedratyuk (2007)- programm for …nding minimal set of generators.
n8 = 9 (Gall, 1888 and Shioda, 1967)
Valentin Lychagin (University of Tromso)
Quotients
Geometry and Lie theory. Dedicated to Eldar
/ 26
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 degree
18 and contains 800 monomials.
n6 = 5, (Gordan, 1885)
n7 33 (Gall, 1888), n7 = 30 (Dixmier and Lazard, 1986).
Bedratyuk (2007)- programm for …nding minimal set of generators.
n8 = 9 (Gall, 1888 and Shioda, 1967)
Kac, V.G. (1982)- number of generators grows exponentially with k
Valentin Lychagin (University of Tromso)
Quotients
Geometry and Lie theory. Dedicated to Eldar
/ 26
Di¤erential contra algebraic invariants
Binary forms are analytic solutions of the Euler equation:
fxux + yuy
Valentin Lychagin (University of Tromso)
ku = 0g
Quotients
J1 .
Geometry and Lie theory. Dedicated to Eldar
/ 26
Di¤erential contra algebraic invariants
Binary forms are analytic solutions of the Euler equation:
fxux + yuy
ku = 0g
J1 .
Gl (2, C) is a symmetry group
Valentin Lychagin (University of Tromso)
Quotients
Geometry and Lie theory. Dedicated to Eldar
/ 26
Di¤erential contra algebraic invariants
Binary forms are analytic solutions of the Euler equation:
fxux + yuy
ku = 0g
J1 .
Gl (2, C) is a symmetry group
di¤erential invariants are polynomials in ux , uy , .... and rational in u.
Valentin Lychagin (University of Tromso)
Quotients
Geometry and Lie theory. Dedicated to Eldar
/ 26
Di¤erential contra algebraic invariants
Binary forms are analytic solutions of the Euler equation:
fxux + yuy
ku = 0g
J1 .
Gl (2, C) is a symmetry group
di¤erential invariants are polynomials in ux , uy , .... and rational in u.
Basic invariant
H=
Valentin Lychagin (University of Tromso)
2
uxx uyy uxy
u2
Quotients
Geometry and Lie theory. Dedicated to Eldar
/ 26
Di¤erential contra algebraic invariants
Binary forms are analytic solutions of the Euler equation:
fxux + yuy
ku = 0g
J1 .
Gl (2, C) is a symmetry group
di¤erential invariants are polynomials in ux , uy , .... and rational in u.
Basic invariant
H=
Invariant derivation
r=
Valentin Lychagin (University of Tromso)
2
uxx uyy uxy
u2
uy d
u dx
Quotients
ux d
u dy
Geometry and Lie theory. Dedicated to Eldar
/ 26
Di¤erential contra algebraic invariants
Binary forms are analytic solutions of the Euler equation:
fxux + yuy
ku = 0g
J1 .
Gl (2, C) is a symmetry group
di¤erential invariants are polynomials in ux , uy , .... and rational in u.
Basic invariant
H=
Invariant derivation
r=
2
uxx uyy uxy
u2
uy d
u dx
ux d
u dy
Regular orbits: H 6= 0, u 6= 0.
Valentin Lychagin (University of Tromso)
Quotients
Geometry and Lie theory. Dedicated to Eldar
/ 26
Di¤erential contra algebraic invariants
Binary forms are analytic solutions of the Euler equation:
fxux + yuy
ku = 0g
J1 .
Gl (2, C) is a symmetry group
di¤erential invariants are polynomials in ux , uy , .... and rational in u.
Basic invariant
H=
Invariant derivation
r=
2
uxx uyy uxy
u2
uy d
u dx
ux d
u dy
Regular orbits: H 6= 0, u 6= 0.
Algebra di¤erential invariants generated by H and r. It separates
regular orbits.
Valentin Lychagin (University of Tromso)
Quotients
Geometry and Lie theory. Dedicated to Eldar
/ 26
On Classi…cation of Binary Forms
Let
and
J1 = H, J2 = r (H ) , J3 = r2 (H )
ji = Ji (P ) ,
i = 1, 2, 3, their values on binary form P.
Valentin Lychagin (University of Tromso)
Quotients
Geometry and Lie theory. Dedicated to Eldar
/ 26
On Classi…cation of Binary Forms
Let
and
J1 = H, J2 = r (H ) , J3 = r2 (H )
ji = Ji (P ) ,
i = 1, 2, 3, their values on binary form P.
j1 , j2 , j3 are rational functions on C2 =) 9 irreducible polynomial
FP (z1 , z2 , z3 ) such that
FP (j1 , j2 , j3 ) = 0.
We call Fp invariant of binary form P.
Valentin Lychagin (University of Tromso)
Quotients
Geometry and Lie theory. Dedicated to Eldar
/ 26
On Classi…cation of Binary Forms
Let
and
J1 = H, J2 = r (H ) , J3 = r2 (H )
ji = Ji (P ) ,
i = 1, 2, 3, their values on binary form P.
j1 , j2 , j3 are rational functions on C2 =) 9 irreducible polynomial
FP (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 j1 j3
Valentin Lychagin (University of Tromso)
Quotients
3 2
2 j2
6= 0.
Geometry and Lie theory. Dedicated to Eldar
/ 26
On Classi…cation of Binary Forms
Let
and
J1 = H, J2 = r (H ) , J3 = r2 (H )
ji = Ji (P ) ,
i = 1, 2, 3, their values on binary form P.
j1 , j2 , j3 are rational functions on C2 =) 9 irreducible polynomial
FP (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 j1 j3
3 2
2 j2
6= 0.
Theorem
Two regular binary forms are Gl (2, C) -equivalent i¤ their invariants
coincide (up to multiplier).
Valentin Lychagin (University of Tromso)
Quotients
Geometry and Lie theory. Dedicated to Eldar
/ 26
Remarks and generalizations
Orbits and automorphic DEs:
xux + yuy
ku = 0
2
FP (H, rH, r H ) = 0
is an Gl (2, C) -automorphic PDEs system of 4th order.
Valentin Lychagin (University of Tromso)
Quotients
Geometry and Lie theory. Dedicated to Eldar
/ 26
Remarks and generalizations
Orbits and automorphic DEs:
xux + yuy
ku = 0
2
FP (H, rH, r H ) = 0
is an Gl (2, C) -automorphic PDEs system of 4th order.
Example: For
P = x 100 + y 100 ,
one has
FP = 9702 J1 J3
Valentin Lychagin (University of Tromso)
9602 J22 + 19208 J13 .
Quotients
Geometry and Lie theory. Dedicated to Eldar
/ 26
Remarks and generalizations
Orbits and automorphic DEs:
xux + yuy
ku = 0
2
FP (H, rH, r H ) = 0
is an Gl (2, C) -automorphic PDEs system of 4th order.
Example: For
P = x 100 + y 100 ,
one has
FP = 9702 J1 J3
9602 J22 + 19208 J13 .
Rational binary forms
Valentin Lychagin (University of Tromso)
Quotients
Geometry and Lie theory. Dedicated to Eldar
/ 26
Remarks and generalizations
Orbits and automorphic DEs:
xux + yuy
ku = 0
2
FP (H, rH, r H ) = 0
is an Gl (2, C) -automorphic PDEs system of 4th order.
Example: For
P = x 100 + y 100 ,
one has
FP = 9702 J1 J3
9602 J22 + 19208 J13 .
Rational binary forms
n -ary forms
Valentin Lychagin (University of Tromso)
Quotients
Geometry and Lie theory. Dedicated to Eldar
/ 26
Remarks and generalizations
Orbits and automorphic DEs:
xux + yuy
ku = 0
2
FP (H, rH, r H ) = 0
is an Gl (2, C) -automorphic PDEs system of 4th order.
Example: For
P = x 100 + y 100 ,
one has
FP = 9702 J1 J3
9602 J22 + 19208 J13 .
Rational binary forms
n -ary forms
Invariants of irreducible representations of semi simple Lie groups
Valentin Lychagin (University of Tromso)
Quotients
Geometry and Lie theory. Dedicated to Eldar
/ 26
Thank you for your attention
Valentin Lychagin (University of Tromso)
Quotients
Geometry and Lie theory. Dedicated to Eldar
/ 26