Parallelisms&Lie Connections

The aim of this article is to study rational parallelisms of algebraic varieties by means of the transcendence of their symmetries. The nature of this transcendence is measured by a Galois group built from the Picard-Vessiot theory of principal connections.


Introduction
The aim of this article is to study rational parallelisms of algebraic varieties by means of the transcendence of their symmetries. Our original motivation was to understand the possible obstructions to the third Lie theorem for algebraic Lie pseudogroups. This article is concerned with the simply transitive case. These obstructions should appear in the Galois group of certain connection associated to a Lie algebroid. However, we have written the article in the language of regular and rational parallelisms of algebraic varieties and their symmetries.
A theorem of P. Deligne says that any Lie algebra can be realized as a parallelism of an algebraic variety. This is a sort of algebraic version of the third Lie theorem. Notwithstanding, there is one main problem: given an algebraic variety with a parallelism, how far is it from being an algebraic group? Is it possible to conjugate this parallelism with the canonical parallelism of invariant vector fields on an algebraic group?
In the analytic context, from the Darboux-Cartan theorem [10, p. 212], a g-parallelized complex manifold M has a natural (G, G) structure where G is a Lie group with lie(G) = g. The obstruction to be a covering of G, as manifold with a (G, G) structure, is contained in a monodromy group [10, p. 130]. In [12], Wang proved that parallelized compact complex manifolds are biholomorphic to quotients of complex Lie groups by discrete cocompact subgroups. This result has been extended by Winkelmann in [13,14] for some open complex manifolds.
In this article we address the problem of classification of rational parallelisms on algebraic varieties up to birational transformations. Such a classification seems impossible in the algebraic category but we prove a criterion to ensure that a parallelized algebraic variety is isogenous to an algebraic group. Summarizing, we pursue the following plan: We regard infinitesimal symmetries of a rational parallelism as horizontal sections of a connection that we call the reciprocal Lie connection. This connection has a Galois group which is represented as a group of internal automorphisms of a Lie algebra. The obstruction to the algebraic conjugation to an algebraic group, under some assumptions, appear in the Lie algebra of this Galois group.
In Section 2 we introduce the basic definitions; several examples of parallelisms are given here. In Section 3 we study the properties of connections on the tangent bundle whose local analytic horizontal sections form a sheaf of Lie algebras of vector fields. We call them Lie connections. They always come by pairs, and they are characterized by having vanishing curvature and constant torsion (Proposition 3.10). We see that each rational parallelism has an attached pair of Lie connections, one of them with trivial Galois group. We compute the Galois groups of some parallelisms given in examples (Proposition 3.14), and prove that any algebraic subgroup of PSL 2 (C) appears as the differential Galois group of a sl 2 (C)-parallelism (Theorem 3.16). Section 4 is devoted to the construction of the isogeny between a g-parallelized variety and an algebraic group G whose Lie algebra is g. In order to do this, we introduce the Darboux-Cartan connection, a G-connection whose horizontal sections are parallelism conjugations. We prove that if g is centerless then the Darboux-Cartan connection and the reciprocal Lie connection have isogenous Galois groups. We prove that the only centerless Lie algebras g such that there exists a g-parallelism with a trivial Galois group are algebraic Lie algebras, i.e., Lie algebras of algebraic groups. In particular this allows us to give a criterion for a parallelized variety to be isogenous to an algebraic group. The vanishing of the Lie algebra of the Galois group of the reciprocal connection is a necessary and sufficient condition for a parallelized variety to be isogenous to an algebraic group: Theorem 4.6. Let g be a centerless Lie algebra. An algebraic variety (M, ω) with a rational parallelism of type g is isogenous to an algebraic group if and only if gal(∇ rec ) = {0}.
Let g be a Lie algebra of dimension r. We fix a basis A 1 , . . . , A r of g, and the following notation for the associated structure constants [A i , A j ] = k λ k ij A k . A parallelism of type g of M is a realization of the Lie algebra g as a Lie algebra of pointwise linearly independent vector fields in M . More precisely: Definition 2.1. A regular parallelism of type g in M is a Lie algebra morphism, ρ : g → X [M ] such that ρA 1 (x), . . . , ρA r (x) form a basis of T x M for any point x of M .
Example 2.2. Let G be an algebraic group and g be its Lie algebra of left invariant vector fields. Then the natural inclusion g ⊂ X[G] is a regular parallelism of G. The Lie algebra g rec of right invariant vector fields is another regular parallelism of the same type. Let invariant and right invariant vector fields commute, hence, an algebraic group is naturally endowed with a pair of commuting parallelisms of the same type.
From Example 2.2, it is clear that any algebraic Lie algebra is realized as a parallelism of some algebraic variety. On the other hand, Theorem B.1 due to P. Deligne and published in [7], ensures that any Lie algebra is realized as a regular parallelism of an algebraic variety. Analogously, we have the definitions of rational and local analytic parallelism. Note that a rational parallelism in M is a regular parallelism in a Zariski open subset M ⊆ M .
There is dual definition, equivalent to that of parallelism. This is more suitable for calculations. Definition 2.3. A regular parallelism form (or coparallelism) of type g in M is a g-valued 1-form ω ∈ Ω 1 [M ] ⊗ C g such that: (1) For any x ∈ M , ω x : T x M → g is a linear isomorphism. Analogously, we define local analytic and rational coparallelism of type g in M . It is clear that each coparallelism induces a parallelism, and reciprocally, by the relation ω(ρ(A)) = A. Thus, there is a natural equivalence between the notions of parallelism and coparallelism. From now on we fix ρ and ω equivalent parallelism and coparallelism of type g on M .
Example 2.4. Let G be an algebraic group and g be the Lie algebra of left invariant vector fields in G. Then the structure form ω is the coparallelism corresponding to the parallelism of Example 2.2.
Example 2.5. Let g = A 1 , A 2 be the 2-dimensional Lie algebra with commutation relation The vector fields define a regular parallelism via ρ(A i ) = X i of C 2 . The associated parallelism form is Example 2.6 (Malgrange). Let g = A 1 , A 2 , A 3 be the 3-dimensional Lie algebra with commutation relations with α, β, non zero complex numbers. In particular, if α/β is not rational then g is not the Lie algebra of an algebraic group. The vector fields define a regular parallelism via ρ(A i ) = X i of C 3 . The associated parallelism form is Definition 2.7. Let (M, ω) and (N, θ) be algebraic manifolds with coparallelisms of type g. We say that they are isogenous if there is an algebraic manifold (P, η) with a coparallelism of type g and dominant maps f : P → M and g : P → N such that f * (ω) = g * (θ) = η.
Clearly, the notion of isogeny of parallelized varieties extends that of isogeny of algebraic groups.
be the quotient by the action of H on the left side. The structure form ω in G is left-invariant and then it is projectable by π. Then, θ = π * (ω) is a regular parallelism form in M .
Example 2.10. Combining Examples 2.8 and 2.9, let H ⊂ G be a finite subgroup and f : M → H \G be a dominant rational map between manifolds of the same dimension. Then θ = f * (π * (ω)) is a rational parallelism form in M .
Example 2.11. By application of Example 2.10 to the case of the multiplicative group we obtain rational multiples of logarithmic forms in CP 1 , p q df f where f ∈ C(z). Thus, rational multiples of logarithmic forms in CP 1 are the rational coparallelisms isogenous to that of the multiplicative group.
Example 2.12. By application of Example 2.10 to the case of the additive group we obtain the exact forms in CP 1 , dF where F ∈ C(z). Thus, the exact forms in CP 1 are the rational coparallelisms isogenous to that of the additive group. Remark 2.14. We can see also Example 2.13 as a coparallelism. Let π : g → h be the projection given by the vector space decomposition g = h ⊕ h . Since π • ω is left invariant form in G, it is projectable by π. Hence, there is a form ω in M such that π * ω = π • ω. This form ω is the corresponding coparallelism.
3 Associated Lie connection

Reciprocal connections
Let ∇ be a linear connection (rational or regular) on T M . The reciprocal connection is defined as From this definition it is clear that the difference ∇ − ∇ rec = Tor ∇ is the torsion tensor, Tor ∇ = − Tor ∇ rec and (∇ rec ) rec = ∇.

Connections and parallelisms
Let ω be a coparallelism of type g in M and ρ its equivalent parallelism. Denote by X i the basis of vector fields in M such that ω( X i ) = A i is a basis of g.
Definition 3.1. The connection ∇ associated to the parallelism ω is the only linear connection in M for which ω is a ∇-horizontal form.
Clearly ∇ is a flat connection and the basis { X i } is a basis of the space of ∇-horizontal vector fields. In this basis ∇ has vanishing Christoffel symbols Let us compute some infinitesimal symmetries of ω. A vector field Y is an infinitesimal symmetry of ω if Lie Y ω = 0, or equivalently, if it commutes with all the vector fields of the parallelism Lemma 3.2. Let ∇ be the connection associated to the parallelism ω. Then for any vector field Y and any j = 1, . . . , r Thus, Y is an infinitesimal symmetry of ω if and only if it is a horizontal vector field for the reciprocal connection ∇ rec .
The above considerations also give us the Christoffel symbols for ∇ rec in the basis { X i } ∇ rec i.e., the Christoffel symbols of ∇ rec are the structure constants of the Lie algebra g.
Lemma 3.3. Let ∇ be the connection associated to a coparallelism in M . Then, ∇ rec is flat, and the Lie bracket of two ∇ rec -horizontal vector fields is a ∇ rec -horizontal vector field.
Proof . The flatness and the preservation of the Lie bracket by ∇ rec are direct consequences of the Jacobi identity. Let us compute the curvature Let us compute the Lie bracket for Y and Z ∇ rec -horizontal vector fields Lemma 3.4. Let x ∈ M be a regular point of the parallelism form ω. The space of germs at x of horizontal vector fields for ∇ rec is a Lie algebra isomorphic to g. Moreover, let Y 1 , . . . , Y r be horizontal vector fields with initial conditions the λ i,j are the structure constants of the Lie algebra generated by the X i .
Proof . We can write the vector fields Y i as linear combinations of the vector fields X i : a ji X j . The matrix (a ij ) satisfies the differential equation On the other hand, a βj a αi λ γ αβ X γ .
Taking values at x, we obtain Example 3.5. Let G be an algebraic group with Lie algebra g. As seen in Example 2.4 the Maurer-Cartan structure form ω is a coparallelism in G. Let ∇ be the connection associated to this coparallelism. There is another canonical coparallelism, the right invariant Maurer-Cartan structure form ω rec , let us consider i : G → G the inversion map, As may be expected, the connection associated to the coparallelism ω rec is ∇ rec . Right invariant vector fields in G are infinitesimal symmetries of left invariant vector fields and vice versa. In this case, the horizontal vector fields of ∇ and ∇ rec are regular vector fields.
As shown in the next three examples, symmetries of a rational parallelism are not in general rational vector fields. Example 3.6. Let us consider the Lie algebra g and the coparallelism ω = A 1 dx+(A 2 −xA 1 )dy, of Example 2.5. Let ∇ be its associated connection. In cartesian coordinates, the only nonvanishing Christoffel symbol of the reciprocal connection is Γ 1 21 = −1. A basis of ∇ rec -horizontal vector fields is Note that they coincide with X 1 , X 2 at the origin point and Y 1 , Example 3.7. Let us consider the Lie algebra g and the coparallelism ω = (A 1 − αyA 2 − βzA 3 )dx + A 2 dy + A 3 dz of Example 2.6. Let ∇ be its associated connection. In cartesian coordinates, the only non-vanishing Christoffel symbols of the reciprocal connection are A basis of ∇ rec -horizontal vector fields is Note that they coincide with X 1 , X 2 , X 3 at the origin point and Example 3.8. Let us consider the Lie algebra g of Example 2.6 and the coparallelism Let ∇ be its associated connection. In cartesian coordinates, the only non-vanishing Christoffel symbols of the reciprocal connection are A basis of ∇ rec -horizontal vector fields on a simply connected open subspace U ⊂ C * × C 2 is

Lie connections
The connections ∇ and ∇ rec associated to a coparallelism ω of type g are particular cases of the following definition. Given a Lie connection ∇ in M , there is a r-dimensional Lie algebra g such that the space of germs of horizontal vector fields at a regular point x is a Lie algebra isomorphic to g. We will say that ∇ is a Lie connection of type g. The following result gives several algebraic characterizations of Lie connections: (1) ∇ is a Lie connection; (2) ∇ rec is a Lie connection; (3) ∇ is flat and has constant torsion, ∇ Tor ∇ = 0; (4) ∇ and ∇ rec are flat.
Proof . Let us first see (1)⇔ (2). Let ∇ be a Lie connection. Around each point of the domain of ∇ there is a parallelism, by possibly transcendental vector fields, such that ∇ is its associated connection. Then, Lemma 3.3 states (1)⇒(2). Taking into account that (∇ rec ) rec = ∇ we have the desired equivalence.
Let us see now that (1)⇔(3). Let us assume that ∇ is a flat connection. For any three vector fields X, Y , Z in M we have Let us assume that Y and Z are ∇-horizontal vector fields. Then, we have and the previous equality yields Thus, we have that ∇ Tor ∇ vanishes if and only if the Lie bracket of any two ∇-horizontal vector fields is also ∇-horizontal. This proves (1)⇔(3).
Finally, let us see (1)⇔(4). It is clear that (1) implies (4) so we only need to see (4)⇒(1). Assume ∇ and ∇ rec are flat. Then, locally, there exist a basis { X i } of ∇-horizontal vector fields and a basis { Y i } of ∇ rec -horizontal vector fields. By the definition of the reciprocal connection, we have that a vector field X is ∇-horizontal if and only if it satisfies [ X, Y i ] = 0 for i = 1, . . . , r. By the Jacobi identity we have The Lie brackets X i , X j are also ∇-horizontal and ∇ is a Lie connection. Lemma 3.11. Let ∇ be a Lie connection on M . Let x be a regular point and X 1 , . . . , X r and Y 1 , . . . , Y r be basis of horizontal vector field germs on M for ∇ and ∇ rec respectively with same initial conditions X i (x) = Y i (x). Then It follows that ∇ and ∇ rec are of the same type g.
Proof . By definition ∇ is the connection associated to the local analytic parallelism given by the basis { X i } of horizontal vector fields. Then we apply Lemma 3.4 in order to obtain the desired conclusion.

Some results on Lie connections by means of Picard-Vessiot theory
Definitions and general results concerning the Picard-Vessiot theory of connections are given in Appendix A. Proof . We will use the notations of Section A.6: R 1 (T M ) is the GL n (C)-principal bundle associated to T M and F is the GL n (C)-invariant foliation on R 1 (T M ) given by graphs of local basis of ∇-horizontal sections. The Galois group Gal(∇ rec ) can be computed as soon as we know the Zariski closure L of a leaf L of the induced foliation F on R 1 (T M ). Gal(∇ rec ) is finite is and only if L = L and is {1} if and only if L is the graph of a rational section M → R 1 (T M ). This means that there exists a basis of rational ∇ rec -horizontal sections. These sections give the desired parallelism. Proof . Let us choose a point x ∈ M regular for ∇ and a basis A 1 , . . . , A r of g, i.e., a basis Y 1 , . . . , Y r of local ∇-horizontal section of T M at x.
Using notation of Section A.6, we will identify R 1 (T x M ) with the set of isomorphisms of linear spaces σ : g → T x M ; now Gal(∇) ⊆ GL(g). Because of the construction of g, we have a canonical point in where X i is the horizontal section such that X i (m) = σA i . These functions are regular functions on R 1 (T M ). Moreover they are constant and equal to the constant structures on the Zariski closure of the leaf passing through σ o . The Galois group is the stabilizer of this leaf then the functions H k i,j are invariant under the action of the Galois group, i.e., the Galois group preserves the Lie bracket.
Proposition 3.14. Let h be a Lie sub-algebra of the Lie algebra of some algebraic group and let G be the smallest algebraic subgroup such that lie(G) = g ⊃ h . Assume the existence of an algebraic subgroup H of G whose Lie algebra h is supplementary to h in g, g = h ⊕ h . Let us consider the following objects: (c) the Lie algebras of right invariant vector fields where i is the inverse map on G.
Then, the following statements are true: (i) h is an ideal of g (equivalently h rec is an ideal of g rec ); (ii) h is commutative (equivalently H is virtually abelian); (iii) the adjoint action of G on g rec preserves h rec and thus gives, by restriction, a morphism Adj : G → Aut(h rec ); (iv) The Galois group of the connection ∇ rec is Adj(H) ⊆ Aut(h rec ) and thus virtually abelian. By continuity of the adjoint action in the Zariski topology, we have that h is preserved by the adjoint action of all elements of G. This proves statement (iii). In order to prove the last statement in the proposition we have to construct a Picard-Vessiot extension for the connection ∇ rec . Let us consider a basis {A 1 , . . . , A m } of h and letĀ i be the projection π * (A i ). We have an extension of differential fields whereD stands for the C(M )-vector space of derivations spanned byĀ 1 , . . . ,Ā m and D stands for the C(G)-vector space of derivations spanned by A 1 , . . . , A m (see Appendix A for our conventions on differential fields). The projection π is a principal H-bundle. Any rational first integral of {A 1 , . . . , A m } is constant along H and thus it is necessarily a complex number. Thus, the above extension has no new constants and it is strongly normal in the sense of Kolchin, with Galois group H. Note that the differential field automorphism corresponding to an element h ∈ H is the pullback of functions by the left translation L −1 h , that is, (hf )(g) = f h −1 g . The horizontal sections for the connection ∇ rec are characterized by the differential equations , and clearly satisfy equation (3.1). Thus, the Picard-Vessiot extension of ∇ rec is spanned by the functions f ij and it is embedded, as a differential field, in C(G). Let us denote such extension by L. We have a chain of extensions By Galois correspondence, the Galois group of ∇ rec is a quotient H/K where K is the subgroup of elements of H that fix, by left translation, the functions f ij . In order to prove statement (iv) we need to check that this group K is the kernel of the morphism Adj. Let us note that the image under the adjoint action by g ∈ G of an element B ∈ h rec is given by the left translation, Adj(g)(B) = L g * (B). This transformation makes sense for any derivation of C[G], and thus we have an action of G on X(G). Let us take h in the kernel of Adj, thus, Adj(h)(B j ) = B j for any index j. Applying the transformation L h * to the expression of B i

as linear combination of the left invariant vector fields
The coefficients of B i as linear combination of the A j are unique, and thus, h(f ij ) = f ij we conclude that h is an automorphism fixing L. On the other hand, let us take

Some examples of sl 2 -parallelisms
We will construct some parallelized varieties as subvarieties of the arc space of the affine line A 1 C . This family of examples show how to realize every subgroup of PSL 2 (C) as the Galois group of the reciprocal Lie connection.
3.5.1 The arc space of the af f ine line and its  In our special case, the arc space of the affine line A 1 C with affine coordinate z, is the space of all formal power series z = z (i) x i i! . It will be denoted by L , its ring of regular functions is Let X be the Lie algebra of formal vector fields C[[x]] ∂ ∂x . One can build a rational form σ : T L → X in following way (see [5,Section 2] This form is rational and is an isomorphism between T p L and X satisfying dσ = − 1 2 [σ, σ] and (L f ) * σ = σ for any biholomorphism f . This means that σ provides an action of X commuting with the lift of biholomorphisms. This form seems to be a coparallelism but it is not compatible with the natural structure of pro-variety of L and X: is a derivation of degree +1 with respect to the pro-variety structure of L . The total derivation above will be denoted by E −1 . This gives a differential structure to the ring C[L ].

The parallelized varieties
Let ν ∈ C(z) be a rational function, f be the rational function on the arc space given by the Schwarzian derivative Lemma 3.15. The zero set V of I is a dimension 3 subvariety of L and ω(T V ) = sl 2 (C) ⊂ X. This provides a sl 2 -parallelism on V .
Proof . One can compute explicitly this parallelism using z (0) , z (1) and z (2) asétale coordinates on a Zariski open subset of V . Let us first compute the sl 2 action on L . The standard inclusion of sl 2 in X is given by Their actions on L are given by (1) and z (2) one gets They form a rational sl 2 -parallelism on C 3 depending on the choice of a rational function in one variable.
3.5.3 Symmetries and the Galois group of the reciprocal connection Theorem 3.16. Any algebraic subgroup of PSL 2 (C) can be realized as the Galois group of the reciprocal connection of a parallelism of C 3 .
Proof . A direct computation shows that z → ϕ(z) is an holomorphic function satisfying if and only if its prolongation L ϕ : z → ϕ( z) on the space L preserves V and preserves each of the vector fields E −1 , E 0 and E 1 .
Taking infinitesimal generators of this pseudogroup, one gets for any local analytic solution of the linear equation The equation (3.2) ensures that L X is tangent to V . The invariance of σ, (L X) * σ = 0, ensures that L X commutes with the sl 2 -parallelism given above. This means that for any solution a of (3.2) the vector field Then the linear differential system of flat section for the reciprocal connection reduces to the linear equation (3.2). This equation is the second symmetric power of y = ν(z)y. If G ⊂ SL 2 (C) is the Galois group of y = ν(z)y then the image of its second symmetric power representation s 2 : G → Sym 2 (C 2 ) is the Galois group of (3.2). The kernel of this representation is {Id, −Id} then the Galois group of (3.2) is an algebraic subgroup of PSL 2 (C).
Let us remark that, as it follows from its definition, the Galois group of an equation contains the monodromy group. Moreover one can determine the monodromy group of classical differential equations. Hypergeometric equations depend on three complex numbers (a, b, c) and is equivalent to and These two equations have the same projectivized Galois group in PGL 2 (C). Any algebraic subgroup of PGL 2 (C) will be realized by an appropriate choice of (a, b, c).

The whole group
For a = b = 1/2, c = 1, the hypergeometric equation is the Picard-Fuchs equation of Legendre family. Its monodromy group is Γ(2) ⊂ SL 2 (Z) and is Zariski dense in SL 2 (C).

The dihedral subgroups
For c = 1/2 and a + b = 0, a basis of solution is given by The monodromy group is a dihedral group in GL 2 (C) whose quotients give dihedral subgroups of PGL 2 (C).

The tetrahedral subgroup
This group is the monodromy group of hypergeometric equation for = 1/3, m = 1/2 and n = 1/3. A basis of solution is given by

The octahedral subgroup
This group is the monodromy group of hypergeometric equation for = 1/2, m = 1/3 and n = 1/4. A basis of solution is given by

The icosaedral subgroup
This group is the monodromy group of hypergeometric equation for = 1/2, m = 1/3 and n = 1/5. As icosahedral group is not solvable, the solution space is not described using formulas as simple as in preceding examples. Let us consider the trivial principal bundle π : P = G × M → M . In this bundle we consider the action of G by right translations (g, x) * g = (gg , x). Let Θ be the g-valued form Θ = θ − ω in P . In any case, the necessary and sufficient condition for (M, ω) and (G, θ) to be isogenous parallelized varieties is that gal(Θ) = {0}.

Darboux-Cartan connection and Picard-Vessiot
Note that the coparallelism ω gives a rational trivialization of T M as the trivial bundle of fiber g.
In T M we have defined the connection ∇ rec whose horizontal vector fields are the symmetries of the parallelism. On the other hand, G acts in g by means of the adjoint action. The Cartan-Darboux connection induces then a connection ∇ adj in the associated trivial bundle g × M of fiber g.
Proposition 4.2. The map is a birational conjugation of the linear connections ∇ rec and ∇ adj .
Proof . It is clear that the mapω is birational. Let us consider {A 1 , . . . , A m } a basis of g. Let ρ : g → X(M ) be the parallelism associated to the parallelism ω and let us define X i = ρ(A i ).
Then {X 1 , . . . , X n } is a rational frame in M and the mapω conjugates the vector field X i with the constant section A i of the trivial bundle of fiber g. By definition of the reciprocal connection On the other hand, by definition of the adjoint action and application of the covariant derivative as in equation (A.1) of Appendix A.7 we obtain Therefore we have thatω is a rational morphism of linear connections that conjugates ∇ rec with ∇ adj .
(c) It is a direct consequence of (b). The kernel of adj : g → End(g) is the center of g.

Algebraic Lie algebras
Let us consider (M, ω) a rational coparallelism of type g with g a centerless Lie algebra. We do not assume a priori that g is an algebraic Lie algebra. The connection ∇ rec is, as said in Proposition 4.2, conjugated to the connection in g × M induced by the adjoint action. Note that, in order to define this connection we do not need the group operation but just the Lie bracket in g. We have an exact sequence where g is the derived algebra [g, g]. Since the Galois group acts by adjoint action, we have that g × M is stabilized by the connection ∇ rec and thus we have an exact sequence of connections Lemma 4.4. The Galois group of ∇ ab is the identity, therefore ∇ ab has a basis of rational horizontal sections.
Proof . By definition, the action of g in g ab vanishes. Thus, the constant functions M → g ab are rational horizontal sections.

Lemma 4.5.
Let ω be a rational coparallelism of M of type g with g a centerless Lie algebra. If gal(∇ rec ) = {0} then g is an algebraic Lie algebra.
Proof . Assume g is a linear Lie algebra and et E be the smallest algebraic subgroup such that Lie(E) = e ⊃ g. We may assume that E is also centerless. Let A 1 , . . . , A r be a basis of g, for i = 1, . . . , r, X i = ω −1 (A i ). Complete with B 1 , . . . , B p in such way that A 1 , . . . , A r , B 1 , . . . , B p is a basis of e. We consider in E × M the distribution spanned by the vector fields A i + X i . This is a E-principal connection called ∇.
If ϕ ∈ gal(∇) then for any X ∈ g, [X, B i ] ∈ g thus 0 = ϕ[X, B i ] = [X, ϕB i ] and ϕB i commute with g. From the second point above ϕB ∈ g. By hypothesis ϕB i = 0 and gal(∇) = {0}. The projection on E of an algebraic leaf of ∇ gives an algebraic leaf for the foliation of E by the left translation by g. This proves the lemma.
Theorem 4.6. Let g be a centerless Lie algebra. An algebraic variety (M, ω) with a rational parallelism of type g is isogenous to an algebraic group if and only if gal(∇ rec ) = {0}.
Proof . It follows directly from Lemma 4.5 and Corollary 4.3.
Corollary 4.7. Let g be a centerless Lie algebra. Any algebraic variety endowed with a pair of commuting rational parallelisms of type g is isogenous to an algebraic group endowed with its two canonical parallelisms of left and right invariant vector fields.
Proof . Just note that to have a pair of commuting parallelism is a more restrictive condition than having a parallelism with vanishing Lie algebra of the Galois group of its reciprocal connection.
This result can be seen as an algebraic version of Wang result in [12]. It gives the classification of algebraic varieties endowed with pairs of commuting parallelisms. Assuming that the Lie algebra is centerless is not a superfluous hypothesis, note that the result clearly does not hold for abelian Lie algebras. There are rational 1-forms in CP 1 that are not exact (isogenous to (C, dz)) nor logarithmic (isogenous to (C * , d log(z))). In these examples, the pair of commuting parallelisms is given by twice the same parallelism.
Remark 4.8. Let (M, ω, ω ) be a manifold endowed with a pair of commuting parallelism forms of type g, a centerless Lie algebra. From Lemma 4.5 we have that g is an algebraic Lie algebra. We can construct the algebraic group enveloping g as follows. We consider the adjoint action adj : g → End(g).
The algebraic group enveloping g is identified with the algebraic subgroup G of Aut(g) whose Lie algebra is adj(g). From Corollary 4.3(a), we have that Gal(Θ) = {e}. Thus, there is a rational map f : M → G such that f * (θ) = ω, where θ is the Maurer-Cartan form of G. We can express explicitly this map in terms of the commuting parallelism forms. For each x ∈ M in the domain of regularity of the parallelisms, ω(x) and ω (x) are isomorphisms of T x M with g. We define  • ∇ F -horizontal sections commute with the parallelism, it is the reciprocal Lie connection; • if leaves of F are algebraic then ∇ F -horizontal section are algebraic.

Some homogeneous varieties
The notion of isogeny can be extended beyond the simply-transitive case. Let us consider a complex Lie algebra g. An infinitesimally homogeneous variety of type g is a pair (M, s) consisting of a complex smooth irreducible variety M and a finite-dimensional Lie algebra s ⊂ X(M ) isomorphic to g.
As before, we are interested in conjugation by rational and algebraic maps so that, whenever necessary, we replace M by a suitable Zariski open subset. In this context, we say that a dominant rational map f : M 1 M 2 between varieties of the same dimension conjugates the infinitesimally homogeneous varieties (M 1 , s 1 ) and (M 2 , s 2 ) if f * (s 2 ) = s 1 . We say that (M 1 , s 1 ) and (M 2 , s 2 ) are isogenous if they are conjugated to the same infinitesimally homogeneous space of type g.
Let G be an algebraic group over C, K an algebraic subgroup, lie(G) its Lie algebra of left invariant vector fields and lie(G) rec its Lie algebra of right invariant vector fields. A natural example of infinitesimally homogeneous space are the homogeneous spaces G/H endowed with the induced action of the Lie algebra lie(G) rec . We want to recognize when a infinitesimally homogeneous space is isogenous to an homogeneous space. We prove that if s ⊂ X(M ) is a normal Lie algebra of vector fields then (M, s) is isogenous to a homogeneous space. In particular, we prove that any n-dimensional infinitesimally homogeneous space of type sl n+1 (C) is isogenous to the projective space. Our answer is based on a generalization of the computations done in Section 3.5. Their proof is elementary. We outline here a more sophisticated proof in the case C = A 1 C that will be generalized in the next section.

The sl 2 case
Proof . Notations are the ones introduced in Section 3.5. L is the space of parameterized arcs z = i z (i) x i i! on C . The vector space CX + CH + CY is denoted by g. Let r o : (C, 0) → A 1 C be an arc with r o (0) = 0 and consider V ⊂ L defined by Claim 5.2. This is a 3-dimensional algebraic variety. Let us describe the canonical structure of L (see [5, pp. 11-12] or next section for a different presentation). For k an integer greater or equal to −1, let us consider the vector field on L We define a morphism of Lie algebra ρ : X → X(L ) by x k+1 ∂ ∂x → E k and the adic continuity. Claim 5.4. The Cartan form σ (as defined in Section 3.5.1) restricted to V takes values in the Lie algebra r * 0 (g). It is the parallelism form reciprocal to the parallelism L X, L H and L Y of V .
Claim 5.5. The actions of K on V and the right action of K on PSL 2 (C) are conjugated by the isogeny.
This induces an isogeny between C and CP 1 . Let π 1 and π 2 be the two maps of the isogeny. A local transformation ϕ such that π 1 • ϕ = π 1 satisfies ϕ * π * 1 (X, H, Y ) = π * 1 (X, H, Y ) and the same is true for the push-forward (π 2 ) * ϕ of ϕ on CP 1 . Then (π 2 ) * ϕ preserves ∂ ∂z and z ∂ ∂z . It is the identity. This finishes the proof. One denotes by z α i : M [n] → C the function defined by z α i (r) = r α i . This function is the element The same procedure can be used to define the prolongation of analytic or formal vector fields on M to M [n] .

The canonical structure
The jet space M [n] is endowed with a differential structure on its coordinate ring and with a group action by "reparameterizations". The compatibility condition between these two structures is well-known (see [5, pp. 11-23]) and is easily obtained using the construction above. . This action is described on the coordinate ring in the following way. For ξ ∈ X, f ∈ C[M ] and P ∈ C[∂ 1 , . . . , ∂ n ], we define ξ · (f ⊗ P ) = f ⊗ (P • ξ)| 0 where the composition is evaluated in 0 in order to get an element of C[∂ 1 , . . . , ∂ n ]. The action of C∂ i is the differential structure. The action of X 0 = lie(Γ), the Lie subalgebra of vector fields vanishing at 0 is the infinitesimal part of the action of Γ. • for any r ∈ M [n] * , σ is a isomorphism from T r M [n] * to X; • for any γ ∈ Γ, (Sγ) * σ = γ * • σ; . These equalities are not compatible with the projective systems.

Normal Lie algebras of vectors f ields
Definition 5.8. A Lie subalgebra g ⊂ X[M ] is said to be normal if for generic p ∈ M on has N (g, p) = g.
Lemma 5.9. If g is transitive then the Lie algebra N (g, p) is finite-dimensional.
Proof . Let k be an integer large enough so that the only element of g vanishing at order k at p is 0. If N (g, p) is not finite-dimensional then there exists a non-zero Y ∈ N (g, p) vanishing at order k + 1 at p. For X ∈ g, the Lie bracket [Y, X] is an element of g vanishing at order k at p. It is zero meaning that Y is invariant under the flows of vector fields in g. The transitivity hypothesis together with Y (p) = 0 proves the lemma.
Lemma 5.10. If there exists a point p ∈ M such that g is maximal among finite-dimensional Lie subalgebra of X(M, p) then g is normal.
Proof . Because of the preceding lemma, if such a point exists then g = N (g, p) in X(M, p). By transitivity, for any couple of points (p 1 , p 2 ) ∈ M 2 there is a composition of flows of elements of g sending p 1 on p 2 . These flows preserve g thus the equality holds at any p.
Example 5.11. Let M be n-dimensional and g be a transitive Lie subalgebra of rational vector fields isomorphic to sl n+1 (C). Then g is normal (see [3]).

5.4
Centerless, transitive and normal ⇒ isogenous to a homogeneous space Theorem 5.12. Let M be a smooth irreducible algebraic variety over C and g be a transitive, centerless, normal, finite-dimensional Lie subalgebra of X(M ). Then there exists an algebraic group G, an algebraic subgroup H ⊂ G and an isogeny between (M, g) and (G/H, lie(G)). Moreover, if N G (lie(H)) = H then the isogeny is a dominant rational map M G/H.
Because of the finiteness and the transitivity, there exists an integer k such that at any p ∈ M and for any Y ∈ N (g, p), j k (Y )(p) = 0, unless Y = 0.
Let r o : (C n , 0) → M be an invertible formal map with r o (0) = p a regular point. Let us consider the subspace of M [n] defined by Proof . If r −1 o • r is tangent to the identity at order k then the induced automorphism of g is the identity. The map r −1 o • r fixes p, thus it is the identity. This proves the lemma.
Using r o one can identify the Lie algebra N (g, p) with a Lie subalgebra of X. The latter acts on M [n] as described in Section 5.2.2. As an application of the Theorem 5.6, one gets: Lemma 5.14. The restriction of the canonical structure of M [n] gives an parallelism called the canonical parallelism. Lemma 5.16. Under the hypothesis of normality of g, V has two commuting parallelisms of type g.
Using Corollary 4.7, g is the Lie algebra of an algebraic group G isogeneous to V . V is foliated by the orbits of the subgroup K of Γ stabilizing V . This group is algebraic with Lie algebra k = r * o (g) ∩ X 0 . Let h ⊂ lie(G) be the Lie algebra corresponding to k by the isogeny. Then the orbits of h are algebraic. This means that h is the Lie algebra of an algebraic subgroup H of G, and that V /K and G/H are isogenous.
Assume that N G (lie(H)) = H. If W is the isogeny between V and G. The push-forward of a local analytic deck transformation of W → V is a transformation of G preserving each element of g, it is a right translation. A deck transformation preserves the orbits of the pull-back of k on W . Its push-forward preserves the orbits of a group containing H with the same Lie algebra. By hypothesis the push-forward is in H and then the isogony obtained by taking the quotient under K and H is the graph of a dominant rational map.

A Picard-Vessiot theory of a principal connection
In the previous reasoning we have used the concept of differential Galois group of a connection. Here, we present a dictionary between invariant connection and strongly normal differential field extension (in the sense of Kolchin). In our setting a differential field is a pair (K, D) where K is a finitely generated field over C and D is a K vector space of derivations of K stable by Lie bracket. The dimension of D is called the rank of the differential field. Note that we can adapt this notion easily to that of a finite number of commuting derivations by taking a suitable basis of D. However we prefer to consider the whole space of derivations. With our definition a differential field extension (K, D) → (K , D ) is a field extension K ⊂ K such that each element of D extends to a unique element of D and such extensions span the space D as K -vector space.

A.1 Dif ferential f ield extensions and foliated varieties
First, let us see that there is a natural dictionary between finitely generated differential fields over C and irreducible foliated varieties over C modulo birational equivalence. Let (M, F) be an irreducible foliated variety of dimension n. The distribution T F ⊂ T M is of rank r ≤ n. We denote by X F the space of rational vector fields in T F; it is a C(M )-Lie algebra of dimension r. Hence, the pair (C(M ), X F ) is a differential field. The field of constants is the field C(M ) F of rational first integrals of the foliation.
Let Example A.1. Let F the foliation of C 2 defined by {dy − ydx = 0}. It corresponds to the differential field C(x, e x ), d dx .

A.2 Invariant F -connections
Let us consider from now a foliated manifold of dimension n and rank r without rational first integrals (M, F), an algebraic group G and a principal irreducible G-bundle π : P → M . A Ginvariant connection in the direction of F is a foliation F of rank r in P such that: (a) π : (P, F ) → (M, F) is a dominant regular map of foliated varieties; (b) The foliation F is invariant by the action of G in P .
With this definition (C(M ), X F ) → (C(P ), X F ) is a differential field extension. Also, each element g ∈ G induces a differential field automorphism of (C(P ), X F ) that fixes (C(M ), X F ) by setting (g · f )(x) = f (x · g). Let g be the Lie algebra of G. There is a way of defining a G-equivariant form Θ F with values in g, and defined in dπ −1 (T F) in such way that T F is the kernel of Θ F . First, there is a canonical form Θ 0 defined in ker(dπ) that sends each vertical vector X p ∈ ker d p π ⊂ T p P to the element g that verifies, This form is G-equivariant in the sense that R * g (Θ 0 ) = Adj g −1 •ω. We have a decomposition of the vector bundle dπ −1 (T F) = ker(dπ) ⊕ T F . This decomposition allows to extend Θ 0 to a form Θ F defined for vectors in dπ −1 (T F) whose kernel is precisely T F. We call horizontal frames to those sections s of π such that s * (Θ F ) = 0.

A.3 Picard-Vessiot bundle
We say that the principal G-bundle with invariant F-connection π : (P, F ) → (M, F) is a Picard-Vessiot bundle if there are no rational first integrals of F . The notion of Picard-Vessiot bundle corresponds exactly to that of primitive extension of Kolchin. In such case G is the group of differential field automorphisms of (C(P ), X F ) that fix (C(M ), X F ) and (C(M ), X F ) → (C(P ), X F ) is a strongly normal extension. Moreover, any strongly normal extension with constant field C can be constructed in this way (see [6, Chapter VI, Section 10, Theorem 9]).
One of the most remarkable properties of strongly normal extensions is the Galois correspondence (from [6, Chapter VI, Section 4]).
Theorem A.3 (Galois correspondence). Assume that (C(M ), X F ) → (C(P ), X F ) is strongly normal with group of automorphisms G. Then, there is a bijection between the set of intermediate differential field extensions and algebraic subgroups of G. To each intermediate differential field extension, it corresponds the group of automorphisms that fix such an extension point-wise. To each subgroup of automorphisms it corresponds its subfield of fixed elements.

A.4 The Picard-Vessiot bundle of an invariant F -connection
Let us consider an irreducible principal G-bundle π : (P, F ) → (M, F) endowed with an invariant F-connection F . We assume that F has no rational first integrals. A result of Bonnet (see [2,Theorem 1.1]) ensures that for a very generic point in M the leaf passing through such point is Zariski dense in M . Let us consider such a Zariski-dense leaf L of F in M . Let us consider any leaf L of F in P that projects by π onto L. Its Zariski closure is unique in the following sense: Theorem A.4. Let L and L two leaves of F whose projections by π are Zariski dense in M . Then, there exist an element g ∈ G such that L · g = L .
Proof . By construction, there is some x ∈ π(L ) ∩ π(L ). Let us consider p ∈ π −1 ({x}) ∩ L and q ∈ π −1 ({x}) ∩ L . Since p and q are in the same fiber, there is a unique element g ∈ G such that p · g = q. By the G-invariance of the connection L · g is the leaf of F that passes through q. The set L is, by construction, union of leaves of F and contains the point q. Thus, L · g ⊆ L , and L · g ⊆ L . Now, by exchanging the roles of L and L , we prove that there is an element h such that L · h ⊆ L . It follows h = g −1 . This finishes the proof.
Let L be the Zariski closure of L . Let us consider the algebraic subgroup H = {g ∈ G : L · g = L} stabilizing L. The projection π restricted to L is dominant, thus there is a Zariski open subset M such that π : L → M is surjective. Let us call F the restriction of F to L . It follows that the bundle: π : (L , F ) → (M , F| M ) is a principal bundle of structure group H called Picard-Vessiot bundle. The differential field extension (C(M ), X F ) → (C(L ), X F ) is the so-called Picard-Vessiot extension associated to the connection. The algebraic group H is the differential Galois group of the connection.

A.5 Split of a connection
Let us consider a pair of morphisms of foliated varieties Then, we can define in M 1 × M M 2 a foliation F 1 × F F 2 in the following way. A vector X = (X 1 , X 2 ) is in T (F 1 × F F 2 ) if and only if dφ 1 (X 1 ) = dφ 2 (X 2 ) ∈ T F. Let us consider (P, F ) a principal F connection. Note that the projection is a principal G-bundle endowed of a F 1 -connection. We call this bundle the pullback of (P, F ) by φ 1 . We also may consider the trivial G-invariant connection F 0 in the trivial principal G-bundle for what the leaves of F 0 are of the form (L, g) where L is a leaf of F and g a fixed element of G. We say that the G-invariant connection (P, F ) is rationally trivial if there is a birational G-equivariant isomorphism of foliated manifolds between (P, F) and (M × G, F 0 ). Invariant connections are always trivialized after pullback; there is a universal G-equivariant isomorphism defined over P that trivializes any G-invariant connection. However, the differential field (C(P ), X F ) may have new constant elements. To avoid this, we replace the pullback to P by a pullback to the Picard-Vessiot bundle L The Picard-Vessiot bundle has some minimality property. It is the smallest bundle on M that trivializes the connection. We have the following result. There is a dominant rational map of foliated varieties ψ :M L such that π • ψ = φ in their common domain.
Proof . Let us take τ :M × G M × M P a birational trivialization, π 2 :M × M P → P be the projection in the second factor, and ι :M →M × G the inclusion p → (p, e). Then, ψ = π 2 • τ • ι is a rational map fromM to P whose differential sends TF to T F. By Bonnet theorem,M is the Zariski closure of a leaf ofF that projects by φ into a Zariski dense leaf of F. From this,ψ contains a dense leaf of F in P . By applying a suitable right translation in P and the uniqueness Theorem A.4, we obtain the desired conclusion.

A.6 Linear connections
Let (M, F) be as above, of dimension n and rank r. Let ξ : E → M be a vector bundle of rank k. A linear integrable F-connection is a foliation F E of rank r which is compatible with the structure of vector bundle in the following sense: the point-wise addition of two leaves of any dilation of a leaf is also a leaf. This can also be stated in terms of a covariant derivative operator ∇ wich is defined only in the direction of F. First, the kernel of dξ is naturally projected onto E itself vert 0 : ker(dξ) → E, where d dε ε=0 v + εw = X v . Then, the decomposition of dξ −1 (T F) as ker(dξ) ⊕ T F E allows us to extend vert 0 to a projection vert : dξ −1 (T F) → E.
Thus, we define for each section s its covariant derivative ∇s = s * (vert•ds| T F ). This is a 1-form on M defined only for vectors in T F. This covariant derivative has the desired properties, it is additive and satisfies the Leibniz formula In general, we write for X a vector in T F, ∇ X s for the contraction of ∇s with the vector X. It is an element of E over the same base point in M that the vector X. We call horizontal sections to those sections s of ξ such that ∇s = 0.
Let π : R 1 (E) → M be the bundle of linear frames in E. It is a principal linear GL k (C)bundle. The foliation F E induces a foliation F in R 1 (E) that is a G-invariant F-connection.

A.7 Associated connections
Let π : (P, F ) → (M, F) be a G-invariant connection, as before, where F is a foliation in M without rational first integrals. Let us consider φ : G → GL(V ) a finite-dimensional linear representation of G. It is well known that the associated bundle π P : V P → M , is a vector bundle with fiber V . Here we represent the action of G in V by the same operation symbol than before. The G-invariant connection F rises to a foliation in P × G and then it is projected to a foliation F V in V P . In this way, the projection turns out to be a linear F-connection. It is called the Lie-Vessiot connection induced in the associated bundle. The Galois group of the principal and the associated Lie-Vessiot connection are linked in the following way.
Theorem A. 6. Let H ⊂ G be the Galois group of the principal connection F . Then, the Galois group of the associated Lie-Vessiot connection Proof . Let us consider the bundle of frames R 1 (V P ), with its induced invariant connection F . Let us fix a basis {v 1 , . . . , v r } of V . Then, we have a map where the pair [p, v] represents the class of the pair (p, v) ∈ P ×V . By construction,π sends T F to T F . It implies that, if L is a Picard-Vessiot bundle for F thenπ(L ) is a Picard-Vessiot bundle for F . Second, if L is a principal H bundle, thenπ(L ) is a principal H/K bundle where K is the subgroup of H that stabilizes the basis {v 1 , . . . , v r }.
Let us discuss how the covariant derivative operator in ∇ is defined in terms of Θ F and the action of G in V . Let us denote by φ : g → gl(V ) the induced Lie algebra morphism. Let s be a local section of ξ. Let us consider the canonical projectionπ : P × V → V (P ). This turns out to be also a principal bundle, here the action on pairs is (p, v) · g = p · g, g −1 · v . Now we can take any section r of this bundle, and defines = r • s. As r takes values in a cartesian product, we obtains = (s 1 , s 2 ) where s 1 is a section of π and s 2 is a function with values in V . Finally we obtain A calculation shows that it does not depend of the choice of r and it is the covariant derivative operator associated to F V . In particular, if s 2 is already an horizontal frame, then the covariant differential is given by the first term ds s | T F .

B Deligne's realization of Lie algebra
The proof of the existence of a regular parallelism for any complex Lie algebra g is written in a set of two letters from P. Deligne to B. Malgrange (dated from November of 2005 and February of 2010 respectively) that are published verbatim in [7]. We reproduce here the proof with some extra details.
Theorem B.1 (Deligne). Given any complex Lie algebra g there exist an algebraic variety endowed with a regular parallelism of type g.
Lemma B.2. Let T be an algebraic torus acting regularly by automophisms in some algebraic group H and let t be the Lie algebra of T . Let us consider the semidirect product t H, (t, h)(t , h ) = (t + t , (exp(t ) · h)h ) as an algebraic variety and analytic Lie group. Its left invariant vector fields form a regular parallelism of t H. The Galois group of this parallelism is a torus.
Proof . Let us denote by α the action of T in H and α : t → X[H] the Lie algebra isomorphism given by the infinitesimal generators (α X) h = d dε ε=0 α exp(εt) (h).
Let X be an invariant vector field in t. Let us compute the left invariant vector field in t H whose value at the identity is (X 0 , 0). In order to perform the computation we write the vector as an infinitesimally near point to (0, e).
And thus (0, Y ) is the left invariant vector field whose value at (0, e) is (0, Y e ). These vector fields of the form (X, α X) and (0, Y ) are regular and span the Lie algebra of left invariant vector fields in t H. Hence, they form a regular parallelism. In order to compute the Galois group of the parallelism, let us compute its reciprocal parallelism. It is formed by the right invariant vector fields in the analytic Lie group t H. A similar computation proves that if X is an invariant vector field in t then (X, 0) is right invariant in t H. For each element τ ∈ T , α τ is a group automorphism of H. Thus, it induces a derived automorphism α τ * of the Lie algebra of regular vector fields in H. Let Y be now a right invariant vector field in H. Let us compute the right invariant vector field Z in t H whose value at (0, e) is (0, Y e ): R (t,h) (0, e + εY e ) = (t, α exp(t) (e + εY e )h) = (t, h + ε(α exp(t) * Y ) h ) and Z t,h = (0, (α exp(t) * Y ) h ). Those analytic vector fields depend on the exponential function in a torus thus we can conclude, by a standard argument of differential Galois theory, that the associated differential Galois group is a torus.
Let us consider g an arbitrary, non algebraic, finite-dimensional complex Lie algebra. We consider an embedding of g in the Lie algebra of general linear group and E the smallest algebraic subgroup whose Lie algebra e contains g. E is a connected linear algebraic group. Proof . Let H be the group of matrices that stabilizes g and acts trivially on g/[g, g]. Its Lie algebra h contains g and thus H ⊇ E and h ⊇ e. By definition of H we have [h, g] = [g, g], therefore [e, g] ⊆ [g, g]. Let us now consider the group H 1 that stabilizes e and g and that acts trivially in e/ [g, g]. This is again an algebraic group containing E, and its Lie algebra h 1 satisfies [h 1 , e] ⊆ [g, g]. Taking into account e ⊆ h 1 we have [e, e] ⊆ [g, g]. The other inclusion is trivial.
Because of Lemma B.3, the abelianized Lie algebra g ab = g/[g, g] is a subspace of e ab = e/[e, e]. Moreover, if we consider the quotient map, π : e → e ab , then g = π −1 (g ab ).
Let us consider an algebraic Levy decomposition E L U (see [9,Chapter 6]). Here, L is reductive and U is the unipotent radical, consisting in all the unipotent elements of E. The semidirect product structure is produced by an action of L in U , so that, (l 1 , u 1 )(l 2 , u 2 ) = (l 1 l 2 , (l 2 · u 1 )u 2 ).
Since L is reductive, its commutator subgroup L is semisimple. Let T be the center of L, which is a torus, the map ϕ : T × L → L, (t, l) → tl, is an isogeny. The isogeny defines an action of T × L in U by (t, l) · u = tl · u. We have found an isogeny The Lie algebra u of U is a nilpotent Lie algebra, so that the exponential map exp : u → U is regular and bijective. In general, if V is an abelian quotient of U with Lie algebra v then the exponential map conjugates the addition law in v with the group law in V .
Lemma B.4. With the above definitions and notation, letū be the biggest quotient of u ab in which L acts by the identity. We have a Lie algebra isomorphism e ab t ×ū.
Proof . Let us compute e ab . We compute the commutators e by means of the isomorphism e (t × l ) u. We obtain [(t 1 , l 1 , u 1 ), (t 2 , l 2 , u 2 )] = (0, [l 1 , l 2 ], a(t 2 , l 2 )u 1 + [u 1 , u 2 ]), where a represents the derivative at (e, e) of the action of L in U . From this we obtain that [e, e] is spanned by ({0} × l ) {0}, {0} [u, u] and {0} × a(l)u . Taking into account that u/( a(l)u + [u, u]) is the biggest quotient of u ab in which L acts trivially, we obtain the result of the lemma.
Let t be the Lie algebra of T . Its exponential map is an analytic group morphism and thus we may consider the analytic action of t × L in U given by (t, l) · u = (exp(t)l) · u. LetẼ be the algebraic variety and analytic Lie group (t × L ) U . By application of Lemma B.2, and taking into account thatẼ t H, where H is the group L · U , we have that the left invariant vector fields inẼ are regular. Let us consider the projection π 1 :Ẽ → e ab = t ×ū, (t, l, u) → (t, [log(u)]), this projection is algebraic by construction, and also a morphism of Lie groups. By Lemmas B.3 and B.4, g ab is a vector subspace of the image. Then, let us takeG the fiber π −1 1 (g ab ). It is an algebraic submanifold ofẼ and an analytic Lie group. The derivative at the identity of π 1 is precisely the abelianization π and it follows that the Lie algebra ofG is precisely g. FinallyG is an algebraic variety with a regular g-parallelism. This finishes the proof of Theorem B.1.
Remark B.5. The right invariant vector fields inG are constructed as in Lemma B.2 by means of the exponential function in the torus. Hence, Galois groups of the parallelisms obtained via this construction are always tori.