Reduced Forms of Linear Differential Systems and the Intrinsic Galois-Lie Algebra of Katz
Generalizing the main result of [Aparicio-Monforte A., Compoint E., Weil J.-A., J. Pure Appl. Algebra 217 (2013), 1504-1516], we prove that a linear differential system is in reduced form in the sense of Kolchin and Kovacic if and only if any differential module in an algebraic construction admits a...
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| Цитувати: | Reduced Forms of Linear Differential Systems and the Intrinsic Galois-Lie Algebra of Katz. Moulay Barkatou, Thomas Cluzeau, Lucia Di Vizio and Jacques-Arthur Weil. SIGMA 16 (2020), 054, 13 pages |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1860191356781068288 |
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| author | Barkatou, Moulay Cluzeau, Thomas Di Vizio, Lucia Weil, Jacques-Arthur |
| author_facet | Barkatou, Moulay Cluzeau, Thomas Di Vizio, Lucia Weil, Jacques-Arthur |
| citation_txt | Reduced Forms of Linear Differential Systems and the Intrinsic Galois-Lie Algebra of Katz. Moulay Barkatou, Thomas Cluzeau, Lucia Di Vizio and Jacques-Arthur Weil. SIGMA 16 (2020), 054, 13 pages |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | Generalizing the main result of [Aparicio-Monforte A., Compoint E., Weil J.-A., J. Pure Appl. Algebra 217 (2013), 1504-1516], we prove that a linear differential system is in reduced form in the sense of Kolchin and Kovacic if and only if any differential module in an algebraic construction admits a constant basis. Then we derive an explicit version of this statement. We finally deduce some properties of the Lie algebra of Katz's intrinsic Galois group.
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 16 (2020), 054, 13 pages
Reduced Forms of Linear Differential Systems
and the Intrinsic Galois–Lie Algebra of Katz
Moulay BARKATOU †, Thomas CLUZEAU †, Lucia DI VIZIO ‡ and Jacques-Arthur WEIL †
† XLIM, UMR7252, Université de Limoges et CNRS,
123 avenue Albert Thomas, 87060 Limoges Cedex, France
E-mail: moulay.barkatou@unilim.fr, thomas.cluzeau@unilim.fr, jacques-arthur.weil@unilim.fr
URL: http://www.unilim.fr/pages_perso/moulay.barkatou/,
http://www.unilim.fr/pages_perso/thomas.cluzeau/,
http://www.unilim.fr/pages_perso/jacques-arthur.weil/
‡ Université Paris-Saclay, UVSQ, CNRS, Laboratoire de mathématiques de Versailles,
78000, Versailles, France
E-mail: lucia.di.vizio@math.cnrs.fr
URL: http://divizio.perso.math.cnrs.fr/
Received January 20, 2020, in final form June 04, 2020; Published online June 17, 2020
https://doi.org/10.3842/SIGMA.2020.054
Abstract. Generalizing the main result of [Aparicio-Monforte A., Compoint E., Weil J.-A.,
J. Pure Appl. Algebra 217 (2013), 1504–1516], we prove that a linear differential system is
in reduced form in the sense of Kolchin and Kovacic if and only if any differential module in
an algebraic construction admits a constant basis. Then we derive an explicit version of this
statement. We finally deduce some properties of the Lie algebra of Katz’s intrinsic Galois
group.
Key words: linear differential systems; differential Galois theory; Lie algebras; reduced forms
2020 Mathematics Subject Classification: 34M03; 34M15; 34C20
1 Introduction
Let us consider the field of rational functions C(x), with the derivation ∂ = d
dx , and a linear
differential system ∂~y = A~y, where A is a square matrix of order n with coefficients in C(x).
One can attach to such an object an algebraic group, called the differential Galois group, whose
geometric properties encode the algebraic properties of the solutions of the linear differential
system. The problem of calculating explicitly the differential Galois group of ∂~y = A~y is old
and still difficult. Among the several references, we cite [1, 8, 10, 12, 20] that do not make any
assumption on the order n of the system. Implemented (or implementable) algorithms exist
only for small dimensions n. See for example [7, 11, 14, 15, 16, 19, 23, 22].
Instead of calculating directly the differential Galois group of ∂~y = A~y, one can try to study,
or calculate, the Lie algebra of the differential Galois group, called Galois–Lie algebra in what
follows. The Galois–Lie algebra already contains a significant part of the information. Kolchin
and Kovacic have proved that one can transform ∂~y = A~y into an equivalent system ∂~y = B~y
defined over a finite extension k of C(x), such that B belongs to the set of k-rational points of the
Galois–Lie algebra (see [21, Proposition 1.31]). One can even prove that the Galois–Lie algebra
is then “generated” by the entries of the matrix B. These ideas are formalized in [3, Section 2.3].
The linear differential system ∂~y = B~y is called a reduced form of ∂~y = A~y. A linear differential
system ∂~y = A~y is said to be in reduced form if A is a k-rational point of the Galois–Lie algebra.
mailto:moulay.barkatou@unilim.fr
mailto:thomas.cluzeau@unilim.fr
mailto:jacques-arthur.weil@unilim.fr
http://www.unilim.fr/pages_perso/moulay.barkatou/
http://www.unilim.fr/pages_perso/thomas.cluzeau/
http://www.unilim.fr/pages_perso/jacques-arthur.weil/
mailto:lucia.di.vizio@math.cnrs.fr
http://divizio.perso.math.cnrs.fr/
https://doi.org/10.3842/SIGMA.2020.054
2 M. Barkatou, T. Cluzeau, L. Di Vizio and J.-A. Weil
Main results. In the present work, we prove three results on reduced forms and their
relation to the Galois–Lie algebra. First of all, we prove that a system is in reduced form if and
only if any differential module in a construction admits a constant basis (see Theorem 3.2, in
particular the equivalence 1 ⇔ 3). This extends the criterion for reduced form from [3] which
concerned only invariant lines (see the equivalence 1⇔ 2 in Theorem 3.2).
Our second contribution is Theorem 3.9 that gives an effective characterization of a gauge
transformation that transforms a linear differential system into a reduced one. It may be consi-
dered as an effective counterpart of Theorem 3.2, based on the “local” data of the semi-invariants.
Compared to the original result by Kolchin and Kovacic on the existence of reduced forms, we
have to perform an algebraic extension of the base field k, which may not be optimal, in order to
gain the effectivity. Theorem 3.9 extends a result which only appeared in the course of the proof
of [3, Proposition 27], under the assumption that the differential system is completely reducible.
Finally, we prove Theorem 4.5 on the Lie algebra of the intrinsic Galois group, introduced
in [13], where the idea of focusing on the Galois–Lie algebra rather than on the differential Galois
group itself is pursued. Indeed, Katz introduces another Galois group for the linear differential
system ∂~y = A~y, called the generic or the intrinsic Galois group. Then he considers the Lie
algebra of such a group, for which he gives a conjectural description equivalent to a well-known
conjecture of Grothendieck on the algebraicity of the solutions of a linear differential system.
We will call it the Katz algebra. In the last section, we gather material from [2, 5, 13, 21] and
show how our criteria for reduced forms, combined with standard Tannakian tools, clarify the
structure of the Katz algebra. Namely, for a reductive group, Theorem 4.5 shows that the Katz
algebra is a k-form of the Galois–Lie algebra.
The algorithm in [4]. In the latter reference, we showed how one can compute the Galois–
Lie algebra of an (absolutely) irreducible linear differential system and hence (a good part of)
its differential Galois group. Notice that in [9], it is shown how to derive the (connected) Galois
group of the reduced form from the Galois–Lie algebra, so that one can actually recover the
connected component of the Galois group of the original system.
The algorithm selects a Lie algebra that is potentially the Katz algebra and checks that there
exists a gauge transformation that transforms its generators into a set of constant generators
of a Lie algebra, that is a candidate for being the Lie algebra of the Galois group. To do so,
it uses Theorem 3.9 in the particular case of a completely reducible differential system, hence
in the case considered in [3]. See [4, Lemma 5.1]. Theorem 4.5 completes the mathematical
background of the algorithm, although it is technically not needed in it.
Organization of the paper. In Section 2, we recall some notions on differential modules,
tensor constructions and differential Galois theory. In Section 3, we prove our two theorems on
the criteria for a linear differential system to be in reduced form. In Section 4, we apply the
previous results to the study of the Katz algebra.
2 Notation and definitions
We consider a characteristic zero differential field (k, ∂), that is a characteristic zero field k with
a derivation ∂ : k → k, such that ∂(a + b) = ∂(a) + ∂(b) and ∂(ab) = ∂(a)b + a∂(b), for any
a, b ∈ k. We suppose that the subfield of constants C := k∂ = {f ∈ k : ∂f = 0} is algebraically
closed.
2.1 Differential modules
A differential moduleM = (M,∇) over k (of rank n) is a k-vector space M of dimension n, with
a C-linear map ∇ : M →M such that ∇(fm) = ∂(f)m+f∇(m) for any f ∈ k and any m ∈M .
For a detailed exposition on differential modules, see [21, Section 2.2]. We denote by M∇
Reduced Forms of Linear Differential Systems and the Intrinsic Galois–Lie Algebra of Katz 3
or ker∇ the set of horizontal elements ofM, that is elements m ∈M satisfying ∇(m) = 0. This
is a C-vector space of dimension at most n.
Main properties of differential modules. Given a basis (denoted as a row) e :=
(e1, . . . , en) of M over k, the action of ∇ with respect to the basis e is described by a square
matrix A ∈Mn(k) as follows
∇e = −eA.
For any ~y ∈ kn such that e~y represents an element of M , we have ∇(e~y) = e(∂~y − A~y). Thus
horizontal elements of M correspond to solutions over k of the differential system
[A] : ∂~y = A~y.
We say that [A] : ∂~y = A~y is the linear differential system associated to M with respect to the
basis e.
If f = eP , with P ∈ GLn(k), is another basis of M , then the horizontal elements of M are
of the form f~z, with ~z ∈ kn, where ~z verifies the linear differential system
∂(~z) = P [A]~z, P [A] := P−1AP − P−1∂(P ).
We say that two matrices A,B ∈ Mn(k) are equivalent over k if there exists a gauge transfor-
mation P ∈ GLn(k) such that B = P [A].
Notice that one can extend the scalars of M to a field extension k′ of k, equipped with an
extension of ∂. The Leibnitz rule allows to extend∇ to M⊗kk
′, so that it makes sense to consider
gauge transformations in GLn(k′), as counterpart of basis changes of M⊗k k
′ = (M ⊗k k
′,∇).
Algebraic constructions. Let us start by formalizing what we mean by construction of
linear algebra.
Definition 2.1. A construction of linear algebra is a finite iteration of the basic constructors ⊕
(direct sum), ⊗ (tensor product), ∗ (dual), Symr (r-th symmetric power, for r ∈ N) and ∧r (r-th
exterior power). Given a construction of linear algebra and a vector space M , we denote by
Constr(M) the finite-dimensional k-vector space obtained by applying the construction to M .
Given vector spaces M1 and M2 with respective bases e and f , an application of each of the
above basic constructors produces canonically a new basis: for instance, (e1, . . . , en, f1, . . . , fm)
is a basis for M1 ⊕M2, (ei ⊗ fj | i = 1, . . . , n; j = 1, . . . ,m) is a basis for M1 ⊗M2, and so
on (see [21, Section 2.2, p. 42], and [3, Section 3]). This way, given a vector space M with
basis e and a construction Constr, we iteratively construct a canonical basis, denoted Constr(e),
of Constr(M). Two different constructions may produce two isomorphic vector spaces. In
this case, there is a canonical isomorphism between the two vector spaces which identifies the
canonical bases.
Remark 2.2. We need to make some comments on our lists of algebraic constructions.
1. Notice that this list implicitly contains Hom (homomorphisms) and End (endomorphisms)
via the canonical identifications Hom(M1,M2) ∼= M1 ⊗M∗2 and End(M) ∼= M ⊗M∗. See
also Example 2.3 below.
2. Some authors use additional basic constructors such as quotients and subspaces. We choose
not to do so to avoid the rising of apparent singularities (see [3, Remark 18]) and because
this list is sufficient for our purposes. See Section 3.2, where we need an ordinary point
for all constructions.
4 M. Barkatou, T. Cluzeau, L. Di Vizio and J.-A. Weil
Let H denote a linear algebraic group, with Lie algebra h, acting on M . Let the linear
map σ ∈ H have a matrix U in the basis e. The morphism induced by σ on Constr(M) is
denoted by Constr(σ). Its matrix in the basis Constr(e) is denoted by Constr(U) and the map
U 7→ Constr(U) is a group morphism. Similarly, for h ∈ h, it acts on M as a linear derivation Dh
with matrix N . The action of this linear derivation Dh on the basis Constr(e) of Constr(M)
induces a matrix constr(N); the map N 7→ constr(N) is a Lie algebra morphism. In what follows,
we will use the gothical letters constr for these constructions “in the sense of Lie algebras”.
Note that the entries of Constr(U) are polynomials in the entries of U and in 1/ det(U); the
1/det(U) is needed to have duals as, when Constr(M) = M∗, Constr(U) =
(
U−1
)T
. The entries
of constr(N) are linear forms in the entries of N . See [21, Section 2.4, p. 53] and [3, Sections 3.1
and 3.2].
Let P ∈ GLn(k). If we consider a change of basis f = eP in the vector space M , then
the corresponding change-of-basis matrix in Constr(M) will be given by Constr(P ) and we
have Constr(f) = Constr(e)Constr(P ). The constructions of linear algebra apply functorially
to differential modules. Let M = (M,∇) be a differential module over k. The operator ∇
induces a C-linear map from Constr(M) to Constr(M), that we will also denote by ∇, defining
a differential module structure over Constr(M). We will call the latter a construction of M (or
tensor construction) and denote it Constr(M) = (Constr(M),∇).
Let −A be the matrix of ∇ with respect to a basis e as defined above. For any construction
Constr(M), the matrix of ∇ with respect to Constr(e) will be −constr(A). For example, in
the case Constr(M) =M∗, constr(A) = −AT, while if Constr(M) =M⊗kM∗, then we have
constr(A) = A⊗ In − In ⊗AT. See [3, Sections 3.1 and 3.2].
Example 2.3. Let M = (M,∇) be a differential module over k and let −A be the matrix
of ∇ with respect to a basis e. We consider the differential module End(M) = (Endk(M),∇). If
ϕ ∈ Endk(M) then ∇(ϕ) is the endomorphism of M defined by ∇(ϕ)(m) = ∇(ϕ(m))−ϕ(∇(m)),
for all m ∈ M . If F is the matrix of ϕ with respect to the basis e, one can check that we have
∇(ϕ)(e) = e(∂F −AF +FA). If we denote by square matrices F the elements of End(M) with
respect to the basis induced by e, then the linear differential system associated to End(M) with
respect to the basis induced by e is ∂F = AF − FA.
The horizontal elements of End(M) are the elements ϕ ∈ Endk(M) such that we have
∇(ϕ(m)) = ϕ(∇(m)), ∀m ∈M. Thus the horizontal elements of the differential module End(M)
are exactly the k-endomorphisms of M which commute with ∇. They are called differential
module endomorphisms of M. They form a C-algebra denoted by E(M) which is called the
eigenring of M.
Among all the possible constructions, the differential module M⊗kM∗ will play a special
role in the exposition below. If we identify it canonically to End(M), then the linear differential
system associated to M⊗k M∗ in the basis induced by e is exactly ∂F = AF − FA. The
set E([A]) of matrices F ∈Mn(k) satisfying the above matrix differential equation is called the
eigenring of the system [A] : ∂~y = A~y. It is isomorphic (as a C-algebra) to E(M).
2.2 Picard–Vessiot extensions
We introduce very briefly some notions of differential Galois theory, with the main purpose of
fixing the notation. There exists several detailed introduction to the topic. We refer to [21] for
a general introduction and to [3] for more specific notions which are needed in this paper.
Let us consider the linear differential system [A] : ∂~y = A~y. To any such system we can attach
a k-algebra R, with an extension of ∂, having the following properties:
1) there exists U ∈ GLn(R) such that ∂U = AU ;
2) the entries of U plus detU−1 generate R over k, namely R = k
[
U,detU−1
]
;
Reduced Forms of Linear Differential Systems and the Intrinsic Galois–Lie Algebra of Katz 5
3) R has no proper non-trivial ideals stable by ∂, i.e., it is a simple differential ring.
We say that R is a Picard–Vessiot ring of k for [A]. It is an integral domain and its ring of
constants R∂ coincides with C. Its quotient field K = Frac(R) is generated (as a field) by the
entries of U and its subfield of constant is again C. We call K a Picard–Vessiot extension of k
for [A].
Remark 2.4. We are going to use several properties of Picard–Vessiot rings and extensions,
namely:
1. If [A] and [B] are two equivalent systems over k then any Picard–Vessiot extension of k
for [A] is a Picard–Vessiot extension of k for [B]. Hence one can define a Picard–Vessiot
extension of k for a given differential module M = (M,∇) as a Picard–Vessiot extension
of k for the differential system associated to M with respect to a basis of M .
2. Let K be a Picard–Vessiot extension of k for differential module M = (M,∇). The
Leibnitz rule allows to endow M ⊗k K with a natural structure of differential module
over K, which will be denoted by M⊗k K. The definition of K implies that M⊗k K is
trivial, i.e., M⊗k K admits a basis over K of horizontal elements. One can show that if
a module is trivial, then all its algebraic constructions and their subquotients are trivial.
See [21, Exercice 2.12,5].
3. Let V := (M⊗k K)∇ be the C-vector space of the horizontal elements of M⊗k K. As
already pointed out, it has dimension n.
We give now a definition that we will use in the main theorem.
Definition 2.5. Let M = (M,∇) be a differential module and K a Picard–Vessiot extension.
A semi-invariant ofM is a horizontal element m⊗g contained in some construction of the form
Constr(M⊗k K) ∼= (Constr(M) ⊗k K,∇), i.e., an element m ⊗ g with m ∈ Constr(M) and
g ∈ K such that ∇(m ⊗ g) = 0. If m is a horizontal element in some construction Constr(M),
then it is called an invariant of M.
For the convenience of the reader, we reprove the following classical lemma that we will use
in this work.
Lemma 2.6. Let M = (M,∇) be a differential module and K a Picard–Vessiot extension of k
for M. Fix a basis e of M and let [A] : ∂~y = A~y be the associated linear differential system.
The following statements are equivalent:
1) there exists m ∈ M such that ∇(m) = fm, for some f ∈ k (i.e., m generates a ∇-stable
line);
2) there exists a solution g~v over K of ∂~y = A~y, with g ∈ K, such that ∂(g)/g = −f ∈ k,
and ~v ∈ kn.
Proof. Let us assume that there exists m ∈ M such that ∇(m) = fm, for some f ∈ k. Then
the line L generated by m over k is a differential module L, and L⊗k K ⊂M⊗k K is a trivial
differential module (see Remark 2.4). Thus there exists g ∈ K, g 6= 0, such that m ⊗ g is
a horizontal element of L ⊗k K. It follows that we have
0 = ∇(m⊗ g) = fm⊗ g +m⊗ ∂(g) = m⊗ (fg + ∂(g)),
which implies ∂(g)/g = −f ∈ k. If we define ~v ∈ kn by the relation m = e~v, then we can check
that g~v is a solution vector of ∂~y = A~y.
Let us now suppose that we are in the situation described in the second assertion. If we define
m := e~v, then we have ∇(m) = e(∂~v − A~v). Moreover, we have ∂(g~v) = ∂(g)~v + g∂~v = Ag~v so
that ∂~v −A~v = −(∂(g)/g)~v = f~v. Finally ∇(m) = fm, with f ∈ k which ends the proof. �
6 M. Barkatou, T. Cluzeau, L. Di Vizio and J.-A. Weil
Remark 2.7. In particular, Lemma 2.6 above implies that, if m ⊗ g is a semi-invariant, then
necessarily ∇(m) = (−∂(g)/g)m, or equivalently m generates a ∇-stable line L contained in
a construction Constr(M) of M . If m′ = hm, for some h ∈ k, then ∇(m′) = (−∂(g)/g +
∂(h)/h)m′ and ∇(m′ ⊗ (g/h)) = 0. Moreover, if m ⊗ g is a semi-invariant of M, such that
m generates a line L in a construction of M , and c ∈ C is a nonzero constant, then c(m ⊗ g)
is another semi-invariant, corresponding to the same line L. One can prove that all the semi-
invariants can be obtained in this way. Roughly speaking, semi-invariants of M correspond to
exponential solutions of ∂~y = constr(A)~y and invariants correspond to rational solutions. For
more details on these definitions, see [3, Section 3.4].
For further reference we recall the following lemma:
Lemma 2.8 ([3, Lemma 29]). Let [A] : ∂~y = A~y be the linear differential system associated
to M with respect to a fixed basis e. We suppose that there exist an algebraic extension k′/k
and a matrix P ∈ GLn(k′) such that, for any invariant of M given by a horizontal element in
some construction Constr(M) of coordinates ~v with respect to the basis Constr(e), the vector
Constr(P )−1~v has constant coordinates. Then the same property holds for any invariant of
M⊗k k, where k is the algebraic closure of k.
2.3 The differential Galois group
The differential Galois group of [A] (or, equivalently, of M) is defined as
Gal∂([A]) := Aut∂(K/k),
where K is a Picard–Vessiot extension of k for [A] and
Aut∂(K/k) :=
{
ϕ : K → K, field automorphism s.t. ∀ f ∈ k,
we have ϕ(f) = f and [ϕ, ∂] = 0
}
.
The first result of the differential Galois theory is that any fundamental matrix of solutions
U ∈ GLn(R) of [A] determines a faithful representation of Gal∂([A]) as a linear algebraic group
defined over C:
Gal∂([A])→ GLn(C),
ϕ 7→ U−1ϕ(U).
In fact, ϕ(U) is a fundamental matrix of solutions of [A], therefore U−1ϕ(U) must be an invertible
matrix with constant coefficients. The choice of another fundamental matrix of solutions leads
to a conjugated representation. We will sometimes simply call G the differential Galois group
Gal∂([A]), identifying it with its image via the morphism above and without mentioning the
matrix U , unless the context makes it necessary.
We are not explaining here any result on the Galois correspondence and we refer the interested
reader to the literature. For the purpose of this paper, we mostly need to know that, if k◦ is
the relative algebraic closure of k in K, then K/k◦ is a Picard–Vessiot extension for [A] over k◦.
Moreover the field of constants is still C and we have Aut∂(K/k◦) = G◦, where G◦ is the
connected component of G containing 1. This means that the differential Galois group of [A]
over k◦ coincides with the automorphisms of G that fix k◦ and can be identified with G◦.
The Galois–Lie algebra gal([A]). Since the differential Galois group G = Gal∂([A]) is an
algebraic group over C, one can naturally consider its Lie algebra g := gal([A]) called Galois–Lie
algebra, that is the tangent space to G◦ at 1. If we look at its C-rational points, we have
gal([A])(C) = {N ∈Mn(C) : 1 + εN ∈ Gal∂([A])(C[ε])},
where Gal∂([A])(C[ε]) are the rational points of Gal∂([A]) over the C-algebra C[ε], with ε2 = 0.
Reduced Forms of Linear Differential Systems and the Intrinsic Galois–Lie Algebra of Katz 7
As it is an algebraic Lie algebra over C, gal([A]) is generated as a C-vector space by a finite
subset of Mn(C). It will be useful to notice that the same subset of matrices of Mn(C) generates,
as a k-vector space, the algebra gal([A])(k) of k-rational points of gal([A]). For further reference
we recall the following result:
Proposition 2.9 ([21, Proposition 1.31]). Let h be an algebraic Lie algebra defined over C and
such that A ∈ h(k). Then gal([A]) ⊂ h.
3 Reduced forms
3.1 Characterization of reduced forms
We keep the notation of the previous section.
Definition 3.1. We say that a linear differential system [A] : ∂~y = A~y is in reduced form when A
belongs to gal([A])(k). Let M = (M,∇) be a differential module and e be a k-basis of M . We
say that e is a reduced basis if the system associated to M with respect to the basis e is in
reduced form.
As pointed out in the introduction, Kolchin and Kovacic has proved that a reduced form
always exists on a finite extension of k. See [21, Proposition 1.31]. Criteria for reduced forms
have been studied by Aparicio, Compoint and Weil in [3]. Their main result is generalized below.
Our contribution in this theorem is the new characterization of reduced form stated in item (3).
Theorem 3.2. Let [A] : ∂~y = A~y be the linear differential system associated to a differential
module M = (M,∇) over k, with respect to a fixed basis e. The following assertions are
equivalent:
1) [A] is in reduced form;
2) for any construction Constr(M) of M, every ∇-stable line of Constr(M) admits a con-
stant basis (i.e., a basis whose elements have constant coordinates with respect to the basis
induced by e);
3) for any construction Constr(M) of M, every ∇-stable sub-k-vector space of Constr(M)
admits a constant basis.
If moreover M is completely reducible (i.e., it is direct sum of irreducibles), then the assertions
above are equivalent to
4) any invariant of M has constant coordinates.
Proof. Lemma 2.6 and Remark 2.7 show that the constant bases of ∇-stable lines correspond to
the semi-invariants considered in [3]. Therefore “(1)⇔ (2)” is a reformulation of [3, Theorem 1].
Since “(3) ⇒ (2)” is tautological, it is enough to prove that “(2) ⇒ (3)”. Let N = (N,∇) be
a construction of M and W be a ∇-stable sub-k-vector space of N of dimension d. It follows
that ∧dW is ∇-stable sub-k-vector space of dimension 1 of ∧dN . By assumption, there exists
a non-zero element of ∧dW , whose coordinates ~w with respect to the basis induced by e on ∧dN
are in C. Hence, in the basis induced by e, the map
Ψ: N → ∧d+1N,
~v 7→ ~w ∧ ~v,
is represented by a matrix with coefficients in C. It follows that ker Ψ has a basis of vectors
with coordinates in C, with respect to the basis induced by e. Since ker Ψ = W , we have proved
“(2)⇒ (3)”.
8 M. Barkatou, T. Cluzeau, L. Di Vizio and J.-A. Weil
By definition, a horizontal element of a construction Constr(M) corresponds to a solution
of the associated differential system ∂~y = constr(A)~y with respect to the basis Constr(e). Def-
inition 2.5 and [21, Exercice 2.38] imply that the equivalence “(1) ⇔ (4)” coincides with [3,
Proposition 27] which ends the proof of Theorem 3.2. �
Remark 3.3. Let m ∈ Constr(M) generate a line which is invariant by ∇ and let e be a reduced
basis. By Theorem 3.2 above, we can choose m such that m = Constr(e)~v, with ~v ∈ Cn.
Moreover since ∇(m) = fm for some f ∈ k, there exists g ∈ K verifying ∂(g) = fg. This means
that g~v is a solution vector of the system associated to Constr(M) with respect to Constr(e).
This is what we mean when we say that invariants and semi-invariants have constant coordinates
with respect to this reduced basis.
Example 3.4. Theorem 3.2 has as corollary the following known fact: in a reduced basis, the
eigenring of M (see Example 2.3 for the definition) can be identified to the ring of matrices
with coefficients in C which commute with A, hence with all the elements of a Wei–Norman
decomposition of A (see [3, Section 2.2] and [24]).
The Kolchin–Kovacic theorem implies in particular that there exists a finite extension k′/k
such thatM⊗k k
′ admits a reduced basis. See [21, Proposition 1.31 and Corollary 1.32] and [3,
Remark 31].
Definition 3.5. We say that k′ is a reduction field when M⊗k k
′ admits a reduced basis.
Remark 3.6. When k′ is a reduction field, then the differential Galois group of M⊗k k
′ is
connected, see [3, Lemma 32], and the Galois correspondence implies that k′∩K = k◦, the fixed
field of G◦ in the Picard–Vessiot extension K.
3.2 Gauge transformation to a reduced form with local conditions
Assumption 3.7. In this section, we suppose that the field k is a subfield of the field of
meromorphic functions over a region D of C in the variable x such that x ∈ k and ∂ = d
dx .
Remark 3.8. The field of rational functions k = C(x) satisfies the assumption above, as well
as most differential fields occurring in the concrete examples. Indeed let k be any differential
field. Then the entries of the matrix A of the linear differential system [A] : ∂~y = A~y, generate
a differential field k̃, which is a finitely generated differential extension of Q. Seidenberg’s
embedding theorem (see [17, 18]) ensures that k̃ can be embedded isomorphically in a differential
field of meromorphic functions on an open region D of C.
We consider a linear differential system [A] : ∂~y = A~y, with coefficients in k. For all points
x0 ∈ D such that x0 is not a pole of A, the system [A] has a fundamental matrix of solu-
tions Ûx0(x) with the following properties:
1) Ûx0(x) ∈ GLn(C[[x− x0]]),
2) Ûx0(x0) = Id, the identity matrix.
The columns of Ûx0(x) generate over C the vector space V of solutions of [A] (contained in
C[[x−x0]]n), and K := k(Ûx0(x)) is a Picard–Vessiot extension of [A]. Let Constr be a construc-
tion. Then Constr
(
Ûx0
)
(x0) is the identity matrix (because Constr acts on matrices as a group
morphism). Furthermore, Constr
(
Ûx0
)
is a fundamental matrix of solutions for [constr(A)].
Consider an invariant of M given by a horizontal element m in Constr(M) such that m has
coordinates ~v(x) ∈ kN with respect to the basis Constr(e). The vector ~v(x) is a solution of
[constr(A)] and hence we have ~v(x) = Constr
(
Ûx0
)
(x)~w for some ~w with constant coefficients
in C. The fact that Constr
(
Ûx0
)
(x0) is the identity matrix allows to conclude that ~w = ~v(x0).
Reduced Forms of Linear Differential Systems and the Intrinsic Galois–Lie Algebra of Katz 9
In the theorem below, we give an algebraic characterization for a reduction matrix. The
original Kolchin–Kovacic theorem ensures that, when the base field k is a C1-field, there exists
a reduced basis defined on the relative algebraic closure k◦ of k in the Picard–Vessiot extensionK.
However, its proof is not effective and it relies on finding rational points on varieties and this part
is not either, to our knowledge, algorithmic yet. In order to gain effectiveness, we enlarge k to
an algebraic extension k′, in which we can compute (without C1-assumptions on k) a reduction
matrix that is characterized by the property of transforming any invariant in its “value at x0”.
The field k′ may depend on the choice of x0. However, as seen in Remark 3.6, we have k′∩K = k◦
so k′ ∩K does not depend on the choice of x0.
The spirit of this result and of our proof appears in the proof of [3, Theorem 3] in a particular
case. It is extended here to all constructions and this is useful for reduced form algorithms; see
[4, Lemma 5.1], where this result was alluded to, without a full proof for lack of space.
Theorem 3.9. Let us consider a linear differential system ∂~y = A~y, defined over a field k,
associated to a completely reducible differential module M with respect to a fixed basis e. We
choose a point x0 ∈ D such that A does not have a pole at x0. Then, there exists a finite
extension k′ of k and a matrix Px0 ∈ GLn(k′) such that f := ePx0 is a reduced basis of M⊗k k
′,
having the following property:
For any invariant of M given by a horizontal element m in some construction
Constr(M) such that m has coordinates ~v(x) ∈ kN with respect to the basis Constr(e),
we have ~v(x) = Constr(Px0)~v(x0).
Remark 3.10. In the last statement, we view Constr(e) as a basis of Constr(M ⊗k k
′) and
identify an element m ∈ Constr(M) with its image m⊗1 ∈ Constr(M)⊗k k
′ ∼= Constr(M⊗k k
′).
Proof. In the new basis f , if it exists, all invariants are constant therefore it is a reduced basis,
thanks to Lemma 2.8 and Theorem 3.2. We now prove the existence. Let X be a matrix with
indeterminate entries. For any invariant in some construction Constr(M) having coordinates
~v(x) ∈ kN with respect to the basis Constr(e), consider the equation ~v(x) = Constr(X)~v(x0). It
provides an infinite set of polynomial equations over k in the entries of X which are all satisfied
by the entries of a fundamental matrix of solutions Ûx0(x) of [A] (see the explanations above).
As a consequence, the latter set of equations generates a proper ideal in the ring of polynomials
in n2 variables, which is finitely generated because of the noetherianity. The Nullstellensatz
then ensures that there exists a solution P ∈ GLn(k′), where k′/k is a finite extension of k. �
Remark 3.11. Assume thatM admits a reduced basis e. Consider an invariant m in some con-
struction Constr(M) having (constant) coordinates ~v ∈ CN with respect to the basis Constr(e).
Then we have ~v = Constr
(
Ûx0(x)
)
~c and the constant vector ~c is an invariant (in the usual sense
of representation theory) of the Galois group Gal∂([A]) in its representation induced by Ûx0(x).
Now, by construction, Ûx0(x0) = Id (the identity matrix); as Constr acts as a group morphism,
we see that Constr
(
Ûx0
)
(x0) = Id. Now, because ~v is constant, we have ~v = Constr
(
Ûx0
)
(x0)~c
so ~v = ~c. In a reduced basis, this observation allows to identify the invariants of M and those
of Gal∂([A]).
4 The intrinsic Galois–Lie algebra of Katz
Let M = (M,∇) be a differential module over k.
Definition 4.1 ([13]). The intrinsic Galois group GKatz of Katz is the set of ϕ ∈ GL(M) such
that, for any ∇-stable sub-k-vector space N of a construction Constr(M), N is set-wise stable
under Constr(ϕ).
The Katz algebra gKatz is the Lie algebra of GKatz.
10 M. Barkatou, T. Cluzeau, L. Di Vizio and J.-A. Weil
Remark 4.2. For ϕ ∈ End(M), the functor constr acts on ϕ as a Lie algebra morphism, see the
explanations before Example 2.3. In particular, for any invariant m ∈ Constr(M) and ϕ ∈ gKatz,
we have constr(ϕ)(m) = 0.
By Noetherianity, gKatz is a stabilizer of a finite family of differential modules contained in
some constructions of M. This shows that gKatz is an algebraic Lie algebra. It is the central
object of the famous Grothendieck–Katz conjecture on p-curvatures. The following properties
of gKatz are known, see [2, 13, 21], but are reproved for self-containedness.
Lemma 4.3. Let M = (M,∇) be a differential module over k, K a Picard–Vessiot extension,
G the differential Galois group of M and g the Galois–Lie algebra.
1. The Lie algebra gKatz can be defined as the stabilizer of a single line in an algebraic con-
struction of M (which can be chosen to be ∇-invariant).
2. The Lie algebra gKatz is a differential module for the structure induced by M⊗kM∗.
3. gKatz = (g⊗C K)G.
Proof. 1. Because of the Noetherianity, a theorem of Chevalley ensures that gKatz is the stabi-
lizer of a finite family Wi of ∇-stable k-vector spaces contained in some constructions of M . A
classical argument then shows that gKatz can be defined as the stabilizer of ∧d⊕iWi, where d is
the dimension of ⊕iWi over k. See [13, proof of Proposition 9.3]
2. If m generates a∇-stable line in some construction of M , which defines gKatz as a stabilizer,
and ϕ ∈ gKatz, then constr(ϕ)(m) = αm and ∇(m) = βm, for some α, β ∈ k. We conclude that
∇(constr(ϕ))(m) = ∇(constr(ϕ)(m))− constr(ϕ)(∇(m)) = ∇(αm)− constr(ϕ)(βm)
= ∂(α)m+ α∇(m)− βconstr(ϕ)(m) = ∂(α)m.
Therefore ∇(constr(ϕ)) belongs to gKatz.
3. The Tannakian correspondence will show that gKatz
∼= (g⊗k K)G. Indeed, we have seen
that gKatz is the stabilizer of a line L. Then LC := (L ⊗C K)∇ is a C-line defined in the
corresponding construction on the C-vector space of solutions V := (M ⊗kK)∇. It is stabilized
by g so g⊗CK stabilizes LC⊗CK. Hence (g⊗CK)G stabilises (LC⊗CK)G. Now (LC⊗CK)G = L
so (g⊗C K)G stabilises L and thus (g⊗C K)G ⊆ gKatz. In fact, the same argument shows that
(g⊗C K)G stabilizes any differential module in a construction and thus (g⊗C K)G = gKatz.
This ends the proof. �
Definition 4.4. An invariant m of M in Constr(M) is called a Chevalley invariant of gKatz
when gKatz = {h ∈ gl(M) |h(m) = 0}.
Theorem 4.5. Let M = (M,∇) be a differential module over k, G the differential Galois group
of M, g the Galois–Lie algebra and gKatz the Katz algebra. Let k′ be a reduction field so that
M⊗k k
′ admits a reduced basis e. If M is completely reducible, then g⊗C k
′ = gKatz ⊗k k
′.
Proof. In a reduced basis ofM⊗k k
′, Remark 3.11 shows that g and gKatz⊗k k
′ have the same
invariants. Now M is completely reducible (equivalently, the Galois group G is reductive) so
both g and gKatz are completely reducible and this remains true after extension of scalars. From
[6, Chapter II, Section 5.5, p. 92] or [3, Lemma 26], it follows that both g and gKatz ⊗k k
′ are
determined by their invariants. These two observations show that a Chevalley invariant of g
is also a Chevalley invariant of gKatz ⊗k k
′. Let m be a Chevalley invariant of gKatz ⊗k k
′ in
some construction Constr(M)⊗k k
′. In the reduced basis, m has constant coefficients: we have
m = Constr(e)~v, with ~v ∈ Cn. Thus, a matrix N = (nij)i,j is in gKatz ⊗k k
′ if and only if
constr(N)~v = 0. This relation yields a system of linear equations L((ni,j)i,j) = 0 for the entries
Reduced Forms of Linear Differential Systems and the Intrinsic Galois–Lie Algebra of Katz 11
(ni,j)i,j of N , with coefficients in C. Now, as seen above, ~v is also a Chevalley invariant of
the Galois–Lie algebra g. So a matrix is in g if and only if its entries satisfy the same linear
equations L((ni,j)i,j) = 0. Consequently, g⊗C k
′ and gKatz ⊗k k
′ share the same constant basis
so that g⊗C k
′ = gKatz ⊗k k
′. �
Definition 4.6. Let k′ be an extension of k and W be a subspace of a construction Constr(M)
⊗k k
′. We say that W is defined over C when it is generated by constant matrices (i.e., when it
is a k′-form of a C-space).
Corollary 4.7. Let k′ be a reduction field so that M⊗k k
′ admits a reduced basis e. Then
gKatz ⊗k k
′ is defined over C. Moreover, if A is the matrix of the associated linear differential
system with respect to e, then A ∈ gKatz ⊗k k
′.
Proof. Let L be a ∇-invariant line in some Constr(M) such that gKatz is the stabilizer of L.
By Theorem 3.2, we can choose a generator m of L⊗C k
′ whose coordinates are constant with
respect to the reduced basis. This shows that gKatz ⊗k k
′ (but not gKatz in general) is defined
over C. The fact that A ∈ gKatz ⊗k k
′ follows from the definition of a reduced basis and from
Theorem 4.5. �
Example 4.8. We illustrate the previous results on an example where everything can be checked
by hand calculations. Let
A =
(
0 1
x 1
2x
)
.
The Galois group G is a central extension of the infinite dihedral group (see [3, Example 6.1]).
The connected component G◦ of G containing 1 is the multiplicative group Gm with Lie alge-
bra gm generated by(
1 0
0 −1
)
.
Furthermore, using [4], we can see that the Katz algebra is 1-dimensional and it is generated by
N1 :=
(
0 1
x
1 0
)
.
We have ∇(N1) := N ′1 − [A,N1] = − 1
2xN1. Over k = C(x), no nonzero multiple of N1 is
conjugated to a constant matrix. However, over the finite extension k′ := C(
√
x),
√
xN1 is
conjugated to a constant matrix. Indeed, we have N1 = P−1DP with
D =
(
1√
x
0
0 − 1√
x
)
=
1√
x
(
1 0
0 −1
)
, P =
(
1 −1√
x
√
x
)
.
We see that gKatz ⊗k k
′ is now generated by the (constant) generator of the Galois–Lie
algebra gm and, applying the gauge transformation P , we have the reduced form
P [A] =
(√
x 0
0 −
√
x
)
=
√
x
(
1 0
0 −1
)
.
This shows that gKatz ⊗k k
′ = gm ⊗C k
′ and P [A] ∈ gKatz ⊗k k
′. Note that gKatz is not defined
over C whereas gKatz ⊗k k
′ is.
12 M. Barkatou, T. Cluzeau, L. Di Vizio and J.-A. Weil
Remark 4.9. Following the definition of the eigenring E([A]) in Example 2.3, any matrix
in E([A]) maps a solution of [A] to a solution of [A]. Example 3.4 showed that, in a reduced
basis, the eigenring is generated by constant matrices which commute with A. In reduced form,
we have A =
d∑
i=1
fiNi where fi ∈ k′, the fi are linearly independent over C and the Ni generate g
(see [3, Definition 6 and Remark 7]). Theorem 4.5 shows that the Ni also generate gKatz ⊗k k
′
as a Lie algebra. Given a constant matrix T ∈ E([A]), we have [A, T ] = 0, i.e.,
d∑
i=1
fi[Ni, T ] = 0.
As the fi are linearly independent over C, it follows that we have [Ni, T ] = 0 for all i. So,
in a reduced basis, the eigenring becomes the set of constant matrices which commute with
gKatz ⊗k k
′ (or, equivalently, with g).
In [2], André warns that spaces which are stable under gKatz may not be stable under ∇.
We can see this easily using reduced forms. Consider a linear differential system [A] : ∂~y = A~y
having all its solutions algebraic over k. Then gKatz = 0 so that anything is stable under gKatz.
In a reduced basis, the matrix of the linear differential system is the zero matrix and ∇ coincides
with ∂ = d
dx . A random vector in kn is not stable under ∇ even though it is stable under gKatz.
However, any line defined over C is clearly stable under ∂ and hence under ∇. The next
result builds on this observation to characterize which spaces, among those which are stable
under gKatz, are stable under ∇.
Proposition 4.10. Let M = (M,∇) be a completely reducible differential module over k and k′
denote a reduction field forM. Let W be a subspace of a construction Constr(M)⊗kk
′. Then W
is stable under ∇ if and only if both conditions below are fulfilled:
1) W is stable under gKatz ⊗k k
′;
2) W is defined over C.
Proof. The “only if” part follows from the definition of the Katz algebra (1) and from Theo-
rem 3.2(2). Now, assume that W is stable under gKatz⊗k k
′ and defined over C. Let C1, . . . , Cs
denote a constant basis of W . Let A be the matrix of the linear differential system associated
with M⊗k k
′ in a reduced basis. We have A =
∑
j fj(x)Nj , where fj(x) ∈ k′ and the Nj
form a (constant) basis of gKatz (and of g). As the Ci are constant, ∇ acts on them via
∇(Ci) = −constr(A)Ci and thus ∇(Ci) = −
∑
j fj(x)constr(Nj)Ci. Now, by hypothesis W is
stable under gKatz ⊗k k
′ so that constr(Nj)Ci is a linear combination (over k′) of the Cl. It
follows that ∇(Ci) is in W as announced. �
Each space in Constr(V ) which is invariant under g is in (Tannakian) correspondence with
a submodule of Constr(M), invariant under ∇ and hence under gKatz. A reciprocal property
would be to characterize, among all subspaces in any Constr(M) that are stable under gKatz,
which ones are stable under ∇ (and hence are in Tannakian correspondence with a g-module in
Constr(V ). Proposition 4.10 shows that this it is possible to do that at the cost of extending
scalars. When we study the subspaces of Constr(M)⊗k k
′ which are invariant under gKatz⊗k k
′,
the ones which are invariant under ∇ (and hence in tannakian correspondence with a g-module)
are exactly those which admit a constant basis.
Acknowledgements
We are grateful to the anonymous referees for their relevant suggestions which helped us to
improve the clarity and quality of this work.
Reduced Forms of Linear Differential Systems and the Intrinsic Galois–Lie Algebra of Katz 13
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1 Introduction
2 Notation and definitions
2.1 Differential modules
2.2 Picard–Vessiot extensions
2.3 The differential Galois group
3 Reduced forms
3.1 Characterization of reduced forms
3.2 Gauge transformation to a reduced form with local conditions
4 The intrinsic Galois–Lie algebra of Katz
References
|
| id | nasplib_isofts_kiev_ua-123456789-210696 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-17T12:04:31Z |
| publishDate | 2020 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Barkatou, Moulay Cluzeau, Thomas Di Vizio, Lucia Weil, Jacques-Arthur 2025-12-15T15:22:51Z 2020 Reduced Forms of Linear Differential Systems and the Intrinsic Galois-Lie Algebra of Katz. Moulay Barkatou, Thomas Cluzeau, Lucia Di Vizio and Jacques-Arthur Weil. SIGMA 16 (2020), 054, 13 pages 1815-0659 2020 Mathematics Subject Classification: 34M03; 34M15; 34C20 arXiv:1912.10567 https://nasplib.isofts.kiev.ua/handle/123456789/210696 https://doi.org/10.3842/SIGMA.2020.054 Generalizing the main result of [Aparicio-Monforte A., Compoint E., Weil J.-A., J. Pure Appl. Algebra 217 (2013), 1504-1516], we prove that a linear differential system is in reduced form in the sense of Kolchin and Kovacic if and only if any differential module in an algebraic construction admits a constant basis. Then we derive an explicit version of this statement. We finally deduce some properties of the Lie algebra of Katz's intrinsic Galois group. We are grateful to the anonymous referees for their relevant suggestions, which helped us to improve the clarity and quality of this work. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Reduced Forms of Linear Differential Systems and the Intrinsic Galois-Lie Algebra of Katz Article published earlier |
| spellingShingle | Reduced Forms of Linear Differential Systems and the Intrinsic Galois-Lie Algebra of Katz Barkatou, Moulay Cluzeau, Thomas Di Vizio, Lucia Weil, Jacques-Arthur |
| title | Reduced Forms of Linear Differential Systems and the Intrinsic Galois-Lie Algebra of Katz |
| title_full | Reduced Forms of Linear Differential Systems and the Intrinsic Galois-Lie Algebra of Katz |
| title_fullStr | Reduced Forms of Linear Differential Systems and the Intrinsic Galois-Lie Algebra of Katz |
| title_full_unstemmed | Reduced Forms of Linear Differential Systems and the Intrinsic Galois-Lie Algebra of Katz |
| title_short | Reduced Forms of Linear Differential Systems and the Intrinsic Galois-Lie Algebra of Katz |
| title_sort | reduced forms of linear differential systems and the intrinsic galois-lie algebra of katz |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/210696 |
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