Parallelisms & Lie Connections

Parallelisms & Lie Connections

David BLÁZQUEZ-SANZ † and Guy CASALE ‡

D. Blázquez-Sanz and G. Casale

† Universidad Nacional de Colombia, Sede Medellín, Facultad de Ciencias,
† Escuela de Matemáticas, Calle 59A No. 63 - 20, Medellín, Antioquia, Colombia
<EMAIL_ADDRESS>
‡ IRMAR, Université de Rennes 1, Campus de Beaulieu, bât. 22-23,
‡ 263 avenue du Général Leclerc, CS 74205, 35042 RENNES Cedex, France
<EMAIL_ADDRESS>
Received September 16, 2016, in final form October 25, 2017; Published online
November 04, 2017

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.

parallelism; isogeny; $G$-structure; linear connection; principal connection;
differential Galois theory

53C05; 14L40; 14E05; 12H05

##  1 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
$\mathfrak{g}$-parallelized complex manifold $M$ has a natural $(G,G)$
structure where $G$ is a Lie group with $\mathfrak{lie}(G)=\mathfrak{g}$. 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 ${\rm
PSL}_{2}(\mathbf{C})$ appears as the differential Galois group of a
$\mathfrak{sl}_{2}(\mathbf{C})$-parallelism (Theorem 3.16). Section 4 is
devoted to the construction of the isogeny between a
$\mathfrak{g}$-parallelized variety and an algebraic group $G$ whose Lie
algebra is $\mathfrak{g}$. In order to do this, we introduce the
Darboux–Cartan connection, a $G$-connection whose horizontal sections are
parallelism conjugations. We prove that if $\mathfrak{g}$ is centerless then
the Darboux–Cartan connection and the reciprocal Lie connection have isogenous
Galois groups. We prove that the only centerless Lie algebras $\mathfrak{g}$
such that there exists a $\mathfrak{g}$-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 $\mathfrak{g}$ be a centerless Lie algebra. An algebraic variety
$(M,\omega)$ with a rational parallelism of type $\mathfrak{g}$ is isogenous
to an algebraic group if and only if $\mathfrak{gal}(\nabla^{\rm
rec})=\\{0\\}$.

The notion of isogeny can be extended beyond the simply-transitive case. Let
us consider a complex Lie algebra $\mathfrak{g}$. An infinitesimally
homogeneous variety of type $\mathfrak{g}$ is a pair $(M,\mathfrak{s})$
consisting of a complex smooth irreducible variety $M$ and a finite-
dimensional Lie algebra $\mathfrak{s}\subset\mathfrak{X}(M)$ isomorphic to
$\mathfrak{g}$ that spans the tangent bundle of $M$ on the generic point.

We are interested in conjugation by rational or by 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 $f\colon M_{1}\dasharrow M_{2}$
between varieties of the same dimension conjugates the infinitesimally
homogeneous varieties $(M_{1},\mathfrak{s}_{1})$ and
$(M_{2},\mathfrak{s}_{2})$ if $f^{*}(\mathfrak{s}_{2})=\mathfrak{s}_{1}$. We
say that $(M_{1},\mathfrak{s}_{1})$ and $(M_{2},\mathfrak{s}_{2})$ are
isogenous if they are conjugated to the same infinitesimally homogeneous space
of type $\mathfrak{g}$.

Under some hypothesis on the Lie algebra $\mathfrak{s}\subset\mathfrak{X}(M)$
one can prove that $(M,\mathfrak{s})$ is isogenous to a homogeneous space
$(G/H,\mathfrak{lie}(G)^{\rm rec})$ with the action of right invariant vector
fields. These hypothesis are satisfied by transitive actions of
$\mathfrak{sl}_{n+1}(\mathbf{C})$ on $n$-dimensional varieties. As a
particular case of Theorem 5.12 one has

######  Theorem.

Let $(M,\mathfrak{s})$ be an infinitesimally homogeneous variety of complex
dimension $n$ such that $\mathfrak{s}$ is isomorphic to
$\mathfrak{sl}_{n+1}(\mathbf{C})$. Then there exists a dominant rational map
$M\dasharrow\mathbf{CP}_{n}$ conjugating $\mathfrak{s}$ with the Lie algebra
$\mathfrak{sl}_{n+1}(\mathbf{C})$ of projective vector fields in
$\mathbf{CP}_{n}$.

Appendix A is devoted to a geometrical presentation of Picard–Vessiot theory
for linear and principal connections. Finally, Appendix B contains a detailed
proof of Deligne’s theorem of the realization of a regular parallelism modeled
over any finite-dimensional Lie algebra. This includes also a computation of
the Galois group that turns out to be, for this particular construction, an
algebraic torus.

##  2 Parallelisms

Let $M$ be a smooth connected affine variety over $\mathbf{C}$ of dimension
$r$. We denote by $\mathbf{C}[M]$ its ring of regular functions and by
$\mathbf{C}(M)$ its field of rational functions. Analogously, we denote by
$\mathfrak{X}[M]$ and $\mathfrak{X}(M)$ respectively the Lie algebras of
regular and rational vector fields in $M$, and so on.

Let $\mathfrak{g}$ be a Lie algebra of dimension $r$. We fix a basis
$A_{1},\ldots,A_{r}$ of $\mathfrak{g}$, and the following notation for the
associated structure constants $[A_{i},A_{j}]=\sum_{k}\lambda_{ij}^{k}A_{k}$.

A parallelism of type $\mathfrak{g}$ of $M$ is a realization of the Lie
algebra $\mathfrak{g}$ as a Lie algebra of pointwise linearly independent
vector fields in $M$. More precisely:

######  Definition 2.1.

A regular parallelism of type $\mathfrak{g}$ in $M$ is a Lie algebra morphism,
$\rho\colon\mathfrak{g}\to\mathfrak{X}[M]$ such that $\rho
A_{1}(x),\ldots,\rho A_{r}(x)$ form a basis of $T_{x}M$ for any point $x$ of
$M$.

######  Example 2.2.

Let $G$ be an algebraic group and $\mathfrak{g}$ be its Lie algebra of left
invariant vector fields. Then the natural inclusion
$\mathfrak{g}\subset\mathfrak{X}[G]$ is a regular parallelism of $G$. The Lie
algebra $\mathfrak{g}^{\rm rec}$ 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
$M^{\star}\subseteq M$.

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 $\mathfrak{g}$ in $M$ is
a $\mathfrak{g}$-valued $1$-form
$\omega\in\Omega^{1}[M]\otimes_{\mathbb{C}}\mathfrak{g}$ such that:

  * (1)

For any $x\in M$, $\omega_{x}\colon T_{x}M\to\mathfrak{g}$ is a linear
isomorphism.

  * (2)

If $A$ and $B$ are in $\mathfrak{g}$ and $X$, $Y$ are vector fields such that
$\omega(X)=A$ and $\omega(Y)=B$ then $\omega[X,Y]=[A,B]$.

Analogously, we define local analytic and rational coparallelism of type
$\mathfrak{g}$ in $M$. It is clear that each coparallelism induces a
parallelism, and reciprocally, by the relation $\omega(\rho(A))=A$. Thus,
there is a natural equivalence between the notions of parallelism and
coparallelism. From now on we fix $\rho$ and $\omega$ equivalent parallelism
and coparallelism of type $\mathfrak{g}$ on $M$.

The Lie algebra structure of $\mathfrak{g}$ forces $\omega$ to satisfy
Maurer–Cartan structure equations

$\displaystyle{\rm d}\omega+\frac{1}{2}[\omega,\omega]=0.$

Taking components $\omega=\sum_{i}\omega_{i}A_{i}$ we have

$\displaystyle{\rm
d}\omega_{i}+\sum_{j,k=1}^{r}\frac{1}{2}\lambda_{jk}^{i}\omega_{j}\wedge\omega_{k}=0.$

######  Example 2.4.

Let $G$ be an algebraic group and $\mathfrak{g}$ be the Lie algebra of left
invariant vector fields in $G$. Then the structure form $\omega$ is the
coparallelism corresponding to the parallelism of Example 2.2.

######  Example 2.5.

Let $\mathfrak{g}=\langle A_{1},A_{2}\rangle$ be the 2-dimensional Lie algebra
with commutation relation

$\displaystyle[A_{1},A_{2}]=A_{1}.$

The vector fields

$\displaystyle X_{1}=\frac{\partial}{\partial x},\qquad
X_{2}=x\frac{\partial}{\partial x}+\frac{\partial}{\partial y},$

define a regular parallelism via $\rho(A_{i})=X_{i}$ of $\mathbf{C}^{2}$. The
associated parallelism form is

$\displaystyle\omega=A_{1}{\rm d}x+(A_{2}-xA_{1}){\rm d}y.$

######  Example 2.6 (Malgrange).

Let $\mathfrak{g}=\langle A_{1},A_{2},A_{3}\rangle$ be the 3-dimensional Lie
algebra with commutation relations

$\displaystyle[A_{1},A_{2}]=\alpha A_{2},\qquad[A_{1},A_{3}]=\beta
A_{3},\qquad[A_{2},A_{3}]=0,$

with $\alpha$, $\beta$, non zero complex numbers. In particular, if
$\alpha/\beta$ is not rational then $\mathfrak{g}$ is not the Lie algebra of
an algebraic group. The vector fields

$\displaystyle X_{1}=\frac{\partial}{\partial x}+\alpha
y\frac{\partial}{\partial y}+\beta z\frac{\partial}{\partial z},\qquad
X_{2}=\frac{\partial}{\partial y},\qquad X_{3}=\frac{\partial}{\partial z},$

define a regular parallelism via $\rho(A_{i})=X_{i}$ of $\mathbf{C}^{3}$. The
associated parallelism form is

$\displaystyle\omega=(A_{1}-A_{2}\alpha y-A_{3}\beta z){\rm d}x+A_{2}{\rm
d}y+A_{3}{\rm d}z.$

######  Definition 2.7.

Let $(M,\omega)$ and $(N,\theta)$ be algebraic manifolds with coparallelisms
of type $\mathfrak{g}$. We say that they are isogenous if there is an
algebraic manifold $(P,\eta)$ with a coparallelism of type $\mathfrak{g}$ and
dominant maps $f\colon P\to M$ and $g\colon P\to N$ such that
$f^{*}(\omega)=g^{*}(\theta)=\eta$.

Clearly, the notion of isogeny of parallelized varieties extends that of
isogeny of algebraic groups.

######  Example 2.8.

Let $f\colon M\dasharrow G$ be a dominant rational map with values in an
algebraic group with $\dim_{\mathbf{C}}M=\dim_{\mathbf{C}}G$. Then
$\theta=f^{*}(\omega)$ is a rational parallelism form in $M$.

######  Example 2.9.

Let $H$ be a finite subgroup of the algebraic group $G$ and

$\displaystyle\pi\colon\ G\to M=H\setminus G=\\{Hg\colon g\in G\\}$

be the quotient by the action of $H$ on the left side. The structure form
$\omega$ in $G$ is left-invariant and then it is projectable by $\pi$. Then,
$\theta=\pi_{*}(\omega)$ is a regular parallelism form in $M$.

######  Example 2.10.

Combining Examples 2.8 and 2.9, let $H\subset G$ be a finite subgroup and
$f\colon M\to H\setminus G$ be a dominant rational map between manifolds of
the same dimension. Then $\theta=f^{*}(\pi_{*}(\omega))$ 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 $\mathbf{CP}_{1}$,
$\frac{p}{q}\frac{{\rm d}f}{f}$ where $f\in\mathbf{C}(z)$. Thus, rational
multiples of logarithmic forms in $\mathbf{CP}_{1}$ 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 $\mathbf{CP}_{1}$, ${\rm d}F$ where $F\in\mathbf{C}(z)$. Thus,
the exact forms in $\mathbf{CP}_{1}$ are the rational coparallelisms isogenous
to that of the additive group.

######  Example 2.13.

Let $H$ be a subgroup of the algebraic group $G$, with Lie algebra
$\mathfrak{h}\subset\mathfrak{g}$. Let us assume that $\mathfrak{h}$ admits a
supplementary Lie algebra $\mathfrak{h}^{\prime}$

$\displaystyle\mathfrak{g}=\mathfrak{h}\oplus\mathfrak{h}^{\prime}\qquad\mbox{(as
vector spaces).}$

We consider the left quotient $M=H\setminus G$ of $G$ by the action of $H$ and
the quotient map $\pi\colon G\to M$. It turns out that $\mathfrak{h}^{\prime}$
is a Lie algebra of vector fields in $G$ projectable by $\pi$, and thus
$\pi_{*}|_{\mathfrak{h}^{\prime}}\colon\mathfrak{h}^{\prime}\to\mathfrak{X}[M]$
gives a parallelism of $M$ that is regular in the open subset

$\displaystyle\\{Hg\in
M\colon\operatorname{Adj}_{g}(\mathfrak{h})\cap\mathfrak{h}^{\prime}=\\{0\\}\\}.$

It turns out to be regular in $M$ if $H\lhd G$. Examples 2.5 and 2.6 are
particular cases where $G$ is $\operatorname{Af\/f}(2,\mathbf{C})$ and
$\operatorname{Af\/f}(3,\mathbf{C})$ respectively.

######  Remark 2.14.

We can see also Example 2.13 as a coparallelism. Let
$\pi^{\prime}\colon\mathfrak{g}\to\mathfrak{h}^{\prime}$ be the projection
given by the vector space decomposition
$\mathfrak{g}=\mathfrak{h}\oplus\mathfrak{h}^{\prime}$. Since
$\pi^{\prime}\circ\omega$ is left invariant form in $G$, it is projectable by
$\pi$. Hence, there is a form $\omega^{\prime}$ in $M$ such that
$\pi^{*}\omega^{\prime}=\pi^{\prime}\circ\omega$. This form $\omega^{\prime}$
is the corresponding coparallelism.

##  3 Associated Lie connection

###  3.1 Reciprocal connections

Let $\nabla$ be a linear connection (rational or regular) on $TM$. The
reciprocal connection is defined as

$\displaystyle\nabla^{\rm
rec}_{\vec{X}}\vec{Y}=\nabla_{\vec{Y}}\vec{X}+\big{[}\vec{X},\vec{Y}\big{]}.$

From this definition it is clear that the difference $\nabla-\nabla^{\rm
rec}=\operatorname{Tor}_{\nabla}$ is the torsion tensor,
$\operatorname{Tor}_{\nabla}=-\operatorname{Tor}_{\nabla^{\rm rec}}$ and
$(\nabla^{\rm rec})^{\rm rec}=\nabla$.

###  3.2 Connections and parallelisms

Let $\omega$ be a coparallelism of type $\mathfrak{g}$ in $M$ and $\rho$ its
equivalent parallelism. Denote by $\vec{X}_{i}$ the basis of vector fields in
$M$ such that $\omega(\vec{X}_{i})=A_{i}$ is a basis of $\mathfrak{g}$.

######  Definition 3.1.

The connection $\nabla$ associated to the parallelism $\omega$ is the only
linear connection in $M$ for which $\omega$ is a $\nabla$-horizontal form.

Clearly $\nabla$ is a flat connection and the basis $\\{\vec{X}_{i}\\}$ is a
basis of the space of $\nabla$-horizontal vector fields. In this basis
$\nabla$ has vanishing Christoffel symbols

$\displaystyle\nabla_{\vec{X}_{i}}\vec{X}_{j}=0.$

Let us compute some infinitesimal symmetries of $\omega$. A vector field
$\vec{Y}$ is an infinitesimal symmetry of $\omega$ if ${\rm
Lie}_{\vec{Y}}\omega=0$, or equivalently, if it commutes with all the vector
fields of the parallelism

$\displaystyle\big{[}\vec{X}_{i},\vec{Y}\big{]}=0,\qquad i=1,\ldots,r.$

######  Lemma 3.2.

Let $\nabla$ be the connection associated to the parallelism $\omega$. Then
for any vector field $\vec{Y}$ and any $j=1,\ldots,r$

$\displaystyle\big{[}\vec{X}_{j},\vec{Y}\big{]}=\nabla^{\rm
rec}_{\vec{X}_{j}}{\vec{Y}}.$

Thus, $\vec{Y}$ is an infinitesimal symmetry of $\omega$ if and only if it is
a horizontal vector field for the reciprocal connection $\nabla^{\rm rec}$.

###### Proof.

A direct computation yields the result. Take
$\vec{Y}=\sum\limits_{k=1}^{r}f_{k}\vec{X}_{k}$, for each $j$ we have

$\displaystyle\nabla^{\rm
rec}_{\vec{X}_{j}}\vec{Y}=\sum_{k=1}^{r}\big{(}\big{(}\vec{X}_{j}f_{k}\big{)}\vec{X}_{k}+f_{k}\big{[}\vec{X}_{j},\vec{X}_{k}\big{]}\big{)}=\big{[}\vec{X}_{j},\vec{Y}\big{]}.$
∎

The above considerations also give us the Christoffel symbols for $\nabla^{\rm
rec}$ in the basis $\\{\vec{X}_{i}\\}$

$\displaystyle\nabla^{\rm
rec}_{\vec{X}_{i}}\vec{X}_{j}=\big{[}\vec{X}_{i},\vec{X}_{j}\big{]}=\sum_{k=1}^{r}\lambda_{ij}^{k}\vec{X}_{k},$

i.e., the Christoffel symbols of $\nabla^{\rm rec}$ are the structure
constants of the Lie algebra $\mathfrak{g}$.

######  Lemma 3.3.

Let $\nabla$ be the connection associated to a coparallelism in $M$. Then,
$\nabla^{\rm rec}$ is flat, and the Lie bracket of two $\nabla^{\rm
rec}$-horizontal vector fields is a $\nabla^{\rm rec}$-horizontal vector
field.

###### Proof.

The flatness and the preservation of the Lie bracket by $\nabla^{\rm rec}$ are
direct consequences of the Jacobi identity. Let us compute the curvature

$\displaystyle R\big{(}\vec{X}_{i},\vec{X}_{j},\vec{X}_{k}\big{)}=\nabla^{\rm
rec}_{\vec{X}_{i}}\big{(}\nabla^{\rm
rec}_{\vec{X}_{j}}X_{k}\big{)}-\nabla^{\rm
rec}_{\vec{X}_{j}}\big{(}\nabla^{\rm
rec}_{\vec{X}_{i}}\vec{X}_{k}\big{)}-\nabla^{\rm
rec}_{[\vec{X}_{i},\vec{X}_{j}]}\vec{X}_{k}$
$\displaystyle\hphantom{R\big{(}\vec{X}_{i},\vec{X}_{j},\vec{X}_{k}\big{)}}{}=\rho([A_{i},[A_{j},A_{k}]]-[A_{j},[A_{i},A_{k}]]-[[A_{i},A_{j}],A_{k}])=0.$

Let us compute the Lie bracket for $\vec{Y}$ and $\vec{Z}$ $\nabla^{\rm
rec}$-horizontal vector fields

$\displaystyle\nabla^{\rm
rec}_{\vec{X}_{i}}\big{[}\vec{Y},\vec{Z}\big{]}\\!=\big{[}\vec{X}_{i},\big{[}\vec{Y},\vec{Z}\big{]}\big{]}\\!=\big{[}\big{[}\vec{X}_{i},\vec{Y}\big{]},\vec{Z}\big{]}\\!+\big{[}\vec{Y},\big{[}\vec{X}_{i},\vec{Z}\big{]}\big{]}\\!=\big{[}\nabla^{\rm
rec}_{\vec{X}_{i}}\vec{Y},\vec{Z}\big{]}\\!+\big{[}\vec{Y},\nabla^{\rm
rec}_{\vec{X}_{i}}\vec{Z}\big{]}\\!=0.\\!\\!\\!\\!\\!\\!$ ∎

######  Lemma 3.4.

Let $x\in M$ be a regular point of the parallelism form $\omega$. The space of
germs at $x$ of horizontal vector fields for $\nabla^{\rm rec}$ is a Lie
algebra isomorphic to $\mathfrak{g}$. Moreover, let
$\vec{Y}_{1},\ldots,\vec{Y}_{r}$ be horizontal vector fields with initial
conditions $\vec{Y}_{i}(x)=\vec{X}_{i}(x)$, then
$\big{[}\vec{Y}_{i},\vec{Y}_{j}\big{]}=-\sum\limits_{k=1}^{r}\lambda_{ij}^{k}\vec{Y}_{k}$,
where the $\lambda_{i,j}$ are the structure constants of the Lie algebra
generated by the $\vec{X}_{i}$.

###### Proof.

We can write the vector fields $\vec{Y}_{i}$ as linear combinations of the
vector fields $\vec{X}_{i}$:
$\vec{Y}_{i}=\sum\limits_{j=1}^{r}a_{ji}\vec{X}_{j}$. The matrix $(a_{ij})$
satisfies the differential equation

$\displaystyle\vec{X}_{k}a_{ij}=-\sum_{\alpha=1}^{r}\lambda_{k\alpha}^{i}a_{\alpha
j},\qquad a_{ij}(x)=\delta_{ij}.$

On the other hand, we have
$\big{[}\vec{Y}_{i},\vec{Y}_{j}\big{]}(x)=\sum\limits_{k=1}^{r}\hat{\lambda}_{ij}^{k}\vec{Y}_{k}(x)$,
for certain unknown structure constants $\hat{\lambda}_{ij}^{k}$. Let us check
that $\hat{\lambda}_{ij}^{k}=\lambda_{ji}^{k}=-\lambda_{ij}^{k}$,

$\displaystyle\big{[}\vec{Y}_{i},\vec{Y}_{j}\big{]}=\left[\sum_{\alpha=1}^{r}a_{\alpha
i}\vec{X}_{\alpha},\sum_{\beta=1}^{r}a_{\beta j}\vec{X}_{\beta}\right]$
$\displaystyle\hphantom{\big{[}\vec{Y}_{i},\vec{Y}_{j}\big{]}}{}=\sum_{\alpha,\beta,\gamma=1}^{r}-a_{\alpha
i}\lambda_{\alpha\gamma}^{\beta}a_{\gamma
j}\vec{X}_{\beta}+\sum_{\alpha,\beta,\gamma=1}^{r}a_{\beta
j}\lambda_{\beta\gamma}^{\alpha}a_{\gamma
i}\vec{X}_{\alpha}+\sum_{\alpha,\beta,\gamma=1}^{r}a_{\beta j}a_{\alpha
i}\lambda_{\alpha\beta}^{\gamma}\vec{X}_{\gamma}.$

Taking values at $x$, we obtain

$\displaystyle\big{[}\vec{Y}_{i},\vec{Y}_{j}\big{]}(x)=\sum_{\beta=1}^{r}-\lambda_{ij}^{\beta}\vec{Y}_{\beta}(x)+\sum_{\alpha=1}^{r}\lambda_{ji}^{\alpha}\vec{Y}_{\alpha}(x)+\sum_{\gamma=1}^{r}\lambda_{ij}^{\gamma}\vec{Y}_{\gamma}(x)=\sum_{\alpha=1}^{r}\lambda_{ji}^{k}\vec{Y}_{k}(x).$
∎

######  Example 3.5.

Let $G$ be an algebraic group with Lie algebra $\mathfrak{g}$. As seen in
Example 2.4 the Maurer–Cartan structure form $\omega$ is a coparallelism in
$G$. Let $\nabla$ be the connection associated to this coparallelism. There is
another canonical coparallelism, the right invariant Maurer–Cartan structure
form $\omega_{\rm rec}$, let us consider ${\bf i}\colon G\to G$ the inversion
map,

$\displaystyle\omega_{\rm rec}=-{\bf i}^{*}(\omega).$

As may be expected, the connection associated to the coparallelism
$\omega_{\rm rec}$ is $\nabla^{\rm rec}$. 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 $\nabla$ and $\nabla^{\rm rec}$
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 $\mathfrak{g}$ and the coparallelism
$\omega=A_{1}{\rm d}x+(A_{2}-xA_{1}){\rm d}y$, of Example 2.5. Let $\nabla$ be
its associated connection. In cartesian coordinates, the only non-vanishing
Christoffel symbol of the reciprocal connection is $\Gamma_{21}^{1}=-1$. A
basis of $\nabla^{\rm rec}$-horizontal vector fields is

$\displaystyle\vec{Y}_{1}=e^{y}\frac{\partial}{\partial
x},\qquad\vec{Y}_{2}=\frac{\partial}{\partial y}.$

Note that they coincide with $\vec{X}_{1}$, $\vec{X}_{2}$ at the origin point
and $\big{[}\vec{Y}_{1},\vec{Y}_{2}\big{]}=-Y_{1}$.

######  Example 3.7.

Let us consider the Lie algebra $\mathfrak{g}$ and the coparallelism
$\omega=(A_{1}-\alpha yA_{2}-\beta zA_{3}){\rm d}x+A_{2}{\rm d}y+A_{3}{\rm
d}z$ of Example 2.6. Let $\nabla$ be its associated connection. In cartesian
coordinates, the only non-vanishing Christoffel symbols of the reciprocal
connection are

$\displaystyle\Gamma_{11}^{2}=-\alpha,\qquad\Gamma_{11}^{3}=-\beta.$

A basis of $\nabla^{\rm rec}$-horizontal vector fields is

$\displaystyle\vec{Y}_{1}=\frac{\partial}{\partial
x},\qquad\vec{Y}_{2}=e^{\alpha x}\frac{\partial}{\partial
y},\qquad\vec{Y}_{3}=e^{\beta x}\frac{\partial}{\partial z}.$

Note that they coincide with $\vec{X}_{1}$, $\vec{X}_{2}$, $\vec{X}_{3}$ at
the origin point and

$\displaystyle\big{[}\vec{Y}_{1},\vec{Y}_{2}\big{]}=-\alpha
Y_{2},\qquad\big{[}\vec{Y}_{1},\vec{Y}_{3}\big{]}=-\beta\vec{Y}_{3}.$

######  Example 3.8.

Let us consider the Lie algebra $\mathfrak{g}$ of Example 2.6 and the
coparallelism

$\displaystyle\omega=(A_{1}-\alpha yA_{2}-\beta zA_{3})\frac{{\rm
d}x}{x}+A_{2}{\rm d}y+A_{3}{\rm d}z.$

Let $\nabla$ be its associated connection. In cartesian coordinates, the only
non-vanishing Christoffel symbols of the reciprocal connection are

$\displaystyle\Gamma_{11}^{2}=-\alpha,\qquad\Gamma_{11}^{3}=-\beta.$

A basis of $\nabla^{\rm rec}$-horizontal vector fields on a simply connected
open subspace $U\subset\mathbb{C}^{\ast}\times\mathbb{C}^{2}$ is

$\displaystyle\vec{Y}_{1}=x\frac{\partial}{\partial
x},\qquad\vec{Y}_{2}=x^{\alpha}\frac{\partial}{\partial
y},\qquad\vec{Y}_{3}=x^{\beta}\frac{\partial}{\partial z}.$

###  3.3 Lie connections

The connections $\nabla$ and $\nabla^{\rm rec}$ associated to a coparallelism
$\omega$ of type $\mathfrak{g}$ are particular cases of the following
definition.

######  Definition 3.9.

A Lie connection (regular or rational) in $M$ is a flat connection $\nabla$ in
$TM$ such that the Lie bracket of any two horizontal vector fields is a
horizontal vector field.

Given a Lie connection $\nabla$ in $M$, there is a $r$-dimensional Lie algebra
$\mathfrak{g}$ such that the space of germs of horizontal vector fields at a
regular point $x$ is a Lie algebra isomorphic to $\mathfrak{g}$. We will say
that $\nabla$ is a Lie connection of type $\mathfrak{g}$. The following result
gives several algebraic characterizations of Lie connections:

######  Proposition 3.10.

Let $\nabla$ be a linear connection in $TM$, the following statements are
equivalent:

  * $(1)$

$\nabla$ is a Lie connection;

  * $(2)$

$\nabla^{\rm rec}$ is a Lie connection;

  * $(3)$

$\nabla$ is flat and has constant torsion,
$\nabla\operatorname{Tor}_{\nabla}=0$;

  * $(4)$

$\nabla$ and $\nabla^{\rm rec}$ are flat.

###### Proof.

Let us first see (1)$\Leftrightarrow$(2). Let $\nabla$ be a Lie connection.
Around each point of the domain of $\nabla$ there is a parallelism, by
possibly transcendental vector fields, such that $\nabla$ is its associated
connection. Then, Lemma 3.3 states (1)$\Rightarrow$(2). Taking into account
that $(\nabla^{\rm rec})^{\rm rec}=\nabla$ we have the desired equivalence.

Let us see now that (1)$\Leftrightarrow$(3). Let us assume that $\nabla$ is a
flat connection. For any three vector fields $X$, $Y$, $Z$ in $M$ we have

$\displaystyle(\nabla_{X}\operatorname{Tor}_{\nabla})(Y,Z)=-\operatorname{Tor}_{\nabla}(\nabla_{X}Y,Z)-\operatorname{Tor}_{\nabla}(Y,\nabla_{X}Z)+\nabla_{X}\operatorname{Tor}_{\nabla}(Y,Z).$

Let us assume that $Y$ and $Z$ are $\nabla$-horizontal vector fields. Then, we
have

$\displaystyle\operatorname{Tor}_{\nabla}(Y,Z)=\nabla_{Y}Z-\nabla_{Z}Y-[Y,Z]=-[Y,Z]$

and the previous equality yields

$\displaystyle(\nabla_{X}\operatorname{Tor}_{\nabla})(Y,Z)=-\nabla_{X}[Y,Z].$

Thus, we have that $\nabla\operatorname{Tor}_{\nabla}$ vanishes if and only if
the Lie bracket of any two $\nabla$-horizontal vector fields is also
$\nabla$-horizontal. This proves (1)$\Leftrightarrow$(3).

Finally, let us see (1)$\Leftrightarrow$(4). It is clear that (1) implies (4)
so we only need to see (4)$\Rightarrow$(1). Assume $\nabla$ and $\nabla^{\rm
rec}$ are flat. Then, locally, there exist a basis $\\{\vec{X}_{i}\\}$ of
$\nabla$-horizontal vector fields and a basis $\\{\vec{Y}_{i}\\}$ of
$\nabla^{\rm rec}$-horizontal vector fields. By the definition of the
reciprocal connection, we have that a vector field $\vec{X}$ is
$\nabla$-horizontal if and only if it satisfies $[\vec{X},\vec{Y}_{i}]=0$ for
$i=1,\ldots,r$. By the Jacobi identity we have

$\displaystyle\big{[}\big{[}\vec{X}_{i},\vec{X}_{j}\big{]},\vec{Y}_{k}\big{]}=0.$

The Lie brackets $\big{[}\vec{X}_{i},\vec{X}_{j}\big{]}$ are also
$\nabla$-horizontal and $\nabla$ is a Lie connection. ∎

######  Lemma 3.11.

Let $\nabla$ be a Lie connection on $M$. Let $x$ be a regular point and
$\vec{X}_{1},\ldots,\vec{X}_{r}$ and $\vec{Y}_{1},\ldots,\vec{Y}_{r}$ be basis
of horizontal vector field germs on $M$ for $\nabla$ and $\nabla^{\rm rec}$
respectively with same initial conditions $\vec{X}_{i}(x)=\vec{Y}_{i}(x)$.
Then

$\displaystyle\big{[}\vec{X}_{i},\vec{X}_{j}\big{]}(x)=-\big{[}\vec{Y}_{i},\vec{Y}_{j}\big{]}(x).$

It follows that $\nabla$ and $\nabla^{\rm rec}$ are of the same type
$\mathfrak{g}$.

###### Proof.

By definition $\nabla$ is the connection associated to the local analytic
parallelism given by the basis $\\{\vec{X}_{i}\\}$ of horizontal vector
fields. Then we apply Lemma 3.4 in order to obtain the desired conclusion. ∎

###  3.4 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.

######  Proposition 3.12.

Let $\nabla$ be a rational Lie connection in $TM$. The $\nabla$-horizontal
vector fields are the symmetries of a rational parallelism of $M$ if and only
if $\operatorname{Gal}(\nabla^{\rm rec})=\\{1\\}$.

###### Proof.

We will use the notations of Section A.6: $R^{1}(TM)$ is the ${\rm
GL}_{n}(\mathbb{C})$-principal bundle associated to $TM$ and
$\mathcal{F}^{\prime}$ is the ${\rm GL}_{n}(\mathbb{C})$-invariant foliation
on $R^{1}(TM)$ given by graphs of local basis of $\nabla$-horizontal sections.
The Galois group $\operatorname{Gal}(\nabla^{\rm rec})$ can be computed as
soon as we know the Zariski closure $\overline{\mathscr{L}}$ of a leaf
$\mathscr{L}$ of the induced foliation $\mathcal{F}^{\prime}$ on $R^{1}(TM)$.
$\operatorname{Gal}(\nabla^{\rm rec})$ is finite is and only if
$\overline{\mathscr{L}}=\mathscr{L}$ and is $\\{1\\}$ if and only if
$\mathscr{L}$ is the graph of a rational section $M\to R^{1}(TM)$. This means
that there exists a basis of rational $\nabla^{\rm rec}$-horizontal sections.
These sections give the desired parallelism. ∎

######  Proposition 3.13.

For any Lie connection $\nabla$,
$\operatorname{Gal}(\nabla)\subseteq\operatorname{Aut}(\mathfrak{g})$.

###### Proof.

Let us choose a point $x\in M$ regular for $\nabla$ and a basis
$A_{1},\ldots,A_{r}$ of $\mathfrak{g}$, i.e., a basis $Y_{1},\ldots,Y_{r}$ of
local $\nabla$-horizontal section of $TM$ at $x$.

Using notation of Section A.6, we will identify $R^{1}(T_{x}M)$ with the set
of isomorphisms of linear spaces $\sigma\colon\mathfrak{g}\to T_{x}M$; now
$\operatorname{Gal}(\nabla)\subseteq{\rm GL}(\mathfrak{g})$. Because of the
construction of $\mathfrak{g}$, we have a canonical point in $R^{1}(TM)$
corresponding to the identity $\sigma_{o}\colon\mathfrak{g}\to T_{x}M$.

For $m\in M$, if $\sigma$ is an isomorphism from $\mathfrak{g}$ to $T_{m}M$
then one defines $H^{k}_{{i,j}}(\sigma)$ to be

$\displaystyle\frac{[X_{i},X_{j}]\wedge
X_{1}\wedge\cdots\wedge\widehat{X_{k}}\wedge\cdots\wedge X_{r}}{X_{k}\wedge
X_{1}\wedge\cdots\wedge\widehat{X_{k}}\wedge\cdots\wedge X_{r}}\Big{|}_{m},$

where $X_{i}$ is the horizontal section such that $X_{i}(m)=\sigma A_{i}$.
These functions are regular functions on $R^{1}(TM)$. Moreover they are
constant and equal to the constant structures on the Zariski closure of the
leaf passing through $\sigma_{o}$. 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 $\mathfrak{h}^{\prime}$ be a Lie sub-algebra of the Lie algebra of some
algebraic group and let $G$ be the smallest algebraic subgroup such that
${\mathfrak{lie}}(G)=\mathfrak{g}\supset\mathfrak{h}^{\prime}$. Assume the
existence of an algebraic subgroup $H$ of $G$ whose Lie algebra $\mathfrak{h}$
is supplementary to $\mathfrak{h}^{\prime}$ in $\mathfrak{g}$,
$\mathfrak{g}=\mathfrak{h}\oplus\mathfrak{h}^{\prime}$. Let us consider the
following objects:

  * $(a)$

the quotient map $\pi\colon G\to M$ where $M$ is the variety of cosets
$H\setminus G$, and $\nabla$ the Lie connection associated to the parallelism
$\pi_{*}\colon\mathfrak{h}^{\prime}\to\mathfrak{X}[M]$ in $M$ $($as given in
Example 2.13);

  * $(b)$

its reciprocal Lie connection $\nabla^{\rm rec}$ on $M$;

  * $(c)$

the Lie algebras of right invariant vector fields

$\displaystyle\mathfrak{g}^{\rm rec}={\bf
i}_{*}(\mathfrak{g}),\qquad\mathfrak{h}^{\prime\rm rec}={\bf
i}_{*}(\mathfrak{h}^{\prime}),$

where ${\bf i}$ is the inverse map on $G$.

Then, the following statements are true:

  * $(i)$

$\mathfrak{h}^{\prime}$ is an ideal of $\mathfrak{g}$ $($equivalently
$\mathfrak{h}^{\prime\rm rec}$ is an ideal of $\mathfrak{g}^{\rm rec})$;

  * $(ii)$

$\mathfrak{h}$ is commutative $($equivalently $H$ is virtually abelian$)$;

  * $(iii)$

the adjoint action of $G$ on $\mathfrak{g}^{\rm rec}$ preserves
$\mathfrak{h}^{\prime\rm rec}$ and thus gives, by restriction, a morphism
$\overline{\operatorname{Adj}}\colon
G\to\operatorname{Aut}(\mathfrak{h}^{\prime\rm rec})$;

  * $(iv)$

The Galois group of the connection $\nabla^{\rm rec}$ is
${\overline{\operatorname{Adj}}}(H)\subseteq\operatorname{Aut}(\mathfrak{h}^{\prime\rm
rec})$ and thus virtually abelian.

###### Proof.

We have that $\mathfrak{g}$ is the algebraic hull of $\mathfrak{h}^{\prime}$.
From Lemma B.3 in Appendix B we obtain
$[\mathfrak{g},\mathfrak{g}]\subseteq\mathfrak{h}^{\prime}$. Statement (i)
follows straightforwardly. Let us consider $A$ and $B$ in $\mathfrak{h}$. Then
$[A,B]$ is in $\mathfrak{h}$ and also in $\mathfrak{h}^{\prime}$ by the
previous argument. Thus, $[A,B]=0$ and this finishes the proof of statement
(ii). Let us denote by $H^{\prime}$ the subgroup of $G$ spanned by the image
of $\mathfrak{h}^{\prime}$ by the exponential map. For each element $h\in
H^{\prime}$, the adjoint action of $h$ preserves the Lie algebra
$\mathfrak{h}^{\prime}$. By continuity of the adjoint action in the Zariski
topology, we have that $\mathfrak{h}^{\prime}$ 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 $\nabla^{\rm rec}$. Let us consider a basis
$\\{A_{1},\ldots,A_{m}\\}$ of $\mathfrak{h}^{\prime}$ and let $\bar{A}_{i}$ be
the projection $\pi_{*}(A_{i})$. We have an extension of differential fields

$\displaystyle(\mathbf{C}(M),\bar{\mathcal{D}})\subseteq(\mathbf{C}(G),\mathcal{D}),$

where $\bar{\mathcal{D}}$ stands for the $\mathbf{C}(M)$-vector space of
derivations spanned by $\bar{A}_{1},\ldots,\bar{A}_{m}$ and $\mathcal{D}$
stands for the $\mathbf{C}(G)$-vector space of derivations spanned by
$A_{1},\ldots,A_{m}$ (see Appendix A for our conventions on differential
fields).

The projection $\pi$ is a principal $H$-bundle. Any rational first integral of
$\\{A_{1},\ldots,A_{m}\\}$ is constant along $H^{\prime}$ 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 $h\in H$
is the pullback of functions by the left translation $L_{h}^{-1}$, that is,
$(hf)(g)=f\big{(}h^{-1}g\big{)}$.

The horizontal sections for the connection $\nabla^{\rm rec}$ are
characterized by the differential equations

$\displaystyle[\bar{A}_{i},X]=0.$ (3.1)

Let us consider $\\{B_{1},\ldots,B_{m}\\}$ a basis of $\mathfrak{h}^{\prime\rm
rec}$. From the Zariski closedness of $H$ in $G$ it follows that there are
regular functions $f_{ij}\in\mathbf{C}[G]$ such that
$B_{i}=\sum\limits_{j=1}^{m}f_{ij}A_{j}$. Thus let us define
$\bar{B}_{i}=\sum\limits_{j=1}^{m}f_{ij}\bar{A}_{j}$. Those objects are vector
fields in $M$ with coefficients in $\mathbf{C}[G]$, and clearly satisfy
equation (3.1). Thus, the Picard–Vessiot extension of $\nabla^{\rm rec}$ is
spanned by the functions $f_{ij}$ and it is embedded, as a differential field,
in $\mathbf{C}(G)$. Let us denote such extension by $\mathbf{L}$. We have a
chain of extensions

$\displaystyle\mathbf{C}(M)\subseteq\mathbf{L}\subseteq\mathbf{C}(G).$

By Galois correspondence, the Galois group of $\nabla^{\rm rec}$ 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
$\overline{\operatorname{Adj}}$.

Let us note that the image under the adjoint action by $g\in G$ of an element
$B\in\mathfrak{h}^{\prime\rm rec}$ is given by the left translation,
$\overline{\operatorname{Adj}}(g)(B)=L_{g*}(B)$. This transformation makes
sense for any derivation of $\mathbf{C}[G]$, and thus we have an action of $G$
on $\mathfrak{X}(G)$. Let us take $h$ in the kernel of
$\overline{\operatorname{Adj}}$, thus,
$\overline{\operatorname{Adj}}(h)(B_{j})=B_{j}$ for any index $j$. Applying
the transformation $L_{h*}$ to the expression of $B_{i}$ as linear combination
of the left invariant vector fields $A_{j}$ we obtain
$B_{i}=\sum\limits_{j=1}^{m}L_{h*}(f_{ij}A_{j})=\sum\limits_{j=1}^{m}h(f_{ij})A_{j}$.
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
$\mathbf{L}$. On the other hand, let us take $h\in H$ fixing $\mathbf{L}$.
Then $L_{h*}\big{(}\sum f_{ij}A_{j}\big{)}=\sum f_{ij}A_{j}$ thus
$\overline{\operatorname{Adj}}(h)(B_{i})=B_{i}$ and then $h$ is in the kernel
of $\overline{\operatorname{Adj}}$. ∎

###  3.5 Some examples of $\boldsymbol{\mathfrak{sl}_{2}}$-parallelisms

We will construct some parallelized varieties as subvarieties of the arc space
of the affine line ${\mathbb{A}}^{1}_{\mathbf{C}}$. This family of examples
show how to realize every subgroup of ${\rm PSL}_{2}(\mathbf{C})$ as the
Galois group of the reciprocal Lie connection.

####  3.5.1 The arc space of the affine line and its Cartan 1-form

In our special case, the arc space of the affine line
${\mathbb{A}}^{1}_{\mathbf{C}}$ with affine coordinate $z$, is the space of
all formal power series $\widehat{z}=\sum z^{(i)}\frac{x^{i}}{i!}$. It will be
denoted by ${\mathscr{L}}$, its ring of regular functions is
$\mathbf{C}[{\mathscr{L}}]=\mathbf{C}\big{[}z^{(0)},z^{(1)},z^{(2)},\ldots\big{]}$.
For an open subset $U\subset\mathbf{C}$ one denotes by ${\mathscr{L}}U$ the
set of power series $\widehat{z}$ with $z^{(0)}\in U$.

A biholomorphism $f\colon U\to V$ between open sets of $\mathbf{C}$ can be
lift to a biholomorphism ${\mathscr{L}}f\colon{\mathscr{L}}U\to{\mathscr{L}}V$
by composition $\widehat{z}\to f\circ\widehat{z}$.

Let $\widehat{\mathfrak{X}}$ be the Lie algebra of formal vector fields
$\mathbf{C}[[x]]\frac{\partial}{\partial x}$. One can build a rational form
$\sigma\colon T{\mathscr{L}}\to\widehat{\mathfrak{X}}$ in following way (see
[5, Section 2]). Let $v=\sum a_{i}\frac{\partial}{\partial z^{(i)}}$ be a
tangent vector at the formal coordinate $\widehat{p}$, i.e., an arc in the
Zariski open subset $\\{z^{(1)}\not=0\\}$. The local coordinate $\widehat{p}$
can be used to have formal coordinates $p_{0},p_{1},p_{2},\ldots$, on
$\mathscr{L}$ and $v$ can be written $v=\sum b_{i}\frac{\partial}{\partial
p_{i}}$. The form $\sigma$ is defined by $\sigma(v)=\sum
b_{i}\frac{x^{i}}{i!}\frac{\partial}{\partial x}$. This form is rational and
is an isomorphism between $T_{p}{\mathscr{L}}$ and $\widehat{\mathfrak{X}}$
satisfying $d\sigma=-\frac{1}{2}[\sigma,\sigma]$ and
$({\mathscr{L}}f)^{\ast}\sigma=\sigma$ for any biholomorphism $f$.

This means that $\sigma$ provides an action of $\widehat{\mathfrak{X}}$
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 ${\mathscr{L}}$ and $\widehat{\mathfrak{X}}$:
$\sigma^{-1}\big{(}\frac{\partial}{\partial x}\big{)}=\sum\limits_{i\geq
0}z^{(i+1)}\frac{\partial}{\partial z^{(i)}}$ is a derivation of degree $+1$
with respect to the pro-variety structure of ${\mathscr{L}}$. The total
derivation above will be denoted by $E_{-1}$. This gives a differential
structure to the ring $\mathbf{C}[{\mathscr{L}}]$.

####  3.5.2 The parallelized varieties

Let $\nu\in\mathbf{C}(z)$ be a rational function, $f$ be the rational function
on the arc space given by the Schwarzian derivative

$\displaystyle
f\big{(}z^{(0)},z^{(1)},z^{(2)},z^{(3)}\big{)}=\frac{z^{(3)}}{z^{(1)}}-\frac{3}{2}\left(\frac{z^{(2)}}{z^{(1)}}\right)^{2}+\nu\big{(}z^{(0)}\big{)}\big{(}z^{(1)}\big{)}^{2},$

and $I\subset\mathbf{C}[{\mathscr{L}}]$ be the $E_{-1}$-invariant ideal
generated by
$p\big{(}z^{(0)}\big{)}{z^{(1)}}^{2}f\big{(}z^{(0)},z^{(1)},z^{(2)},z^{(3)}\big{)}$
where $p$ is a minimal denominator of $\nu$.

######  Lemma 3.15.

The zero set $V$ of $I$ is a dimension $3$ subvariety of ${\mathscr{L}}$ and
$\omega(TV)={\mathfrak{sl}}_{2}(\mathbf{C})\subset\widehat{\mathfrak{X}}$.
This provides a ${\mathfrak{sl}}_{2}$-parallelism on $V$.

###### Proof.

One can compute explicitly this parallelism using $z^{(0)}$, $z^{(1)}$ and
$z^{(2)}$ as étale coordinates on a Zariski open subset of $V$. Let us first
compute the $\mathfrak{sl}_{2}$ action on ${\mathscr{L}}$. The standard
inclusion of $\mathfrak{sl}_{2}$ in $\widehat{\mathfrak{X}}$ is given by
$E_{-1}=\frac{\partial}{\partial x}$, $E_{0}=x\frac{\partial}{\partial x}$ and
$E_{1}=x^{2}\frac{\partial}{\partial x}$. Their actions on ${\mathscr{L}}$ are
given by $E_{-1}=\sum z^{(i+1)}\frac{\partial}{\partial z^{(i)}}$, $E_{0}=\sum
iz^{(i)}\frac{\partial}{\partial z^{(i)}}$ and $E_{1}=\sum
i(i-1)z^{(i-1)}\frac{\partial}{\partial z^{(i)}}$. The ideal $I$ is generated
by the functions $E_{-1}^{n}\cdot f$. By definition $E_{-1}\cdot f\in I$, a
direct computation gives that $E_{0}\cdot f=2f\in I$, $E_{1}\cdot f=0\in I$.
The relations in $\mathfrak{sl}_{2}$ give that $E_{-1}\cdot I\subset I$,
$E_{0}\cdot I\subset I$ and $E_{1}\cdot I\subset I$, i.e., the vector fields
$E_{-1}$, $E_{0}$ and $E_{1}$ are tangent to $V$. ∎

Now parameterizing $V$ by $z^{(0)}$, $z^{(1)}$ and $z^{(2)}$ one gets

$\displaystyle E_{-1}|_{\mathbf{C}^{3}}=z^{(1)}\frac{\partial}{\partial
z^{(0)}}+z^{(2)}\frac{\partial}{\partial
z^{(1)}}+\left(-\nu\big{(}z^{(0)}\big{)}\big{(}z^{(1)}\big{)}^{3}+\frac{3}{2}\frac{(z^{(2)})^{2}}{z^{(1)}}\right)\frac{\partial}{\partial
z^{(2)}},$ $\displaystyle
E_{0}|_{\mathbf{C}^{3}}=z^{(1)}\frac{\partial}{\partial
z^{(1)}}+2z^{(2)}\frac{\partial}{\partial z^{(2)}},\qquad
E_{1}|_{\mathbf{C}^{3}}=2z^{(1)}\frac{\partial}{\partial z^{(2)}}.$

They form a rational $\mathfrak{sl}_{2}$-parallelism on $\mathbf{C}^{3}$
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 ${\rm PSL}_{2}(\mathbf{C})$ can be realized as the
Galois group of the reciprocal connection of a parallelism of
$\mathbf{C}^{3}$.

###### Proof.

A direct computation shows that $z\mapsto\varphi(z)$ is an holomorphic
function satisfying

$\displaystyle\frac{\varphi^{\prime\prime\prime}}{\varphi^{\prime}}-\frac{3}{2}\left(\frac{\varphi^{\prime\prime}}{\varphi^{\prime}}\right)^{2}+\nu(\varphi)(\varphi^{\prime})^{2}=\nu(z),$

if and only if its prolongation
${\mathscr{L}}\varphi\colon\widehat{z}\mapsto\varphi(\widehat{z})$ on the
space ${\mathscr{L}}$ 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

$\displaystyle a^{\prime\prime\prime}+2\nu a^{\prime}+\nu^{\prime}a=0,$ (3.2)

a vector field $X=a(z)\frac{\partial}{\partial z}$ whose prolongation on
${\mathscr{L}}$ is

$\displaystyle{\mathscr{L}}X=a\big{(}z^{(0)}\big{)}\frac{\partial}{\partial
z^{(0)}}+a^{\prime}\big{(}z^{(0)}\big{)}z^{(1)}\frac{\partial}{\partial
z^{(1)}}+\big{(}a^{\prime\prime}\big{(}z^{(0)}\big{)}\big{(}z^{(1)}\big{)}^{2}+a^{\prime}\big{(}z^{(0)}\big{)}z^{(2)}\big{)}\frac{\partial}{\partial
z^{(2)}}+\cdots.$

The equation (3.2) ensures that ${\mathscr{L}}{X}$ is tangent to $V$. The
invariance of $\sigma$, $({\mathscr{L}}X)_{\ast}\sigma=0$, ensures that
${\mathscr{L}}X$ commutes with the ${\mathfrak{sl}}_{2}$-parallelism given
above. This means that for any solution $a$ of (3.2) the vector field

$\displaystyle a\big{(}z^{(0)}\big{)}\frac{\partial}{\partial
z^{(0)}}+a^{\prime}\big{(}z^{(0)}\big{)}z^{(1)}\frac{\partial}{\partial
z^{(1)}}+\big{(}a^{\prime\prime}\big{(}z^{(0)}\big{)}\big{(}z^{(1)}\big{)}^{2}+a^{\prime}\big{(}z^{(0)}\big{)}z^{(2)}\big{)}\frac{\partial}{\partial
z^{(2)}},$

commutes with $E_{-1}|_{\mathbf{C}^{3}}$, $E_{0}|_{\mathbf{C}^{3}}$ and
$E_{1}|_{\mathbf{C}^{3}}$.

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 $y^{\prime\prime}=\nu(z)y$. If $G\subset{\rm
SL}_{2}(\mathbf{C})$ is the Galois group of $y^{\prime\prime}=\nu(z)y$ then
the image of its second symmetric power representation $s^{2}\colon
G\to\operatorname{Sym}^{2}(\mathbf{C}^{2})$ is the Galois group of (3.2). The
kernel of this representation is $\\{{\rm Id},-{\rm Id}\\}$ then the Galois
group of (3.2) is an algebraic subgroup of ${\rm PSL}_{2}(\mathbf{C})$.

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 $(a,b,c)$

$\displaystyle z(1-z)F^{\prime\prime}+(c-(a+b+1)z)F^{\prime}-abF=0,$

and is equivalent to

$\displaystyle y^{\prime\prime}=\nu(\ell,n,m;z)y,$

with

$\displaystyle\nu(\ell,m,n;z)=\frac{\big{(}1-\ell^{2}\big{)}}{4z^{2}}+\frac{1-m^{2}}{4(1-z)^{2}}+\frac{1-\ell^{2}-m^{2}+n^{2}}{4z(1-z)},$

and

$\displaystyle F=z^{-c/2}(1-z)^{(c-a-b-1)/2}y,\qquad\ell=1-c,\qquad
m=c-a-b,\qquad n=a-b.$

These two equations have the same projectivized Galois group in ${\rm
PGL}_{2}(\mathbf{C})$. Any algebraic subgroup of ${\rm PGL}_{2}(\mathbf{C})$
will be realized by an appropriate choice of $(a,b,c)$.

####  3.5.4 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 $\Gamma(2)\subset{\rm
SL}_{2}(\mathbf{Z})$ and is Zariski dense in ${\rm SL}_{2}(\mathbf{C})$.

####  3.5.5 The triangular subgroups

For $b=0$ and $a=-1$ one can compute a basis of solutions of the equation: $1$
and $\int\big{(}\frac{1-z}{z}\big{)}^{c}{\rm d}z$. If $c$ is not rational, the
Galois group is the group of invertible matrices
$\left[\begin{smallmatrix}u&v\\\ 0&1\end{smallmatrix}\right]$. When $c$ is
rational then $u$ must be a root of the unity of order the denominators of
$c$. When $c\in\mathbf{Z}$, the Galois group is the group of matrices
$\left[\begin{smallmatrix}1&v\\\ 0&1\end{smallmatrix}\right]$.

For $b=0$ and $c=a+1$ a basis of solution is given by $z^{-a}$ and $1$. Its
Galois group is a subgroup of the group of matrices
$\left[\begin{smallmatrix}u&0\\\ 0&1\end{smallmatrix}\right]$. The parameter
$a$ is rational if and only if it is a finite subgroup.

####  3.5.6 The dihedral subgroups

For $c=1/2$ and $a+b=0$, a basis of solution is given by
$\big{(}\sqrt{z}+\sqrt{1-z}\big{)}^{a}$ and
$\big{(}\sqrt{z}-\sqrt{1-z}\big{)}^{a}$. The monodromy group is a dihedral
group in ${\rm GL}_{2}(\mathbf{C})$ whose quotients give dihedral subgroups of
${\rm PGL}_{2}(\mathbf{C})$.

####  3.5.7 The tetrahedral subgroup

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

$\displaystyle(z-1)^{-1/12}\Big{(}\sqrt{3}\big{(}z^{1/3}+1\big{)}\pm
2\sqrt{z^{2/3}+z^{1/3}+1}\Big{)}^{1/4}.$

####  3.5.8 The octahedral subgroup

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

$\displaystyle(z-1)^{-1/24}\Big{[}\sqrt{3}\big{(}\big{(}\sqrt{z}-1\big{)}^{1/3}+\big{(}\sqrt{z}+1\big{)}^{1/3}\big{)}^{1/3}$
$\displaystyle\qquad{}\pm
2\sqrt{\big{(}\sqrt{z}-1\big{)}^{2/3}+(z-1)^{1/3}+\big{(}\sqrt{z}+1\big{)}^{2/3}}\Big{]}^{1/4}.$

####  3.5.9 The icosaedral subgroup

This group is the monodromy group of hypergeometric equation for $\ell=1/2$,
$m=1/3$ and $n=1/5$. As icosahedral group is not solvable, the solution space
is not described using formulas as simple as in preceding examples. ∎

##  4 Darboux–Cartan connections

###  4.1 Connection of parallelism conjugations

Let $\omega$ be a rational coparallelism on $M$ of type $\mathfrak{g}$ and $G$
an algebraic group with Lie algebra of left invariant vector fields
$\mathfrak{g}$ and Maurer–Cartan form $\theta$. Denote by $M^{\star}$ the open
subset of $M$ in wich $\omega$ is regular. We will study the contruction of
conjugating maps between the parallelisms $(M,\omega)$ and $(G,\theta)$.

Let us consider the trivial principal bundle $\pi\colon P=G\times M\to M$. In
this bundle we consider the action of $G$ by right translations
$(g,x)*g^{\prime}=(gg^{\prime},x)$. Let $\Theta$ be the $\mathfrak{g}$-valued
form $\Theta=\theta-\omega$ in $P$.

######  Definition 4.1.

The kernel of $\Theta$ is a rational flat invariant connection on the
principal bundle $\pi\colon P\to M$. We call it the Darboux–Cartan connection
of parallelism conjugations from $(M,\omega)$ to $(G,\theta)$.

The equation $\Theta=0$ defines a foliation on $P$ transversal to the fibers
at regular points of $\omega$. The leaves of the foliation are the graphs of
analytic parallelism conjugations from $(M,\omega)$ to $(G,\theta)$. By means
of differential Galois theory the Darboux–Cartan connection has a Galois group
$\operatorname{Gal}(\Theta)$ with Lie algebra $\mathfrak{gal}(\Theta)$. The
following facts are direct consequences of the definition of the Galois group:

  * (a)

there is a regular covering map $c\colon(M^{\star},\omega)\to(U,\theta)$ with
$U$ an open subset of $G$, and $c^{*}(\theta)=\omega$ if and only if
$\operatorname{Gal}(\Theta)=\\{1\\}$;

  * (b)

there is a regular covering map $c\colon(M^{\star},\omega)\to(U,q_{*}\theta)$
with $U$ an open subset of $G/H$, $H$ a group of finite index, and
$c^{*}(q_{*}\theta|_{U})=\omega$ if and only if
$\mathfrak{gal}(\Theta)=\\{0\\}$.

In any case, the necessary and sufficient condition for $(M,\omega)$ and
$(G,\theta)$ to be isogenous parallelized varieties is that
$\mathfrak{gal}(\Theta)=\\{0\\}$.

###  4.2 Darboux–Cartan connection and Picard–Vessiot

Note that the coparallelism $\omega$ gives a rational trivialization of $TM$
as the trivial bundle of fiber $\mathfrak{g}$. In $TM$ we have defined the
connection $\nabla^{\rm rec}$ whose horizontal vector fields are the
symmetries of the parallelism. On the other hand, $G$ acts in $\mathfrak{g}$
by means of the adjoint action. The Cartan-Darboux connection induces then a
connection $\nabla^{\rm adj}$ in the associated trivial bundle
$\mathfrak{g}\times M$ of fiber $\mathfrak{g}$.

######  Proposition 4.2.

The map

$\displaystyle\tilde{\omega}\colon\ \big{(}TM,\nabla^{\rm
rec}\big{)}\to\big{(}\mathfrak{g}\times M,\nabla^{\rm adj}\big{)},\qquad
X_{x}\mapsto(\omega_{x}(X_{x}),x)$

is a birational conjugation of the linear connections $\nabla^{\rm rec}$ and
$\nabla^{\rm adj}$.

###### Proof.

It is clear that the map $\tilde{\omega}$ is birational. Let us consider
$\\{A_{1},\ldots,A_{m}\\}$ a basis of $\mathfrak{g}$. Let
$\rho\colon\mathfrak{g}\to\mathfrak{X}(M)$ be the parallelism associated to
the parallelism $\omega$ and let us define $X_{i}=\rho(A_{i})$. Then
$\\{X_{1},\ldots,X_{n}\\}$ is a rational frame in $M$ and the map
$\tilde{\omega}$ conjugates the vector field $X_{i}$ with the constant section
$A_{i}$ of the trivial bundle of fiber $\mathfrak{g}$. By definition of the
reciprocal connection

$\displaystyle\nabla^{\rm rec}_{X_{i}}X_{j}=[X_{i},X_{j}].$

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

$\displaystyle\nabla^{\rm adj}_{X_{i}}A_{j}=[A_{i},A_{j}].$

Therefore we have that $\tilde{\omega}$ is a rational morphism of linear
connections that conjugates $\nabla^{\rm rec}$ with $\nabla^{\rm adj}$. ∎

The following facts follow directly from Proposition 4.2, and basic properties
of the Galois group.

######  Corollary 4.3.

Let us consider the adjoint action $\rm{Adj}\colon G\to{\rm GL}(\mathfrak{g})$
and its derivative ${\rm
adj}\colon\mathfrak{g}\to\operatorname{End}(\mathfrak{g})$. The following
facts hold:

  * $(a)$

$\operatorname{Gal}(\nabla^{\rm rec})=\operatorname{Adj}({\rm Gal(\Theta)})$;

  * $(b)$

$\mathfrak{gal}(\nabla^{\rm rec})={\rm adj}(\mathfrak{gal}(\Theta))$;

  * $(c)$

if $\mathfrak{g}$ is centerless then $\mathfrak{gal}(\nabla^{\rm rec})$ is
isomorphic to $\mathfrak{gal}(\Theta)$;

  * $(d)$

assume $\mathfrak{g}$ is centerless, then the necessary and sufficient
condition for $(M,\omega)$ and $(G,\theta)$ to be isogenous is that
$\mathfrak{gal}(\nabla^{\rm rec})=\\{0\\}$.

###### Proof.

(a) and (b). First, by Proposition 4.2 we have that
$\operatorname{Gal}(\nabla^{\rm rec})=\operatorname{Gal}(\nabla^{\rm adj})$
and so $\mathfrak{gal}(\nabla^{\rm rec})=\mathfrak{gal}(\nabla^{\rm adj})$. By
definition $\nabla^{\rm adj}$ is the associated connection induced by $\Theta$
in the associated bundle $\mathfrak{g}\times M$. This trivial bundle is the
associated bundle induced by the adjoint representation
$\operatorname{Adj}\colon G\to\operatorname{End}(\mathfrak{g})$. Then, by
Theorem A.6, we have $\operatorname{Gal}(\nabla^{\rm
rec})=\operatorname{Adj}({\rm Gal(\Theta)})$ and
$\operatorname{Gal}(\nabla^{\rm rec})=\operatorname{Adj}({\rm Gal(\Theta)})$.

(c) It is a direct consequence of (b). The kernel of ${\rm
adj}\colon\mathfrak{g}\to\operatorname{End}(\mathfrak{g})$ is the center of
$\mathfrak{g}$.

(d) It follows from the definition of Darboux–Cartan connection (see remarks
after Definition 4.1) that the necessary and sufficient condition for
$(M,\omega)$ and $(G,\theta)$ to be isogenous is that
$\mathfrak{gal}(\nabla^{\rm rec})=\\{0\\}$. By point (b) we conclude. ∎

###  4.3 Algebraic Lie algebras

Let us consider $(M,\omega)$ a rational coparallelism of type $\mathfrak{g}$
with $\mathfrak{g}$ a centerless Lie algebra. We do not assume _a priori_ that
$\mathfrak{g}$ is an algebraic Lie algebra. The connection $\nabla^{\rm rec}$
is, as said in Proposition 4.2, conjugated to the connection in
$\mathfrak{g}\times M$ 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 $\mathfrak{g}$. We have an exact sequence

$\displaystyle 0\to\mathfrak{g}^{\prime}\to\mathfrak{g}\to\mathfrak{g}^{ab}\to
0,$

where $\mathfrak{g}^{\prime}$ is the derived algebra
$[\mathfrak{g},\mathfrak{g}]$. Since the Galois group acts by adjoint action,
we have that $\mathfrak{g}^{\prime}\times M$ is stabilized by the connection
$\nabla^{\rm rec}$ and thus we have an exact sequence of connections

$\displaystyle 0\to(\mathfrak{g}^{\prime}\times
M,\nabla^{\prime})\to\big{(}\mathfrak{g}\times M,\nabla^{\rm
rec}\big{)}\to\big{(}\mathfrak{g}^{ab}\times M,\nabla^{ab}\big{)}\to 0.$

######  Lemma 4.4.

The Galois group of $\nabla^{ab}$ is the identity, therefore $\nabla^{ab}$ has
a basis of rational horizontal sections.

###### Proof.

By definition, the action of $\mathfrak{g}$ in $\mathfrak{g}^{ab}$ vanishes.
Thus, the constant functions $M\to\mathfrak{g}^{ab}$ are rational horizontal
sections. ∎

######  Lemma 4.5.

Let $\omega$ be a rational coparallelism of $M$ of type $\mathfrak{g}$ with
$\mathfrak{g}$ a centerless Lie algebra. If $\mathfrak{gal}(\nabla^{\rm
rec})=\\{0\\}$ then $\mathfrak{g}$ is an algebraic Lie algebra.

###### Proof.

Assume $\mathfrak{g}$ is a linear Lie algebra and et $E$ be the smallest
algebraic subgroup such that
$\operatorname{Lie}(E)=\mathfrak{e}\supset\mathfrak{g}$. We may assume that
$E$ is also centerless. Let $A_{1},\ldots,A_{r}$ be a basis of $\mathfrak{g}$,
for $i=1,\ldots,r$, $X_{i}=\omega^{-1}(A_{i})$. Complete with
$B_{1},\ldots,B_{p}$ in such way that $A_{1},\ldots,A_{r},B_{1},\ldots,B_{p}$
is a basis of $\mathfrak{e}$. We consider in $E\times M$ the distribution
spanned by the vector fields $A_{i}+X_{i}$. This is a $E$-principal connection
called $\nabla$.

Let $\overline{\nabla}$ be the induced connection via the adjoint
representation on $\mathfrak{e}\times M$ then

  1. 1)

$\overline{\nabla}$ preserves $\mathfrak{g}$ and
$\overline{\nabla}|_{\mathfrak{g}}=\nabla^{\rm rec}$, by hypothesis
$\mathfrak{gal}(\overline{\nabla}|_{\mathfrak{g}})=\\{0\\}$;

  2. 2)

if $\widetilde{\nabla}$ is the quotient connection on
${\mathfrak{e}}/\mathfrak{g}$ then
$\mathfrak{gal}(\widetilde{\nabla})=\\{0\\}$.

If $\varphi\in\mathfrak{gal}(\overline{\nabla})$ then for any
$X\in\mathfrak{g}$, $[X,B_{i}]\in\mathfrak{g}$ thus
$0=\varphi[X,B_{i}]=[X,\varphi B_{i}]$ and $\varphi B_{i}$ commute with
$\mathfrak{g}$. From the second point above $\varphi B\in\mathfrak{g}$. By
hypothesis $\varphi B_{i}=0$ and $\mathfrak{gal}(\overline{\nabla})=\\{0\\}$.
The projection on $E$ of an algebraic leaf of $\nabla$ gives an algebraic leaf
for the foliation of $E$ by the left translation by $\mathfrak{g}$. This
proves the lemma. ∎

######  Theorem 4.6.

Let $\mathfrak{g}$ be a centerless Lie algebra. An algebraic variety
$(M,\omega)$ with a rational parallelism of type $\mathfrak{g}$ is isogenous
to an algebraic group if and only if $\mathfrak{gal}(\nabla^{\rm
rec})=\\{0\\}$.

###### Proof.

It follows directly from Lemma 4.5 and Corollary 4.3. ∎

######  Corollary 4.7.

Let $\mathfrak{g}$ be a centerless Lie algebra. Any algebraic variety endowed
with a pair of commuting rational parallelisms of type $\mathfrak{g}$ 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 $\mathbf{CP}_{1}$ that are not
exact (isogenous to $(\mathbf{C},{\rm d}z)$) nor logarithmic (isogenous to
$(\mathbf{C}^{*},{\rm d}\log(z))$). In these examples, the pair of commuting
parallelisms is given by twice the same parallelism.

######  Remark 4.8.

Let $(M,\omega,\omega^{\prime})$ be a manifold endowed with a pair of
commuting parallelism forms of type $\mathfrak{g}$, a centerless Lie algebra.
From Lemma 4.5 we have that $\mathfrak{g}$ is an algebraic Lie algebra. We can
construct the algebraic group enveloping $\mathfrak{g}$ as follows. We
consider the adjoint action

$\displaystyle{\rm adj}\colon\
\mathfrak{g}\hookrightarrow\operatorname{End}(\mathfrak{g}).$

The algebraic group enveloping $\mathfrak{g}$ is identified with the algebraic
subgroup $G$ of $\operatorname{Aut}(\mathfrak{g})$ whose Lie algebra is ${\rm
adj}(\mathfrak{g})$. From Corollary 4.3(a), we have that
$\operatorname{Gal}(\Theta)=\\{e\\}$. Thus, there is a rational map $f\colon
M\to G$ such that $f^{*}(\theta)=\omega$, where $\theta$ is the Maurer–Cartan
form of $G$. We can express explicitly this map in terms of the commuting
parallelism forms. For each $x\in M$ in the domain of regularity of the
parallelisms, $\omega(x)$ and $\omega^{\prime}(x)$ are isomorphisms of
$T_{x}M$ with $\mathfrak{g}$. We define

$\displaystyle f(x)=-\omega(x)\circ\omega^{\prime}(x)^{-1}.$

######  Remark 4.9.

In virtue of Corollary 4.7, if $\mathfrak{g}$ is a non-algebraic centerless
Lie algebra, there is no algebraic variety endowed with a pair of regular
commuting parallelisms of type $\mathfrak{g}$. This limits the possible
generalizations of Theorem B.1.

######  Remark 4.10.

B. Malgrange has given in [8] another criterion: If $(M,\omega)$ is a
parallelized variety and $\mathcal{F}$ is the foliation on $M\times M$ given
by $\operatorname{pr}_{1}^{\ast}\omega-\operatorname{pr}_{2}^{\ast}\omega=0$.
Then $(M,\omega)$ is birational to an algebraic group if and only if leaves of
$\mathcal{F}$ are graphs of rational maps. The relations with Theorem 4.6 and
Corollary 4.7 are the following. One can identify $TM$ with the vertical
tangent (i.e., the kernel of ${\rm d}\operatorname{pr}_{2}$) along the
diagonal in $M\times M$. The diagonal is a leaf of $\mathcal{F}$ and the
linearization of $\mathcal{F}$ along the diagonal defines a connection
$\nabla_{\mathcal{F}}$ on $TM$. By construction:

  * •

$\nabla_{\mathcal{F}}$-horizontal sections commute with the parallelism, it is
the reciprocal Lie connection;

  * •

if leaves of $\mathcal{F}$ are algebraic then
$\nabla_{\mathcal{F}}$-horizontal section are algebraic.

##  5 Some homogeneous varieties

The notion of isogeny can be extended beyond the simply-transitive case. Let
us consider a complex Lie algebra $\mathfrak{g}$. An infinitesimally
homogeneous variety of type $\mathfrak{g}$ is a pair $(M,\mathfrak{s})$
consisting of a complex smooth irreducible variety $M$ and a finite-
dimensional Lie algebra $\mathfrak{s}\subset\mathfrak{X}(M)$ isomorphic to
$\mathfrak{g}$.

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 $f\colon M_{1}\dasharrow
M_{2}$ between varieties of the same dimension conjugates the infinitesimally
homogeneous varieties $(M_{1},\mathfrak{s}_{1})$ and
$(M_{2},\mathfrak{s}_{2})$ if $f^{*}(\mathfrak{s}_{2})=\mathfrak{s}_{1}$. We
say that $(M_{1},\mathfrak{s}_{1})$ and $(M_{2},\mathfrak{s}_{2})$ are
isogenous if they are conjugated to the same infinitesimally homogeneous space
of type $\mathfrak{g}$.

Let $G$ be an algebraic group over $\mathbf{C}$, $K$ an algebraic subgroup,
$\mathfrak{lie}(G)$ its Lie algebra of left invariant vector fields and
$\mathfrak{lie}(G)^{\rm rec}$ 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
$\mathfrak{lie}(G)^{\rm rec}$. We want to recognize when a infinitesimally
homogeneous space is isogenous to an homogeneous space. We prove that if
$\mathfrak{s}\subset\mathfrak{X}(M)$ is a normal Lie algebra of vector fields
then $(M,\mathfrak{s})$ is isogenous to a homogeneous space. In particular, we
prove that any $n$-dimensional infinitesimally homogeneous space of type
$\mathfrak{sl}_{n+1}(\mathbf{C})$ is isogenous to the projective space. Our
answer is based on a generalization of the computations done in Section 3.5.

###  5.1 The $\boldsymbol{\mathfrak{sl}_{2}}$ case

######  Theorem 5.1 (Loray–Pereira–Touzet (private communication)).

Let $\mathscr{C}$ be a curve with $X$, $Y$, $H$ three rational vector fields
such that $[X,Y]=H$, $[H,X]=-X$ and $[H,Y]=Y$. Then there exists a rational
dominant map $h\colon\mathscr{C}\dasharrow\mathbf{CP}_{1}$ such that
$X=h^{\ast}\big{(}\frac{\partial}{\partial z}\big{)}$,
$H=h^{\ast}\big{(}z\frac{\partial}{\partial z}\big{)}$ and
$Y=h^{\ast}\big{(}z^{2}\frac{\partial}{\partial z}\big{)}$.

Their proof is elementary. We outline here a more sophisticated proof in the
case $\mathscr{C}=A^{1}_{\mathbf{C}}$ that will be generalized in the next
section.

###### Proof.

Notations are the ones introduced in Section 3.5. $\mathscr{L}$ is the space
of parameterized arcs $\widehat{z}=\sum_{i}z^{(i)}\frac{x^{i}}{i!}$ on
$\mathscr{C}$. The vector space $\mathbf{C}X+\mathbf{C}H+\mathbf{C}Y$ is
denoted by $\mathfrak{g}$. Let $r_{o}\colon(\mathbf{C},0)\to
A^{1}_{\mathbf{C}}$ be an arc with $r_{o}^{\prime}(0)\not=0$ and consider
$V\subset\mathscr{L}$ defined by

$\displaystyle
V=\\{\widehat{z}\in\mathscr{L}\,|\,\widehat{z}^{\ast}\mathfrak{g}=r_{o}^{\ast}\mathfrak{g}\\}.$

######  Claim 5.2.

This is a $3$-dimensional algebraic variety.

######  Claim 5.3.

The prolongations $\mathscr{L}X$, $\mathscr{L}Y$ and $\mathscr{L}H$ define a
$\mathfrak{sl}_{2}$-parallelism on $V$.

Let us describe the canonical structure of $\mathscr{L}$ (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 $\mathscr{L}$

$\displaystyle E_{k}=\sum_{i\geq
k}\frac{i!}{(i-k-1)!}z^{(i-k)}\frac{\partial}{\partial z^{(i)}}.$

We define a morphism of Lie algebra
$\rho\colon\widehat{\mathfrak{X}}\to\mathfrak{X}(\mathscr{L})$ by
$x^{k+1}\frac{\partial}{\partial x}\mapsto E_{k}$ and the adic continuity.

######  Claim 5.4.

The Cartan form $\sigma$ $($as defined in Section 3.5.1) restricted to $V$
takes values in the Lie algebra $r_{0}^{*}(\mathfrak{g})$. It is the
parallelism form reciprocal to the parallelism $\mathscr{L}X$, $\mathscr{L}H$
and $\mathscr{L}Y$ of $V$.

Using Corollary 4.7, $V$ is isogeneous to ${\rm PSL}_{2}(\mathbf{C})$ as
defined in Definition 2.7. For $p\in M$, $V_{p}=\\{\widehat{z}\in V\ |\
\widehat{z}(0)=p\\}$ are homogeneous spaces for the action of
$\widetilde{K}=\\{\varphi\colon(\mathbf{C},0)\to(\mathbf{C},0)\,|\,r_{o}\circ\varphi\in
V\\}$, i.e., $\mathscr{C}=V/\widetilde{K}$. Let $K$ be the subgroup of ${\rm
PSL}_{2}(\mathbf{C})$ of upper triangular matrices.

######  Claim 5.5.

The actions of $\widetilde{K}$ on $V$ and the right action of $K$ on ${\rm
PSL}_{2}(\mathbf{C})$ are conjugated by the isogeny.

This induces an isogeny between $\mathscr{C}$ and $\mathbf{CP}_{1}$. Let
$\pi_{1}$ and $\pi_{2}$ be the two maps of the isogeny. A local transformation
$\varphi$ such that $\pi_{1}\circ\varphi=\pi_{1}$ satisfies
$\varphi^{\ast}\pi_{1}^{\ast}(X,H,Y)=\pi_{1}^{\ast}(X,H,Y)$ and the same is
true for the push-forward $(\pi_{2})_{\ast}\varphi$ of $\varphi$ on
$\mathbf{CP}_{1}$. Then $(\pi_{2})_{\ast}\varphi$ preserves
$\frac{\partial}{\partial z}$ and $z\frac{\partial}{\partial z}$. It is the
identity. This finishes the proof. ∎

###  5.2 Some jet spaces

Let $M$ be a $n$-dimensional affine variety. The space of parameterized
subspaces of $M$ is the set of formal maps:
$M^{[n]}=\\{r\colon(\mathbf{C}^{n},0)\to M\\}$. Like the arc space, it has a
natural structure of pro-algebraic variety. We will give the construction of
its coordinate ring following [1, Section 2.3.2, p. 80]. Let
$\mathbf{C}[\partial_{1},\ldots,\partial_{n}]$ be the $\mathbf{C}$-vector
space of linear partial differential operators with constant coefficients. The
coordinate ring of $M^{[n]}$ is
$\operatorname{Sym}(\mathbf{C}[M]\otimes\mathbf{C}[\partial_{1},\ldots,\partial_{n}])/\mathcal{L}$
where

  * •

the tensor product is a tensor product of $\mathbf{C}$-vector spaces;

  * •

$\operatorname{Sym}(V)$ is the $\mathbf{C}$-algebra generated by the vector
space $V$;

  * •

$\mathbf{C}[M]\otimes\mathbf{C}[\partial_{1},\ldots,\partial_{n}]$ has a
structure of $\mathbf{C}[\partial_{1},\ldots,\partial_{n}]$-module via the
right composition of differential operators;

  * •

$\operatorname{Sym}(\mathbf{C}[M]\otimes\mathbf{C}[\partial_{1},\ldots,\partial_{n}])$
has the induced structure of
$\mathbf{C}[\partial_{1},\ldots,\partial_{n}]$-algebra;

  * •

the Leibniz ideal $\mathcal{L}$ is the
$\mathbf{C}[\partial_{1},\ldots,\partial_{n}]$-ideal generated by $fg\otimes
1-(f\otimes 1)(g\otimes 1)$ for all $(f,g)\in\mathbf{C}[M]^{2}$ and by
$1-1\otimes 1$.

Local coordinates $(z_{1},\ldots,z_{n})$ on $M$ induce local coordinates on
$M^{[n]}$ via the Taylor expansion of maps $r$ at $0$

$\displaystyle
r(x_{1}\ldots,x_{n})=\left(\sum_{\alpha\in\mathbf{N}^{n}}r_{1}^{\alpha}\frac{x^{\alpha}}{\alpha!},\ldots,\sum_{\alpha\in\mathbf{N}^{n}}r_{n}^{\alpha}\frac{x^{\alpha}}{\alpha!}\right).$

One denotes by $z_{i}^{\alpha}\colon M^{[n]}\to\mathbf{C}$ the function
defined by $z_{i}^{\alpha}(r)=r_{i}^{\alpha}$. This function is the element
$z_{i}\otimes\partial^{\alpha}$ in $\mathbf{C}[M^{[n]}]$.

####  5.2.1 Prolongation of vector fields

Any derivation $Y$ of $\mathbf{C}[M]$ can be trivially extended to a
derivation of
$\operatorname{Sym}(\mathbf{C}[M]\otimes\mathbf{C}[\partial_{1},\ldots,\partial_{n}])$.
It preserves the ideal generated by $fg\otimes 1-(f\otimes 1)(g\otimes 1)$ for
all $(f,g)\in\mathbf{C}[M]^{2}$ and by $1-1\otimes 1$ and commutes with the
action of $\mathbf{C}[\partial_{1},\ldots,\partial_{n}]$ then it preserves the
Leibniz ideal and defines a derivation of $\mathbf{C}[M^{[n]}]$. This
derivation is called the prolongation of $Y$, and it is denoted by $Y^{[n]}$.

The same procedure can be used to define the prolongation of analytic or
formal vector fields on $M$ to $M^{[n]}$.

####  5.2.2 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.

The action of $\partial_{j}\colon\mathbf{C}[M^{[n]}]\to\mathbf{C}[M^{[n]}]$
can be written in local coordinates and gives the total derivative operator
$\sum_{i,\alpha}z_{i}^{\alpha+1_{j}}\frac{\partial}{\partial z_{i}^{\alpha}}$.
It is the differential structure of the jet space. The pro-algebraic group

$\displaystyle\Gamma=\big{\\{}\gamma\colon(\mathbf{C}^{n},0)\overset{\sim}{\rightarrow}(\mathbf{C}^{n},0);\text{
formal invertible}\big{\\}}$

acts on $M^{[n]}$.This action is denoted by $S\gamma(r)=r\circ\gamma$.

These two actions arise from the action of the Lie algebra
$\widehat{\mathfrak{X}}=\bigoplus\mathbf{C}[[x_{1},\ldots,x_{n}]]\partial_{i}$
on $M^{[n]}$. This action is described on the coordinate ring in the following
way. For $\xi\in\widehat{\mathfrak{X}}$, $f\in\mathbf{C}[M]$ and
$P\in\mathbf{C}[\partial_{1},\ldots,\partial_{n}]$, we define
$\xi\cdot(f\otimes P)=f\otimes(P\circ\xi)|_{0}$ where the composition is
evaluated in $0$ in order to get an element of
$\mathbf{C}[\partial_{1},\ldots,\partial_{n}]$. The action of
$\bigoplus\mathbf{C}\partial_{i}$ is the differential structure. The action of
$\widehat{\mathfrak{X}}^{0}=\mathfrak{lie}(\Gamma)$, the Lie subalgebra of
vector fields vanishing at $0$ is the infinitesimal part of the action of
$\Gamma$.

######  Theorem 5.6 ([5]).

Let $M^{[n]\ast}$ be the open subset of submersions. The action above gives a
canonical form $\sigma\colon TM^{[n]\ast}\to\widehat{\mathfrak{X}}$
satisfying:

  * •

for any $r\in M^{[n]\ast}$, $\sigma$ is a isomorphism from $T_{r}M^{[n]\ast}$
to $\widehat{\mathfrak{X}}$;

  * •

for any $\gamma\in\Gamma$, $(S\gamma)^{\ast}\sigma=\gamma^{\ast}\circ\sigma$;

  * •

$d\sigma=-\frac{1}{2}[\sigma,\sigma]$.

These equalities are not compatible with the projective systems.

###  5.3 Normal Lie algebras of vectors fields

Without lost of generality, we should

  1. 1)

identify $\mathfrak{g}$ with its image in $\mathfrak{X}(M)$;

  2. 2)

replace $M$ by a Zariski open subvariety on which $\mathfrak{g}$ is defined
and of maximal rank at any point.

If $p\in M$ one can identify $\mathfrak{g}$ with a Lie subalgebra of
$\widehat{\mathfrak{X}}(M,p)$, the Lie algebra of formal vector fields on $M$
at $p$.

######  Definition 5.7.

For a Lie subalgebra $\mathfrak{g}\subset\mathfrak{X}[M]$, its normalizer at
$p\in M$ is

$\displaystyle\widehat{N}(\mathfrak{g},p)=\big{\\{}Y\in\widehat{\mathfrak{X}}(M,p)\,|\,Y,\mathfrak{g}]\subset\mathfrak{g}\big{\\}}.$

######  Definition 5.8.

A Lie subalgebra $\mathfrak{g}\subset\mathfrak{X}[M]$ is said to be normal if
for generic $p\in M$ on has $\widehat{N}(\mathfrak{g},p)=\mathfrak{g}$.

######  Lemma 5.9.

If $\mathfrak{g}$ is transitive then the Lie algebra
$\widehat{N}(\mathfrak{g},p)$ is finite-dimensional.

###### Proof.

Let $k$ be an integer large enough so that the only element of $\mathfrak{g}$
vanishing at order $k$ at $p$ is $0$. If $\widehat{N}(\mathfrak{g},p)$ is not
finite-dimensional then there exists a non-zero
$Y\in\widehat{N}(\mathfrak{g},p)$ vanishing at order $k+1$ at $p$. For
$X\in\mathfrak{g}$, the Lie bracket $[Y,X]$ is an element of $\mathfrak{g}$
vanishing at order $k$ at $p$. It is zero meaning that $Y$ is invariant under
the flows of vector fields in $\mathfrak{g}$. The transitivity hypothesis
together with $Y(p)=0$ proves the lemma. ∎

######  Lemma 5.10.

If there exists a point $p\in M$ such that $\mathfrak{g}$ is maximal among
finite-dimensional Lie subalgebra of $\widehat{\mathfrak{X}}(M,p)$ then
$\mathfrak{g}$ is normal.

###### Proof.

Because of the preceding lemma, if such a point exists then
$\mathfrak{g}=\widehat{N}(\mathfrak{g},p)$ in $\widehat{\mathfrak{X}}(M,p)$.
By transitivity, for any couple of points $(p_{1},p_{2})\in M^{2}$ there is a
composition of flows of elements of $\mathfrak{g}$ sending $p_{1}$ on $p_{2}$.
These flows preserve $\mathfrak{g}$ thus the equality holds at any $p$. ∎

######  Example 5.11.

Let $M$ be $n$-dimensional and $\mathfrak{g}$ be a transitive Lie subalgebra
of rational vector fields isomorphic to $\mathfrak{sl}_{n+1}(\mathbf{C})$.
Then $\mathfrak{g}$ is normal (see [3]).

###  5.4 Centerless, transitive and normal $\boldsymbol{\Rightarrow}$
isogenous to a homogeneous space

######  Theorem 5.12.

Let $M$ be a smooth irreducible algebraic variety over $\mathbf{C}$ and
$\mathfrak{g}$ be a transitive, centerless, normal, finite-dimensional Lie
subalgebra of $\mathfrak{X}(M)$. Then there exists an algebraic group $G$, an
algebraic subgroup $H\subset G$ and an isogeny between $(M,\mathfrak{g})$ and
$(G/H,\mathfrak{lie}(G))$. Moreover, if $N_{G}(\mathfrak{lie}(H))=H$ then the
isogeny is a dominant rational map $M\dasharrow G/H$.

Because of the finiteness and the transitivity, there exists an integer $k$
such that at any $p\in M$ and for any $Y\in\widehat{N}(\mathfrak{g},p)$,
$j_{k}(Y)(p)\not=0$, unless $Y=0$.

Let $r_{o}\colon(\mathbf{C}^{n},0)\to M$ be an invertible formal map with
$r_{o}(0)=p$ a regular point. Let us consider the subspace of $M^{[n]}$
defined by

$\displaystyle V=\\{r\colon(\mathbf{C}^{n},0)\to
M\,|\,r^{\ast}\mathfrak{g}=r_{o}^{\ast}\mathfrak{g}\\}.$

######  Lemma 5.13.

$V$ is finite-dimensional.

###### Proof.

If $r_{o}^{-1}\circ r$ is tangent to the identity at order $k$ then the
induced automorphism of $\mathfrak{g}$ is the identity. The map
$r_{o}^{-1}\circ r$ fixes $p$, thus it is the identity. This proves the lemma.
∎

Using $r_{o}$ one can identify the Lie algebra $\widehat{N}(\mathfrak{g},p)$
with a Lie subalgebra of $\widehat{\mathfrak{X}}$. 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

$\displaystyle TV=r_{o}^{\ast}(\widehat{N}(\mathfrak{g},p))\times V,$

called the canonical parallelism.

######  Lemma 5.15.

The horizontal sections of the reciprocal Lie connection of the canonical
parallelism are $Y^{[n]}$ for $Y\in\widehat{N}(\mathfrak{g},q)$ for $q\in M$.

######  Lemma 5.16.

Under the hypothesis of normality of $\mathfrak{g}$, $V$ has two commuting
parallelisms of type $\mathfrak{g}$.

Using Corollary 4.7, $\mathfrak{g}$ is the Lie algebra of an algebraic group
$G$ isogeneous to $V$. $V$ is foliated by the orbits of the subgroup $K$ of
$\Gamma$ stabilizing $V$. This group is algebraic with Lie algebra
$\mathfrak{k}=r_{o}^{\ast}(\mathfrak{g})\cap\widehat{\mathfrak{X}}^{0}$. Let
$\mathfrak{h}\subset\mathfrak{lie}(G)$ be the Lie algebra corresponding to
$\mathfrak{k}$ by the isogeny. Then the orbits of $\mathfrak{h}$ are
algebraic. This means that $\mathfrak{h}$ is the Lie algebra of an algebraic
subgroup $H$ of $G$, and that $V/K$ and $G/H$ are isogenous.

Assume that $N_{G}(\mathfrak{lie}(H))=H$. If $W$ is the isogeny between $V$
and $G$. The push-forward of a local analytic deck transformation of $W\to V$
is a transformation of $G$ preserving each element of $\mathfrak{g}$, it is a
right translation. A deck transformation preserves the orbits of the pull-back
of $\mathfrak{k}$ 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.

##  Appendix 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
$(\mathcal{K},\mathcal{D})$ where $\mathcal{K}$ is a finitely generated field
over $\mathbf{C}$ and $\mathcal{D}$ is a $\mathcal{K}$ vector space of
derivations of $\mathcal{K}$ stable by Lie bracket. The dimension of
$\mathcal{D}$ 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 $\mathcal{D}$. However we prefer to consider the
whole space of derivations. With our definition a differential field extension
$(\mathcal{K},\mathcal{D})\to(\mathcal{K}^{\prime},\mathcal{D}^{\prime})$ is a
field extension $\mathcal{K}\subset\mathcal{K}^{\prime}$ such that each
element of $\mathcal{D}$ extends to a unique element of $\mathcal{D}^{\prime}$
and such extensions span the space $\mathcal{D}^{\prime}$ as
$\mathcal{K}^{\prime}$-vector space.

###  A.1 Differential field extensions and foliated varieties

First, let us see that there is a natural dictionary between finitely
generated differential fields over $\mathbf{C}$ and irreducible foliated
varieties over $\mathbf{C}$ modulo birational equivalence. Let
$(M,\mathcal{F})$ be an irreducible foliated variety of dimension $n$. The
distribution $T\mathcal{F}\subset TM$ is of rank $r\leq n$. We denote by
$\mathfrak{X}_{\mathcal{F}}$ the space of rational vector fields in
$T\mathcal{F}$; it is a $\mathbf{C}(M)$-Lie algebra of dimension $r$. Hence,
the pair $(\mathbf{C}(M),\mathfrak{X}_{\mathcal{F}})$ is a differential field.
The field of constants is the field $\mathbf{C}(M)^{\mathcal{F}}$ of rational
first integrals of the foliation.

Let $(M,\mathcal{F})$ and $(M^{\prime},\mathcal{F}^{\prime})$ be foliated
varieties. A regular (rational) map
$\phi\colon(M^{\prime},\mathcal{F}^{\prime})\dasharrow(M,\mathcal{F})$ is a
regular (rational) morphism of foliated varieties if ${\rm d}\phi$ induces an
isomorphism between $T_{x}\mathcal{F}^{\prime}$ and $T_{\phi(x)}\mathcal{F}$
for (generic values of) $x\in M^{\prime}$. It is clear that
$\mathcal{F}^{\prime}$ and $\mathcal{F}$ have the same rank.

A differential field extension, correspond here to a dominant rational map of
irreducible foliated varieties
$\phi\colon(M^{\prime},\mathcal{F}^{\prime})\dasharrow(M,\mathcal{F})$. It
induces the extension
$\phi^{*}\colon(\mathbf{C}(M),\mathfrak{X}_{\mathcal{F}})\to(\mathbf{C}(M^{\prime}),\mathfrak{X}_{\mathcal{F}^{\prime}})$
by composition with $\phi$.

######  Example A.1.

Let $\mathcal{F}$ the foliation of $\mathbf{C}^{2}$ defined by $\\{{\rm
d}y-y{\rm d}x=0\\}$. It corresponds to the differential field
$\big{(}\mathbf{C}(x,e^{x}),\big{\langle}\frac{{\rm d}}{{\rm
d}x}\big{\rangle}\big{)}$.

######  Remark A.2.

Throughout this appendix “connection” means “flat connection”.

###  A.2 Invariant $\boldsymbol{\mathcal{F}}$-connections

Let us consider from now a foliated manifold of dimension $n$ and rank $r$
without rational first integrals $(M,\mathcal{F})$, an algebraic group $G$ and
a principal irreducible $G$-bundle $\pi\colon P\to M$. A $G$-invariant
connection in the direction of $\mathcal{F}$ is a foliation
$\mathcal{F}^{\prime}$ of rank $r$ in $P$ such that:

  * (a)

$\pi\colon(P,\mathcal{F}^{\prime})\to(M,\mathcal{F})$ is a dominant regular
map of foliated varieties;

  * (b)

The foliation $\mathcal{F}^{\prime}$ is invariant by the action of $G$ in $P$.

With this definition
$(\mathbf{C}(M),\mathfrak{X}_{\mathcal{F}})\to(\mathbf{C}(P),\mathfrak{X}_{\mathcal{F}^{\prime}})$
is a differential field extension. Also, each element $g\in G$ induces a
differential field automorphism of $(\mathbf{C}(P),\mathfrak{X}_{F^{\prime}})$
that fixes $(\mathbf{C}(M),\mathfrak{X}_{F})$ by setting $(g\cdot
f)(x)=f(x\cdot g)$.

Let $\mathfrak{g}$ be the Lie algebra of $G$. There is a way of defining a
$G$-equivariant form $\Theta_{\mathcal{F}^{\prime}}$ with values in
$\mathfrak{g}$, and defined in ${\rm d}\pi^{-1}(T\mathcal{F})$ in such way
that $T\mathcal{F}^{\prime}$ is the kernel of $\Theta_{\mathcal{F}^{\prime}}$.
First, there is a canonical form $\Theta_{0}$ defined in $\ker(d\pi)$ that
sends each vertical vector $X_{p}\in\ker d_{p}\pi\subset T_{p}P$ to the
element $\mathfrak{g}$ that verifies,

$\displaystyle\left.\frac{{\rm d}}{{\rm
d}\varepsilon}\right|_{\varepsilon=0}p\cdot\exp{\varepsilon A}=X_{p}.$

This form is $G$-equivariant in the sense that
$R_{g}^{*}(\Theta_{0})=\operatorname{Adj}_{g^{-1}}\circ\omega$. We have a
decomposition of the vector bundle ${\rm d}\pi^{-1}(T\mathcal{F})=\ker({\rm
d}\pi)\oplus T\mathcal{F}^{\prime}$. This decomposition allows to extend
$\Theta_{0}$ to a form $\Theta_{\mathcal{F}^{\prime}}$ defined for vectors in
${\rm d}\pi^{-1}(T\mathcal{F})$ whose kernel is precisely $T\mathcal{F}$. We
call _horizontal frames_ to those sections $s$ of $\pi$ such that
$s^{*}(\Theta_{\mathcal{F}})=0$.

###  A.3 Picard–Vessiot bundle

We say that the principal $G$-bundle with invariant $\mathcal{F}$-connection
$\pi\colon(P,\mathcal{F}^{\prime})\to(M,\mathcal{F})$ is a  _Picard–Vessiot_
bundle if there are no rational first integrals of $\mathcal{F}^{\prime}$. 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 $(\mathbf{C}(P),\mathfrak{X}_{F^{\prime}})$ that fix
$(\mathbf{C}(M),\mathfrak{X}_{F})$ and
$(\mathbf{C}(M),\mathfrak{X}_{F})\to(\mathbf{C}(P),\mathfrak{X}_{F^{\prime}})$
is a strongly normal extension. Moreover, any strongly normal extension with
constant field $\mathbf{C}$ 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
$(\mathbf{C}(M),\mathfrak{X}_{\mathcal{F}})\to(\mathbf{C}(P),\mathfrak{X}_{\mathcal{F}^{\prime}})$
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
$\boldsymbol{\mathcal{F}}$-connection

Let us consider an irreducible principal $G$-bundle
$\pi\colon(P,\mathcal{F}^{\prime})\to(M,\mathcal{F})$ endowed with an
invariant $\mathcal{F}$-connection $\mathcal{F}^{\prime}$. We assume that
$\mathcal{F}$ 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 $\mathcal{L}$ of $\mathcal{F}$ in $M$. Let us consider any leaf
$\mathcal{L}^{\prime}$ of $\mathcal{F}^{\prime}$ in $P$ that projects by $\pi$
onto $\mathcal{L}$. Its Zariski closure is unique in the following sense:

######  Theorem A.4.

Let $\mathcal{L}^{\prime}$ and $\mathcal{L}^{\prime\prime}$ two leaves of
$\mathcal{F}^{\prime}$ whose projections by $\pi$ are Zariski dense in $M$.
Then, there exist an element $g\in G$ such that
$\overline{\mathcal{L}^{\prime}}\cdot
g=\overline{\mathcal{L}^{\prime\prime}}$.

###### Proof.

By construction, there is some
$x\in\pi(\mathcal{L}^{\prime})\cap\pi(\overline{\mathcal{L}^{\prime\prime}})$.
Let us consider $p\in\pi^{-1}(\\{x\\})\cap\mathcal{L}^{\prime}$ and
$q\in\pi^{-1}(\\{x\\})\cap\overline{\mathcal{L}^{\prime\prime}}$. Since $p$
and $q$ are in the same fiber, there is a unique element $g\in G$ such that
$p\cdot g=q$. By the $G$-invariance of the connection
$\mathcal{L}^{\prime}\cdot g$ is the leaf of $\mathcal{F}^{\prime}$ that
passes through $q$. The set $\overline{\mathcal{L}^{\prime\prime}}$ is, by
construction, union of leaves of $\mathcal{F}^{\prime}$ and contains the point
$q$. Thus, $\overline{\mathcal{L}^{\prime}\cdot
g}\subseteq\overline{\mathcal{L}^{\prime\prime}}$, and
$\overline{\mathcal{L}^{\prime}}\cdot
g\subseteq\overline{\mathcal{L}^{\prime\prime}}$. Now, by exchanging the roles
of $\mathcal{L}^{\prime}$ and $\mathcal{L}^{\prime\prime}$, we prove that
there is an element $h$ such that $\overline{\mathcal{L}^{\prime\prime}}\cdot
h\subseteq\overline{\mathcal{L}^{\prime}}$. It follows $h=g^{-1}$. This
finishes the proof. ∎

Let $L$ be the Zariski closure of $\mathcal{L}^{\prime}$. Let us consider the
algebraic subgroup

$\displaystyle H=\\{g\in G\colon L\cdot g=L\\}$

stabilizing $L$. The projection $\pi$ restricted to $L$ is dominant, thus
there is a Zariski open subset $M^{\star}$ such that $\pi^{\star}\colon
L^{\star}\to M^{\star}$ is surjective. Let us call $\mathcal{F}^{\star}$ the
restriction of $\mathcal{F}^{\prime}$ to $L^{\star}$. It follows that the
bundle:
$\pi^{\star}\colon(L^{\star},\mathcal{F}^{\star})\to(M^{\star},\mathcal{F}|_{M^{\star}})$
is a principal bundle of structure group $H$ called Picard–Vessiot bundle. The
differential field extension
$(\mathbf{C}(M),\mathfrak{X}_{\mathcal{F}})\to(\mathbf{C}(L^{\star}),\mathfrak{X}_{\mathcal{F}^{\star}})$
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

$\displaystyle\phi_{j}\colon\
(M_{j},\mathcal{F}_{j})\to(M,\mathcal{F}),\qquad\mbox{for}\quad j=1,2.$

Then, we can define in $M_{1}\times_{M}M_{2}$ a foliation
$\mathcal{F}_{1}\times_{\mathcal{F}}\mathcal{F}_{2}$ in the following way. A
vector $X=(X_{1},X_{2})$ is in
$T(\mathcal{F}_{1}\times_{\mathcal{F}}\mathcal{F}_{2})$ if and only if ${\rm
d}\phi_{1}(X_{1})={\rm d}\phi_{2}(X_{2})\in T\mathcal{F}$. Let us consider
$(P,\mathcal{F}^{\prime})$ a principal $\mathcal{F}$ connection. Note that the
projection

$\displaystyle\pi_{1}\colon\
(M_{1}\times_{M}P,\mathcal{F}_{1}\times_{\mathcal{F}}\mathcal{F}^{\prime})\to(M_{1},\mathcal{F}_{1})$

is a principal $G$-bundle endowed of a $\mathcal{F}_{1}$-connection. We call
this bundle the pullback of $(P,\mathcal{F}^{\prime})$ by $\phi_{1}$.

We also may consider the trivial $G$-invariant connection $\mathcal{F}_{0}$ in
the trivial principal $G$-bundle

$\displaystyle\pi_{0}\colon\ (M\times G,\mathcal{F}_{0})\to(M,\mathcal{F}),$

for what the leaves of $\mathcal{F}_{0}$ are of the form $(\mathcal{L},g)$
where $\mathcal{L}$ is a leaf of $\mathcal{F}$ and $g$ a fixed element of $G$.
We say that the $G$-invariant connection $(P,\mathcal{F}^{\prime})$ is
rationally trivial if there is a birational $G$-equivariant isomorphism of
foliated manifolds between $(P,\mathcal{F})$ and $(M\times
G,\mathcal{F}_{0})$.

Invariant connections are always trivialized after pullback; there is a
universal $G$-equivariant isomorphism defined over $P$

$\displaystyle(P\times
G,\mathcal{F}^{\prime}\times_{\mathcal{F}}\mathcal{F}_{0})\to(P\times_{M}P,\mathcal{F}^{\prime}\times_{\mathcal{F}}\mathcal{F}^{\prime}),\qquad(p,g)\mapsto(p,p\cdot
g),$

that trivializes any $G$-invariant connection. However, the differential field
$(\mathbf{C}(P),\mathfrak{X}_{\mathcal{F}^{\prime}})$ may have new constant
elements. To avoid this, we replace the pullback to $P$ by a pullback to the
Picard–Vessiot bundle $L^{\star}$

$\displaystyle(L^{\star}\times
G,\mathcal{F}^{\star}\times_{\mathcal{F}}\mathcal{F}_{0})\to(L^{\star}\times_{M}P,\mathcal{F}^{\star}\times_{\mathcal{F}}\mathcal{F}^{\prime}),\qquad(p,g)\mapsto(p,p\cdot
g).$

The Picard–Vessiot bundle has some minimality property. It is the smallest
bundle on $M$ that trivializes the connection. We have the following result.

######  Theorem A.5.

Let us consider $\pi\colon(P,\mathcal{F}^{\prime})\to(M,\mathcal{F})$ be as
above, $\pi^{\star}\colon(L^{\star},\mathcal{F}^{\star})\to(M,\mathcal{F})$
the Picard–Vessiot bundle, and and
$\phi\colon\big{(}\tilde{M},\tilde{\mathcal{F}}\big{)}\to(M,\mathcal{F})$ any
dominant rational map of foliated varieties such that:

  * $(a)$

$\tilde{\mathcal{F}}$ has no rational first integrals in $\tilde{M}$;

  * $(b)$

the pullback
$\big{(}\tilde{M}\times_{M}P,\tilde{F}\times_{\mathcal{F}}\mathcal{F}^{\prime}\big{)}\to\big{(}\tilde{M},\tilde{\mathcal{F}}\big{)}$
is rationally trivial.

There is a dominant rational map of foliated varieties
$\psi\colon\tilde{M}\dasharrow L^{\star}$ such that
$\pi^{\star}\circ\psi=\phi$ in their common domain.

###### Proof.

Let us take $\tau\colon\tilde{M}\times G\dasharrow\tilde{M}\times_{M}P$ a
birational trivialization, $\pi_{2}\colon\tilde{M}\times_{M}P\to P$ be the
projection in the second factor, and $\iota\colon\tilde{M}\to\tilde{M}\times
G$ the inclusion $p\mapsto(p,e)$. Then,
$\tilde{\psi}=\pi_{2}\circ\tau\circ\iota$ is a rational map from $\tilde{M}$
to $P$ whose differential sends $T\tilde{\mathcal{F}}$ to $T\mathcal{F}$. By
Bonnet theorem, $\tilde{M}$ is the Zariski closure of a leaf of
$\tilde{\mathcal{F}}$ that projects by $\phi$ into a Zariski dense leaf of
$\mathcal{F}$. From this, $\tilde{\psi}$ contains a dense leaf of
$\mathcal{F}^{\prime}$ 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 $(M,\mathcal{F})$ be as above, of dimension $n$ and rank $r$. Let
$\xi\colon E\to M$ be a vector bundle of rank $k$. A linear integrable
$\mathcal{F}$-connection is a foliation $\mathcal{F}_{E}$ 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 $\nabla$
wich is defined only in the direction of $\mathcal{F}$. First, the kernel of
${\rm d}\xi$ is naturally projected onto $E$ itself

$\displaystyle{\rm vert}_{0}\colon\ \ker({\rm d}\xi)\to E,\qquad X_{v}\mapsto
w,$

where $\left.\frac{{\rm d}}{{\rm
d}\varepsilon}\right|_{\varepsilon=0}v+\varepsilon w=X_{v}$. Then, the
decomposition of ${\rm d}\xi^{-1}(T\mathcal{F})$ as $\ker({\rm d}\xi)\oplus
T\mathcal{F}_{E}$ allows us to extend ${\rm vert_{0}}$ to a projection

$\displaystyle{\rm vert}\colon\ {\rm d}\xi^{-1}(T\mathcal{F})\to E.$

Thus, we define for each section $s$ its covariant derivative $\nabla
s=s^{*}({\rm vert}\circ{\rm d}s|_{T\mathcal{F}})$. This is a $1$-form on $M$
defined only for vectors in $T\mathcal{F}$. This covariant derivative has the
desired properties, it is additive and satisfies the Leibniz formula

$\displaystyle\nabla(fs)={\rm d}f|_{T\mathcal{F}}\otimes s+f\nabla s.$

In general, we write for $X$ a vector in $T\mathcal{F}$, $\nabla_{X}s$ for the
contraction of $\nabla s$ 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 $\xi$ such that $\nabla s=0$.

Let $\pi\colon R^{1}(E)\to M$ be the bundle of linear frames in $E$. It is a
principal linear ${\rm GL}_{k}(\mathbf{C})$-bundle. The foliation
$\mathcal{F}_{E}$ induces a foliation $\mathcal{F}^{\prime}$ in $R^{1}(E)$
that is a $G$-invariant $\mathcal{F}$-connection. Let us consider the
Picard–Vessiot bundle, $(L^{\star},\mathcal{F}^{\star})$. The uniqueness
Theorem A.5 on the Picard–Vessiot bundle, can be rephrased algebraically in
the following way. The Picard–Vessiot extension
$(\mathbf{C}(M),\mathfrak{X}_{\mathcal{F}})\to(\mathbf{C}(L^{\star}),\mathfrak{X}_{\mathcal{F}^{\star}})$
is characterized by the following properties (cf. [11, Section 1.3]):

  * (a)

there are no new constants, $\mathbf{C}(L^{\star})=\mathbf{C}$;

  * (b)

it is spanned, as a field extension of $\mathbf{C}(M)$, by the coefficients of
a fundamental matrix of solutions of the differential equation of the
horizontal sections.

###  A.7 Associated connections

Let $\pi\colon(P,\mathcal{F}^{\prime})\to(M,\mathcal{F})$ be a $G$-invariant
connection, as before, where $\mathcal{F}$ is a foliation in $M$ without
rational first integrals. Let us consider $\phi\colon G\to{\rm GL}(V)$ a
finite-dimensional linear representation of $G$. It is well known that the
associated bundle $\pi_{P}\colon V_{P}\to M$,

$\displaystyle V_{P}=P\times_{G}V=(P\times V)/G,\qquad(p\cdot
g,v)\sim(p,g\cdot v),$

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
$\mathcal{F}^{\prime}$ rises to a foliation in $P\times G$ and then it is
projected to a foliation $\mathcal{F}_{V}$ in $V_{P}$. In this way, the
projection

$\displaystyle\pi_{P}\colon\ (V_{P},\mathcal{F}_{V})\to(M,\mathcal{F}),$

turns out to be a linear $\mathcal{F}$-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 $H\subset G$ be the Galois group of the principal connection
$\mathcal{F}^{\prime}$. Then, the Galois group of the associated Lie–Vessiot
connection $\mathcal{F}_{V}$ is $\phi(H)\subseteq{\rm GL}(V)$.

###### Proof.

Let us consider the bundle of frames $R^{1}(V_{P})$, with its induced
invariant connection $\mathcal{F}^{\prime\prime}$. Let us fix a basis
$\\{v_{1},\ldots,v_{r}\\}$ of $V$. Then, we have a map

$\displaystyle\tilde{\pi}\colon\ P\to R^{1}(V_{P}),\qquad
p\mapsto([p,v_{1}],\ldots,[p,v_{r}]),$

where the pair $[p,v]$ represents the class of the pair $(p,v)\in P\times V$.
By construction, $\tilde{\pi}$ sends $T\mathcal{F}^{\prime}$ to
$T\mathcal{F}^{\prime\prime}$. It implies that, if $\mathcal{L}^{\star}$ is a
Picard–Vessiot bundle for $\mathcal{F}^{\prime}$ then $\tilde{\pi}(L^{\star})$
is a Picard–Vessiot bundle for $\mathcal{F}^{\prime\prime}$. Second, if
$\mathcal{L}^{\star}$ is a principal $H$ bundle, then $\tilde{\pi}(L^{\star})$
is a principal $H/K$ bundle where $K$ is the subgroup of $H$ that stabilizes
the basis $\\{v_{1},\ldots,v_{r}\\}$. ∎

Let us discuss how the covariant derivative operator in $\nabla$ is defined in
terms of $\Theta_{\mathcal{F}^{\prime}}$ and the action of $G$ in $V$. Let us
denote by $\phi^{\prime}\colon\mathfrak{g}\to\mathfrak{gl}(V)$ the induced Lie
algebra morphism. Let $s$ be a local section of $\xi$. Let us consider the
canonical projection $\bar{\pi}\colon P\times V\to V(P)$. This turns out to be
also a principal bundle, here the action on pairs is $(p,v)\cdot
g=\big{(}p\cdot g,g^{-1}\cdot v\big{)}$. Now we can take any section $r$ of
this bundle, and define $\tilde{s}=r\circ s$. As $r$ takes values in a
cartesian product, we obtain $\tilde{s}=(s_{1},s_{2})$ where $s_{1}$ is a
section of $\pi$ and $s_{2}$ is a function with values in $V$. Finally we
obtain

$\displaystyle\nabla s={\rm
d}s_{2}|_{T\mathcal{F}}-\phi^{\prime}(s_{1}^{*}(\Theta_{\mathcal{F}^{\prime}}))(s_{2}).$
(A.1)

A calculation shows that it does not depend of the choice of $r$ and it is the
covariant derivative operator associated to $\mathcal{F}_{V}$. In particular,
if $s_{2}$ is already an horizontal frame, then the covariant differential is
given by the first term ${\rm d}s_{s}|_{T\mathcal{F}}$.

##  Appendix B Deligne’s realization of Lie algebra

The proof of the existence of a regular parallelism for any complex Lie
algebra $\mathfrak{g}$ 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 $\mathfrak{g}$ there exist an algebraic variety
endowed with a regular parallelism of type $\mathfrak{g}$.

######  Lemma B.2.

Let $T$ be an algebraic torus acting regularly by automophisms in some
algebraic group $H$ and let $\mathfrak{t}$ be the Lie algebra of $T$. Let us
consider the semidirect product

$\displaystyle\mathfrak{t}\ltimes
H,\qquad(t,h)(t^{\prime},h^{\prime})=(t+t^{\prime},(\exp(t^{\prime})\cdot
h)h^{\prime})$

as an algebraic variety and analytic Lie group. Its left invariant vector
fields form a regular parallelism of $\mathfrak{t}\ltimes H$. The Galois group
of this parallelism is a torus.

###### Proof.

Let us denote by $\alpha$ the action of $T$ in $H$ and
$\alpha^{\prime}\colon\mathfrak{t}\mapsto\mathfrak{X}[H]$ the Lie algebra
isomorphism given by the infinitesimal generators

$\displaystyle(\alpha^{\prime}X)_{h}=\left.\frac{{\rm d}}{{\rm
d}\varepsilon}\right|_{\varepsilon=0}\alpha_{\exp(\varepsilon t)}(h).$

Let $X$ be an invariant vector field in $\mathfrak{t}$. Let us compute the
left invariant vector field in $t\ltimes H$ 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 $(0,e)$.

$\displaystyle L_{(t,h)}(0+\varepsilon X_{0},e)=(t+\varepsilon
X_{t},\alpha_{\exp(\varepsilon X)}(h))=(t+\varepsilon
X_{t},h+\varepsilon(\alpha^{\prime}X)_{h}).$

And thus $dL_{(t,h)}(X_{0},0)=(X_{t},(\alpha^{\prime}X)_{h})$. We conclude
that $(X,\alpha^{\prime}X)\in\mathfrak{X}[\mathfrak{t}\ltimes H]$ is the left
invariant vector field whose value at $(0,e)$ is $(X_{0},0)$. Let us consider
now $Y$ a left invariant vector field in $H$. Let us compute, as before, the
left invariant vector field whose value at $(t,h)$ is $(0,Y_{h})$

$\displaystyle L_{(t,h)}(0,e+\varepsilon Y_{e})=(t,L_{h}(e+\varepsilon
Y_{e}))=(t,h+\varepsilon Y_{h}).$

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 $(X,\alpha^{\prime}X)$ and
$(0,Y)$ are regular and span the Lie algebra of left invariant vector fields
in $\mathfrak{t}\ltimes H$. 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 $\mathfrak{t}\ltimes H$. A similar computation proves
that if $X$ is an invariant vector field in $\mathfrak{t}$ then $(X,0)$ is
right invariant in $\mathfrak{t}\ltimes H$. For each element $\tau\in T$,
$\alpha_{\tau}$ is a group automorphism of $H$. Thus, it induces a derived
automorphism $\alpha_{\tau*}$ 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 $\mathfrak{t}\ltimes H$ whose value at
$(0,e)$ is $(0,Y_{e})$:

$\displaystyle R_{(t,h)}(0,e+\varepsilon
Y_{e})=(t,\alpha_{\exp(t)}(e+\varepsilon
Y_{e})h)=(t,h+\varepsilon(\alpha_{\exp(t)*}Y)_{h})$

and $Z_{t,h}=(0,(\alpha_{\exp(t)*}Y)_{h})$. 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 $\mathfrak{g}$ an arbitrary, non algebraic, finite-dimensional
complex Lie algebra. We consider an embedding of $\mathfrak{g}$ in the Lie
algebra of general linear group and $E$ the smallest algebraic subgroup whose
Lie algebra $\mathfrak{e}$ contains $\mathfrak{g}$. $E$ is a connected linear
algebraic group.

######  Lemma B.3 (also in [4, Proposition 1]).

With the above definitions and notation
$[\mathfrak{e},\mathfrak{e}]=[\mathfrak{g},\mathfrak{g}]$.

###### Proof.

Let $H$ be the group of matrices that stabilizes $\mathfrak{g}$ and acts
trivially on $\mathfrak{g}/[\mathfrak{g},\mathfrak{g}]$. Its Lie algebra
$\mathfrak{h}$ contains $\mathfrak{g}$ and thus $H\supseteq E$ and
$\mathfrak{h}\supseteq\mathfrak{e}$. By definition of $H$ we have
$[\mathfrak{h},\mathfrak{g}]=[\mathfrak{g},\mathfrak{g}]$, therefore
$[\mathfrak{e},\mathfrak{g}]\subseteq[\mathfrak{g},\mathfrak{g}]$. Let us now
consider the group $H_{1}$ that stabilizes $\mathfrak{e}$ and $\mathfrak{g}$
and that acts trivially in $\mathfrak{e}/[\mathfrak{g},\mathfrak{g}]$. This is
again an algebraic group containing $E$, and its Lie algebra
$\mathfrak{h}_{1}$ satisfies
$[\mathfrak{h}_{1},\mathfrak{e}]\subseteq[\mathfrak{g},\mathfrak{g}]$. Taking
into account $\mathfrak{e}\subseteq\mathfrak{h}_{1}$ we have
$[\mathfrak{e},\mathfrak{e}]\subseteq[\mathfrak{g},\mathfrak{g}]$. The other
inclusion is trivial. ∎

Because of Lemma B.3, the abelianized Lie algebra
$\mathfrak{g}^{ab}=\mathfrak{g}/[\mathfrak{g},\mathfrak{g}]$ is a subspace of
$\mathfrak{e}^{ab}=\mathfrak{e}/[\mathfrak{e},\mathfrak{e}]$. Moreover, if we
consider the quotient map, $\pi\colon\mathfrak{e}\to\mathfrak{e}^{ab}$, then
$\mathfrak{g}=\pi^{-1}(\mathfrak{g}^{ab})$.

Let us consider an algebraic Levy decomposition $E\simeq L\ltimes U$ (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,
$(l_{1},u_{1})(l_{2},u_{2})=(l_{1}l_{2},(l_{2}\cdot u_{1})u_{2})$.

Since $L$ is reductive, its commutator subgroup $L^{\prime}$ is semisimple.
Let $T$ be the center of $L$, which is a torus, the map

$\displaystyle\varphi\colon\ T\times L^{\prime}\to L,\qquad(t,l)\mapsto tl,$

is an isogeny. The isogeny defines an action of $T\times L^{\prime}$ in $U$ by
$(t,l)\cdot u=tl\cdot u$. We have found an isogeny

$\displaystyle(T\times L^{\prime})\ltimes U\to E.$

The Lie algebra $\mathfrak{u}$ of $U$ is a nilpotent Lie algebra, so that the
exponential map $\exp\colon\mathfrak{u}\to U$ is regular and bijective. In
general, if $V$ is an abelian quotient of $U$ with Lie algebra $\mathfrak{v}$
then the exponential map conjugates the addition law in $\mathfrak{v}$ with
the group law in $V$.

######  Lemma B.4.

With the above definitions and notation, let $\bar{\mathfrak{u}}$ be the
biggest quotient of $\mathfrak{u}^{ab}$ in which $L$ acts by the identity. We
have a Lie algebra isomorphism
$\mathfrak{e}^{ab}\simeq\mathfrak{t}\times\bar{\mathfrak{u}}$.

###### Proof.

Let us compute $\mathfrak{e}^{ab}$. We compute the commutators $\mathfrak{e}$
by means of the isomorphism
$\mathfrak{e}\simeq(\mathfrak{t}\times\mathfrak{l}^{\prime})\ltimes\mathfrak{u}$.
We obtain

$\displaystyle[(t_{1},l_{1},u_{1}),(t_{2},l_{2},u_{2})]=(0,[l_{1},l_{2}],a(t_{2},l_{2})u_{1}+[u_{1},u_{2}]),$

where $a$ represents the derivative at $(e,e)$ of the action of $L$ in $U$.
From this we obtain that $[\mathfrak{e},\mathfrak{e}]$ is spanned by
$(\\{0\\}\times\mathfrak{l}^{\prime})\ltimes\mathfrak{\\{}0\\}$,
$\\{0\\}\ltimes[\mathfrak{u},\mathfrak{u}]$ and $\\{0\\}\times\langle
a(\mathfrak{l})\mathfrak{u}\rangle$. Taking into account that
$\bar{\mathfrak{u}}/(\langle
a(\mathfrak{l})\mathfrak{u}\rangle+[\mathfrak{u},\mathfrak{u}])$ is the
biggest quotient of $\mathfrak{u}^{ab}$ in which $L$ acts trivially, we obtain
the result of the lemma. ∎

Let $\mathfrak{t}$ be the Lie algebra of $T$. Its exponential map is an
analytic group morphism and thus we may consider the analytic action of
$\mathfrak{t}\times L^{\prime}$ in $U$ given by $(t,l)\cdot u=(\exp(t)l)\cdot
u$. Let $\tilde{E}$ be the algebraic variety and analytic Lie group
$(\mathfrak{t}\times L^{\prime})\ltimes U$. By application of Lemma B.2, and
taking into account that $\tilde{E}\simeq\mathfrak{t}\ltimes H$, where $H$ is
the group $L^{\prime}\cdot U$, we have that the left invariant vector fields
in $\tilde{E}$ are regular. Let us consider the projection

$\displaystyle\pi_{1}\colon\
\tilde{E}\to\mathfrak{e}^{ab}=\mathfrak{t}\times\bar{\mathfrak{u}},\qquad(t,l,u)\mapsto(t,[\log(u)]),$

this projection is algebraic by construction, and also a morphism of Lie
groups. By Lemmas B.3 and B.4, $\mathfrak{g}^{ab}$ is a vector subspace of the
image. Then, let us take $\tilde{G}$ the fiber
$\pi^{-1}_{1}(\mathfrak{g}^{ab})$. It is an algebraic submanifold of
$\tilde{E}$ and an analytic Lie group. The derivative at the identity of
$\pi_{1}$ is precisely the abelianization $\pi$ and it follows that the Lie
algebra of $\tilde{G}$ is precisely $\mathfrak{g}$. Finally $\tilde{G}$ is an
algebraic variety with a regular $\mathfrak{g}$-parallelism. This finishes the
proof of Theorem B.1.

######  Remark B.5.

The right invariant vector fields in $\tilde{G}$ 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.

### Acknowledgements

The authors thank the ECOS-Nord program C12M01 and the project “IsoGalois”
ANR-13-JS01-0002-01. They also thank the “Universidad Nacional de
Colombia”(project HERMES code 37243) and the “Université de Rennes 1” (Actions
Internationales 2016) for supporting this reseach, and also the Centre Henri
Lebesgue ANR-11-LABX-0020-01 for creating an attractive mathematical
environment.

The authors thank Juan Diego Vélez for his help with the final redaction of
the manuscript and the anonymous referees who gave relevant contributions to
improve the paper.

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