The classification of three-dimensionalhomogeneous spaces with non-solvable

transformation groups

Boris Doubrov

Belarusian State University

Geometry and Lie theory, Trondheim, 03/11/2016

Outline

History

Structure of 1-dimensional invariant foliations

Algebraic techniques

Classification of 3-dimensional homogeneous spaces

Historical remarks

I One-dimensional homogeneous spaces:

R1 :〈∂x〉;A1 :〈∂x , x∂x〉;

RP1 :〈∂x , x∂x , x2∂x〉;

I Two-dimensional homogeneous spaces: 18 cases (S. Lie)

I Three-dimensional homogeneous spaces: classification of allspaces that do not admit 1-dimensional invariant folations(S. Lie, Theorie der Transformationgruppen, Band 3, 1893).

Historical remarks

I One-dimensional homogeneous spaces:

R1 :〈∂x〉;A1 :〈∂x , x∂x〉;

RP1 :〈∂x , x∂x , x2∂x〉;

I Two-dimensional homogeneous spaces: 18 cases (S. Lie)

I Three-dimensional homogeneous spaces: classification of allspaces that do not admit 1-dimensional invariant folations(S. Lie, Theorie der Transformationgruppen, Band 3, 1893).

Historical remarks

I One-dimensional homogeneous spaces:

R1 :〈∂x〉;A1 :〈∂x , x∂x〉;

RP1 :〈∂x , x∂x , x2∂x〉;

I Two-dimensional homogeneous spaces: 18 cases (S. Lie)

I Three-dimensional homogeneous spaces: classification of allspaces that do not admit 1-dimensional invariant folations(S. Lie, Theorie der Transformationgruppen, Band 3, 1893).

Lie, Theorie der Transformationgruppen, Band 3, 1893

English translation

Lie has already carried out the determination of all the primitivegroups of the space x, y, z in the years 1878 and 1879, though byextraordinarily extensive calculations, and the correspondingstatement on the transitive groups has been found.

By the methods developed here, which Lie first found in thesecalculations, the description of all the imprimitive groups of thespace x , y , z is greatly simplified. Using these methods the wholetask is split into a great number of individual tasks, each ofwhich can be done independently without the need toperform the rest in advance.

By means of the developments in this text the reader is able toeasily determine any category of imprimitive groups of the space x,y, z, whose knowledge is desirable to him.

Classification of all primitive homogeneous spaces

A Lie algebra of vector fields g is called primitive, if it does notadmit any invariant foliations. Locally this is equivalent to the factthat the stationary subalgebra g0 is maximal.

I Primitive actions in dimensions 1,2,3: Sophus Lie.

I Reduction to the case of a simple transformation group:V. Morozov.

I Classification of maximal subalgebras in complex simple Liealgebras: E. Dynkin.

I Real case: F. Karpelevich, B. Komrakov.

Classification of all primitive homogeneous spaces

A Lie algebra of vector fields g is called primitive, if it does notadmit any invariant foliations. Locally this is equivalent to the factthat the stationary subalgebra g0 is maximal.

I Primitive actions in dimensions 1,2,3: Sophus Lie.

I Reduction to the case of a simple transformation group:V. Morozov.

I Classification of maximal subalgebras in complex simple Liealgebras: E. Dynkin.

I Real case: F. Karpelevich, B. Komrakov.

Classification of all primitive homogeneous spaces

A Lie algebra of vector fields g is called primitive, if it does notadmit any invariant foliations. Locally this is equivalent to the factthat the stationary subalgebra g0 is maximal.

I Primitive actions in dimensions 1,2,3: Sophus Lie.

I Reduction to the case of a simple transformation group:V. Morozov.

I Classification of maximal subalgebras in complex simple Liealgebras: E. Dynkin.

I Real case: F. Karpelevich, B. Komrakov.

Classification of all primitive homogeneous spaces

A Lie algebra of vector fields g is called primitive, if it does notadmit any invariant foliations. Locally this is equivalent to the factthat the stationary subalgebra g0 is maximal.

I Primitive actions in dimensions 1,2,3: Sophus Lie.

I Reduction to the case of a simple transformation group:V. Morozov.

I Classification of maximal subalgebras in complex simple Liealgebras: E. Dynkin.

I Real case: F. Karpelevich, B. Komrakov.

How to “glue” primitive actions?

I Let M = G/G0 be a homogeneous space with an invariantfoliation. Any invariant foliation on M has the form{(gH)G0}, where H is some subgroup in G containing G0:

G/G0 → G/H

I Then we have two homogeneous spaces: G/H (action on theset of fibers fibers) and H/G0 (action on each fiber). None ofthem needs to be effective.

I Question: how to reconstruct M, if we have only“effectivized” spaces G/H and H/G0?

How to “glue” primitive actions?

I Let M = G/G0 be a homogeneous space with an invariantfoliation. Any invariant foliation on M has the form{(gH)G0}, where H is some subgroup in G containing G0:

G/G0 → G/H

I Then we have two homogeneous spaces: G/H (action on theset of fibers fibers) and H/G0 (action on each fiber). None ofthem needs to be effective.

I Question: how to reconstruct M, if we have only“effectivized” spaces G/H and H/G0?

How to “glue” primitive actions?

I Let M = G/G0 be a homogeneous space with an invariantfoliation. Any invariant foliation on M has the form{(gH)G0}, where H is some subgroup in G containing G0:

G/G0 → G/H

I Then we have two homogeneous spaces: G/H (action on theset of fibers fibers) and H/G0 (action on each fiber). None ofthem needs to be effective.

I Question: how to reconstruct M, if we have only“effectivized” spaces G/H and H/G0?

One-dimensional invariant foliations

I In case of 1-dimensional invariant foliations we have:

M = G/G0 7→ N = G/H, dimH/G0 = 1

I Here N = G/H = G/H be the space of all fibers.

I Examples of invariant foliations on R2 → R:

〈∂x , 2x∂x+ny∂y , x2∂x+nxy∂y , ∂y , x∂y , . . . , x

n∂y 〉,〈∂x , 2x∂x−2y∂y , x

2∂x−(1 + 2xy)∂y 〉,

〈∂x , 2x∂x , x2∂x〉.

One-dimensional invariant foliations

I In case of 1-dimensional invariant foliations we have:

M = G/G0 7→ N = G/H, dimH/G0 = 1

I Here N = G/H = G/H be the space of all fibers.

I Examples of invariant foliations on R2 → R:

〈∂x , 2x∂x+ny∂y , x2∂x+nxy∂y , ∂y , x∂y , . . . , x

n∂y 〉,〈∂x , 2x∂x−2y∂y , x

2∂x−(1 + 2xy)∂y 〉,

〈∂x , 2x∂x , x2∂x〉.

One-dimensional invariant foliations

I In case of 1-dimensional invariant foliations we have:

M = G/G0 7→ N = G/H, dimH/G0 = 1

I Here N = G/H = G/H be the space of all fibers.

I Examples of invariant foliations on R2 → R:

〈∂x , 2x∂x+ny∂y , x2∂x+nxy∂y , ∂y , x∂y , . . . , x

n∂y 〉,〈∂x , 2x∂x−2y∂y , x

2∂x−(1 + 2xy)∂y 〉,

〈∂x , 2x∂x , x2∂x〉.

Local structure of one-dimensional invariant folations

Assume that N = G/H is known (but not G/H!).

TheoremLocally we have one and only one of the following cases:

I G/H is effective. Then to construct G/G0 from G/H we justneed to describe all subgroups of codimension 1 in G0.

I M is the direct product of N and RP1, G = G × PSL(2,R).

I M is an invariant a vector bundle over N, and G is anextension of G :

{e} → V → G → G → {e}

by some finite-dimensional G -invariant space of sections.

I M is an invariant a vector bundle over N, and G is a trivialextension of G × R∗:

{e} → V → G → G × R∗ → {e}.

Extensions by abelian subgroups

I Let N = G/H and π : E → N be an invariant one-dimensionalvector bundle (determined by 1-dimensional representations ofH).

I Let F(π) be the space of all smooth sections of π. Considerthe natural action of G on F(π):

(g .α)(x) = g .α(g−1x), α ∈ F(π), g ∈ G , x ∈ N.

I Let V be any finite-dimensional submodule of the G -moduleF(π) and let G = G n V be the semidirect product of G andV . The group G acts naturally on E :

(g , σ).p = g .p + σ(g .π(p)).

This action is transitive on E , if V contains any non-zerosection. The stationary subgroup G0 of this action is equal toG0 n V0, where V0 is the set of all sections in V vanishing at0 = eG0.

I The semidirect product G n V can be replaced by anyextension of G by means of the abelian subgroup V .

Extensions by abelian subgroups

I Let N = G/H and π : E → N be an invariant one-dimensionalvector bundle (determined by 1-dimensional representations ofH).

I Let F(π) be the space of all smooth sections of π. Considerthe natural action of G on F(π):

(g .α)(x) = g .α(g−1x), α ∈ F(π), g ∈ G , x ∈ N.

I Let V be any finite-dimensional submodule of the G -moduleF(π) and let G = G n V be the semidirect product of G andV . The group G acts naturally on E :

(g , σ).p = g .p + σ(g .π(p)).

This action is transitive on E , if V contains any non-zerosection. The stationary subgroup G0 of this action is equal toG0 n V0, where V0 is the set of all sections in V vanishing at0 = eG0.

I The semidirect product G n V can be replaced by anyextension of G by means of the abelian subgroup V .

Extensions by abelian subgroups

I Let N = G/H and π : E → N be an invariant one-dimensionalvector bundle (determined by 1-dimensional representations ofH).

I Let F(π) be the space of all smooth sections of π. Considerthe natural action of G on F(π):

(g .α)(x) = g .α(g−1x), α ∈ F(π), g ∈ G , x ∈ N.

I Let V be any finite-dimensional submodule of the G -moduleF(π) and let G = G n V be the semidirect product of G andV . The group G acts naturally on E :

(g , σ).p = g .p + σ(g .π(p)).

This action is transitive on E , if V contains any non-zerosection. The stationary subgroup G0 of this action is equal toG0 n V0, where V0 is the set of all sections in V vanishing at0 = eG0.

I The semidirect product G n V can be replaced by anyextension of G by means of the abelian subgroup V .

Extensions by abelian subgroups

I Let N = G/H and π : E → N be an invariant one-dimensionalvector bundle (determined by 1-dimensional representations ofH).

I Let F(π) be the space of all smooth sections of π. Considerthe natural action of G on F(π):

(g .α)(x) = g .α(g−1x), α ∈ F(π), g ∈ G , x ∈ N.

I Let V be any finite-dimensional submodule of the G -moduleF(π) and let G = G n V be the semidirect product of G andV . The group G acts naturally on E :

(g , σ).p = g .p + σ(g .π(p)).

This action is transitive on E , if V contains any non-zerosection. The stationary subgroup G0 of this action is equal toG0 n V0, where V0 is the set of all sections in V vanishing at0 = eG0.

I The semidirect product G n V can be replaced by anyextension of G by means of the abelian subgroup V .

Proofs: algebraic techniques

I Transitive actions are encoded as pairs of Lie algebras (g, g0),where g0 is effective, ie. does not contain any non-zero idealsof g.

I A subalgebra g0 is called almost effective, if it does notcontain non-zero characteristic ideals of g.

I Frattini subalgebra φ(g) is an intersection of all maximalsubalgebras in g. It is a characteristic ideal.

I Based on the theory of Frattini subalgebras (E. Stitzinger,1970), we prove that if g0 is maximal and almost effective,then it is either effective (= the pair (g, g0) corresponds to alocally primitive action), or it has a very special structure:

g = an (V ⊗ kn), g0 = an (V ⊗ kn−1),

where V is an irreducible a-module. This result holds over anyfield k of characteristic 0.

I Using this technique we prove the above description of allinvariant 1-dimensional foliations on homogeneous spaces.

Proofs: algebraic techniques

I Transitive actions are encoded as pairs of Lie algebras (g, g0),where g0 is effective, ie. does not contain any non-zero idealsof g.

I A subalgebra g0 is called almost effective, if it does notcontain non-zero characteristic ideals of g.

I Frattini subalgebra φ(g) is an intersection of all maximalsubalgebras in g. It is a characteristic ideal.

I Based on the theory of Frattini subalgebras (E. Stitzinger,1970), we prove that if g0 is maximal and almost effective,then it is either effective (= the pair (g, g0) corresponds to alocally primitive action), or it has a very special structure:

g = an (V ⊗ kn), g0 = an (V ⊗ kn−1),

where V is an irreducible a-module. This result holds over anyfield k of characteristic 0.

I Using this technique we prove the above description of allinvariant 1-dimensional foliations on homogeneous spaces.

Proofs: algebraic techniques

I Transitive actions are encoded as pairs of Lie algebras (g, g0),where g0 is effective, ie. does not contain any non-zero idealsof g.

I A subalgebra g0 is called almost effective, if it does notcontain non-zero characteristic ideals of g.

I Frattini subalgebra φ(g) is an intersection of all maximalsubalgebras in g. It is a characteristic ideal.

I Based on the theory of Frattini subalgebras (E. Stitzinger,1970), we prove that if g0 is maximal and almost effective,then it is either effective (= the pair (g, g0) corresponds to alocally primitive action), or it has a very special structure:

g = an (V ⊗ kn), g0 = an (V ⊗ kn−1),

where V is an irreducible a-module. This result holds over anyfield k of characteristic 0.

I Using this technique we prove the above description of allinvariant 1-dimensional foliations on homogeneous spaces.

Proofs: algebraic techniques

I Transitive actions are encoded as pairs of Lie algebras (g, g0),where g0 is effective, ie. does not contain any non-zero idealsof g.

I A subalgebra g0 is called almost effective, if it does notcontain non-zero characteristic ideals of g.

I Frattini subalgebra φ(g) is an intersection of all maximalsubalgebras in g. It is a characteristic ideal.

I Based on the theory of Frattini subalgebras (E. Stitzinger,1970), we prove that if g0 is maximal and almost effective,then it is either effective (= the pair (g, g0) corresponds to alocally primitive action), or it has a very special structure:

g = an (V ⊗ kn), g0 = an (V ⊗ kn−1),

where V is an irreducible a-module. This result holds over anyfield k of characteristic 0.

I Using this technique we prove the above description of allinvariant 1-dimensional foliations on homogeneous spaces.

Proofs: algebraic techniques

I Transitive actions are encoded as pairs of Lie algebras (g, g0),where g0 is effective, ie. does not contain any non-zero idealsof g.

I A subalgebra g0 is called almost effective, if it does notcontain non-zero characteristic ideals of g.

I Frattini subalgebra φ(g) is an intersection of all maximalsubalgebras in g. It is a characteristic ideal.

I Based on the theory of Frattini subalgebras (E. Stitzinger,1970), we prove that if g0 is maximal and almost effective,then it is either effective (= the pair (g, g0) corresponds to alocally primitive action), or it has a very special structure:

g = an (V ⊗ kn), g0 = an (V ⊗ kn−1),

where V is an irreducible a-module. This result holds over anyfield k of characteristic 0.

I Using this technique we prove the above description of allinvariant 1-dimensional foliations on homogeneous spaces.

Describing the extensions: cohomology

I Let (g, g0) be an effective pair of Lie algebras. A pair (V ,V0)is called an effective (g , g0)-module, if V is any g-module,g0V0 ⊂ V0 and V0 does not contain any non-zero submodulesof the g-module V .

I Extensions of (g, g0) by means of (V ,V0) are defined as exactsequences:

0→ (V ,V0)→ (g, g0)→ (g, g0)→ 0.

I By analogy to the standard extensions, they are described bythe 2nd cohomology H2((g, g0), (V ,V0)) of the subcomplex:

C k((g, g0), (V ,V0)) ⊂ C k(g,V ),

α ∈ C k((g, g0), (V ,V0)) iff α(g0, . . . , g0) ⊂ V0.

Describing the extensions: cohomology

I Let (g, g0) be an effective pair of Lie algebras. A pair (V ,V0)is called an effective (g , g0)-module, if V is any g-module,g0V0 ⊂ V0 and V0 does not contain any non-zero submodulesof the g-module V .

I Extensions of (g, g0) by means of (V ,V0) are defined as exactsequences:

0→ (V ,V0)→ (g, g0)→ (g, g0)→ 0.

I By analogy to the standard extensions, they are described bythe 2nd cohomology H2((g, g0), (V ,V0)) of the subcomplex:

C k((g, g0), (V ,V0)) ⊂ C k(g,V ),

α ∈ C k((g, g0), (V ,V0)) iff α(g0, . . . , g0) ⊂ V0.

Describing the extensions: cohomology

I Let (g, g0) be an effective pair of Lie algebras. A pair (V ,V0)is called an effective (g , g0)-module, if V is any g-module,g0V0 ⊂ V0 and V0 does not contain any non-zero submodulesof the g-module V .

I Extensions of (g, g0) by means of (V ,V0) are defined as exactsequences:

0→ (V ,V0)→ (g, g0)→ (g, g0)→ 0.

I By analogy to the standard extensions, they are described bythe 2nd cohomology H2((g, g0), (V ,V0)) of the subcomplex:

C k((g, g0), (V ,V0)) ⊂ C k(g,V ),

α ∈ C k((g, g0), (V ,V0)) iff α(g0, . . . , g0) ⊂ V0.

Classification of 3-dimensional homogeneous spaces

I All spaces without invariant one-dimensional foliations:classified by Sophus Lie.

I Spaces with a one-dimensional invariant foliation: constructedfrom (known) 2-dimensional homogeneous spaces M2 = G/H:

(a) M3 = M2 ×H M1, G = G ;

(b) M3 → M2 is an invariant line bundle, G is an extension of Gor G × R∗;

(c) M3 = M2 × RP1, G = G × SL(2,R);

I Completely classified for all non-solvable G . We need onlyto consider cases (a) and (b), when G is non-solvable (10cases over C and 3 additional cases over R), or the trivial case(c) for all possible G .

Classification of 3-dimensional homogeneous spaces

I All spaces without invariant one-dimensional foliations:classified by Sophus Lie.

I Spaces with a one-dimensional invariant foliation: constructedfrom (known) 2-dimensional homogeneous spaces M2 = G/H:

(a) M3 = M2 ×H M1, G = G ;

(b) M3 → M2 is an invariant line bundle, G is an extension of Gor G × R∗;

(c) M3 = M2 × RP1, G = G × SL(2,R);

I Completely classified for all non-solvable G . We need onlyto consider cases (a) and (b), when G is non-solvable (10cases over C and 3 additional cases over R), or the trivial case(c) for all possible G .

Classification of 3-dimensional homogeneous spaces

I All spaces without invariant one-dimensional foliations:classified by Sophus Lie.

I Spaces with a one-dimensional invariant foliation: constructedfrom (known) 2-dimensional homogeneous spaces M2 = G/H:

(a) M3 = M2 ×H M1, G = G ;

(b) M3 → M2 is an invariant line bundle, G is an extension of Gor G × R∗;

(c) M3 = M2 × RP1, G = G × SL(2,R);

I Completely classified for all non-solvable G . We need onlyto consider cases (a) and (b), when G is non-solvable (10cases over C and 3 additional cases over R), or the trivial case(c) for all possible G .

Example I

I One of the SL(2)-actions on the plane is infinitesimally givenby the Lie algebra g:

〈∂x , 2x∂x − ∂y , x2∂x − x∂y 〉

I There is a one-parameter family of invariant line bundles Lα.The action of g on the sections of Lα is generated by lineardifferential operators:

〈∂x , 2x∂x − ∂y , x2∂x − x∂y+αe−2y 〉.I Finite-dimensional invariant subspaces exist only for α = 0

and have the form:

Vm = 〈x iemy | 0 ≤ i ≤ m〉, m ≥ 0.

I We get the following Lie algebras of vector fields:

〈∂x , 2x∂x − ∂y , x2∂x − x∂y+εe−2y∂z , xiemy∂z | 0 ≤ i ≤ m〉.

Iε = 0: trivial extension;ε = 1: non-trivial extension.

Example I

I One of the SL(2)-actions on the plane is infinitesimally givenby the Lie algebra g:

〈∂x , 2x∂x − ∂y , x2∂x − x∂y 〉I There is a one-parameter family of invariant line bundles Lα.

The action of g on the sections of Lα is generated by lineardifferential operators:

〈∂x , 2x∂x − ∂y , x2∂x − x∂y+αe−2y 〉.

I Finite-dimensional invariant subspaces exist only for α = 0and have the form:

Vm = 〈x iemy | 0 ≤ i ≤ m〉, m ≥ 0.

I We get the following Lie algebras of vector fields:

〈∂x , 2x∂x − ∂y , x2∂x − x∂y+εe−2y∂z , xiemy∂z | 0 ≤ i ≤ m〉.

Iε = 0: trivial extension;ε = 1: non-trivial extension.

Example I

I One of the SL(2)-actions on the plane is infinitesimally givenby the Lie algebra g:

〈∂x , 2x∂x − ∂y , x2∂x − x∂y 〉I There is a one-parameter family of invariant line bundles Lα.

The action of g on the sections of Lα is generated by lineardifferential operators:

〈∂x , 2x∂x − ∂y , x2∂x − x∂y+αe−2y 〉.I Finite-dimensional invariant subspaces exist only for α = 0

and have the form:

Vm = 〈x iemy | 0 ≤ i ≤ m〉, m ≥ 0.

I We get the following Lie algebras of vector fields:

〈∂x , 2x∂x − ∂y , x2∂x − x∂y+εe−2y∂z , xiemy∂z | 0 ≤ i ≤ m〉.

Iε = 0: trivial extension;ε = 1: non-trivial extension.

Example I

I One of the SL(2)-actions on the plane is infinitesimally givenby the Lie algebra g:

〈∂x , 2x∂x − ∂y , x2∂x − x∂y 〉I There is a one-parameter family of invariant line bundles Lα.

The action of g on the sections of Lα is generated by lineardifferential operators:

〈∂x , 2x∂x − ∂y , x2∂x − x∂y+αe−2y 〉.I Finite-dimensional invariant subspaces exist only for α = 0

and have the form:

Vm = 〈x iemy | 0 ≤ i ≤ m〉, m ≥ 0.

I We get the following Lie algebras of vector fields:

〈∂x , 2x∂x − ∂y , x2∂x − x∂y+εe−2y∂z , xiemy∂z | 0 ≤ i ≤ m〉.

Iε = 0: trivial extension;ε = 1: non-trivial extension.

Example I

I One of the SL(2)-actions on the plane is infinitesimally givenby the Lie algebra g:

〈∂x , 2x∂x − ∂y , x2∂x − x∂y 〉I There is a one-parameter family of invariant line bundles Lα.

The action of g on the sections of Lα is generated by lineardifferential operators:

〈∂x , 2x∂x − ∂y , x2∂x − x∂y+αe−2y 〉.I Finite-dimensional invariant subspaces exist only for α = 0

and have the form:

Vm = 〈x iemy | 0 ≤ i ≤ m〉, m ≥ 0.

I We get the following Lie algebras of vector fields:

〈∂x , 2x∂x − ∂y , x2∂x − x∂y+εe−2y∂z , xiemy∂z | 0 ≤ i ≤ m〉.

Iε = 0: trivial extension;ε = 1: non-trivial extension.

Example II

I Another SL(2)-action on the plane:

〈∂x , 2x∂x − 2y∂y , x2∂x − (1 + 2xy)∂y 〉

I There is a one-parameter family of invariant line bundles Ln.The action of g on the sections of Ln is generated by lineardifferential operators:

〈∂x , 2x∂x − 2y∂y , x2∂x − (1 + 2xy)∂y+nxz〉.

I Finite-dimensional invariant subspaces exist only for n is anon-negative integer and have the form:

Vn,m = 〈∂ix(xn+m(1 + xy)m), 0 ≤ i ≤ n + 2m〉, m ≥ 0.

I Non-trivial extensions exist only if n = 0. In this case we getthe following families of Lie algebras:

〈∂x , x∂x − y∂y+∂z , x2∂x − (1 + 2xy)∂y+2x∂z , f (x , y)∂z

f ∈ V0,m1 + · · ·+ V0,mk〉.

Example II

I Another SL(2)-action on the plane:

〈∂x , 2x∂x − 2y∂y , x2∂x − (1 + 2xy)∂y 〉

I There is a one-parameter family of invariant line bundles Ln.The action of g on the sections of Ln is generated by lineardifferential operators:

〈∂x , 2x∂x − 2y∂y , x2∂x − (1 + 2xy)∂y+nxz〉.

I Finite-dimensional invariant subspaces exist only for n is anon-negative integer and have the form:

Vn,m = 〈∂ix(xn+m(1 + xy)m), 0 ≤ i ≤ n + 2m〉, m ≥ 0.

I Non-trivial extensions exist only if n = 0. In this case we getthe following families of Lie algebras:

〈∂x , x∂x − y∂y+∂z , x2∂x − (1 + 2xy)∂y+2x∂z , f (x , y)∂z

f ∈ V0,m1 + · · ·+ V0,mk〉.

Example II

I Another SL(2)-action on the plane:

〈∂x , 2x∂x − 2y∂y , x2∂x − (1 + 2xy)∂y 〉

I There is a one-parameter family of invariant line bundles Ln.The action of g on the sections of Ln is generated by lineardifferential operators:

〈∂x , 2x∂x − 2y∂y , x2∂x − (1 + 2xy)∂y+nxz〉.

I Finite-dimensional invariant subspaces exist only for n is anon-negative integer and have the form:

Vn,m = 〈∂ ix(xn+m(1 + xy)m), 0 ≤ i ≤ n + 2m〉, m ≥ 0.

I Non-trivial extensions exist only if n = 0. In this case we getthe following families of Lie algebras:

〈∂x , x∂x − y∂y+∂z , x2∂x − (1 + 2xy)∂y+2x∂z , f (x , y)∂z

f ∈ V0,m1 + · · ·+ V0,mk〉.

Example II

I Another SL(2)-action on the plane:

〈∂x , 2x∂x − 2y∂y , x2∂x − (1 + 2xy)∂y 〉

I There is a one-parameter family of invariant line bundles Ln.The action of g on the sections of Ln is generated by lineardifferential operators:

〈∂x , 2x∂x − 2y∂y , x2∂x − (1 + 2xy)∂y+nxz〉.

I Finite-dimensional invariant subspaces exist only for n is anon-negative integer and have the form:

Vn,m = 〈∂ ix(xn+m(1 + xy)m), 0 ≤ i ≤ n + 2m〉, m ≥ 0.

I Non-trivial extensions exist only if n = 0. In this case we getthe following families of Lie algebras:

〈∂x , x∂x − y∂y+∂z , x2∂x − (1 + 2xy)∂y+2x∂z , f (x , y)∂z

f ∈ V0,m1 + · · ·+ V0,mk〉.

Solvable case

I Consider N = R2 with the group of parallel translations:〈∂x , ∂y 〉.

I Every invariant vector bundle is trivial.

I The set of all sections is C∞(R2).

I Finite-dimensional invariant subspaces are of the formPα,βe

αx+βy , where Pα,β is some invariant subspace consistingof polynomials in x and y .

I Finite-dimensional invariant subspaces of polynomials is x , yare in one-to-one correspondence with ideals of finitecodimension R[∂x , ∂y ].

I Such ideals can not be explicitly parametrized by a finitenumber of parameters!

Solvable case

I Consider N = R2 with the group of parallel translations:〈∂x , ∂y 〉.

I Every invariant vector bundle is trivial.

I The set of all sections is C∞(R2).

I Finite-dimensional invariant subspaces are of the formPα,βe

αx+βy , where Pα,β is some invariant subspace consistingof polynomials in x and y .

I Finite-dimensional invariant subspaces of polynomials is x , yare in one-to-one correspondence with ideals of finitecodimension R[∂x , ∂y ].

I Such ideals can not be explicitly parametrized by a finitenumber of parameters!

Solvable case

I Consider N = R2 with the group of parallel translations:〈∂x , ∂y 〉.

I Every invariant vector bundle is trivial.

I The set of all sections is C∞(R2).

I Finite-dimensional invariant subspaces are of the formPα,βe

αx+βy , where Pα,β is some invariant subspace consistingof polynomials in x and y .

I Finite-dimensional invariant subspaces of polynomials is x , yare in one-to-one correspondence with ideals of finitecodimension R[∂x , ∂y ].

I Such ideals can not be explicitly parametrized by a finitenumber of parameters!

Solvable case

I Consider N = R2 with the group of parallel translations:〈∂x , ∂y 〉.

I Every invariant vector bundle is trivial.

I The set of all sections is C∞(R2).

I Finite-dimensional invariant subspaces are of the formPα,βe

αx+βy , where Pα,β is some invariant subspace consistingof polynomials in x and y .

I Finite-dimensional invariant subspaces of polynomials is x , yare in one-to-one correspondence with ideals of finitecodimension R[∂x , ∂y ].

I Such ideals can not be explicitly parametrized by a finitenumber of parameters!

Solvable case

I Consider N = R2 with the group of parallel translations:〈∂x , ∂y 〉.

I Every invariant vector bundle is trivial.

I The set of all sections is C∞(R2).

I Finite-dimensional invariant subspaces are of the formPα,βe

αx+βy , where Pα,β is some invariant subspace consistingof polynomials in x and y .

I Finite-dimensional invariant subspaces of polynomials is x , yare in one-to-one correspondence with ideals of finitecodimension R[∂x , ∂y ].

I Such ideals can not be explicitly parametrized by a finitenumber of parameters!

Solvable case

I Consider N = R2 with the group of parallel translations:〈∂x , ∂y 〉.

I Every invariant vector bundle is trivial.

I The set of all sections is C∞(R2).

I Finite-dimensional invariant subspaces are of the formPα,βe

αx+βy , where Pα,β is some invariant subspace consistingof polynomials in x and y .

I Finite-dimensional invariant subspaces of polynomials is x , yare in one-to-one correspondence with ideals of finitecodimension R[∂x , ∂y ].

I Such ideals can not be explicitly parametrized by a finitenumber of parameters!

Happy birthday, Eldar!