Differential Algebraic Groups Research Articles

Zariski closures of reductive linear differential algebraic groups

Linear differential algebraic groups (LDAGs) appear as Galois groups of systems of linear differential and difference equations with parameters. These groups measure differential-algebraic dependencies among solutions of the equations. LDAGs are now also used in factoring partial differential operators. In this paper, we study Zariski closures of LDAGs. In particular, we give a Tannakian characterization of algebraic groups that are Zariski closures of a given LDAG. Moreover, we show that the Zariski closures that correspond to representations of minimal dimension of a reductive LDAG are all isomorphic. In addition, we give a Tannakian description of simple LDAGs. This substantially extends the classical results of P. Cassidy and, we hope, will have an impact on developing algorithms that compute differential Galois groups of the above equations and factoring partial differential operators.

A Jordan–Hölder Theorem for differential algebraic groups

We show that a differential algebraic group can be filtered by a finite subnormal series of differential algebraic groups such that successive quotients are almost simple, that is have no normal subgroups of the same type. We give a uniqueness result, prove several properties of almost simple groups and, in the ordinary differential case, classify almost simple linear differential algebraic groups.

Differential Tannakian categories

We define a differential Tannakian category and show that under a natural assumption it has a fibre functor. If in addition this category is neutral, that is, the target category for the fibre functor are finite dimensional vector spaces over the base field, then it is equivalent to the category of representations of a (pro-)linear differential algebraic group. Our treatment of the problem is via differential Hopf algebras and Deligne's fibre functor construction [P. Deligne, Catégories tannakiennes, in: The Grothendieck Festschrift, vol. II, in: Progr. Math., vol. 87, Birkhäuser Boston, Boston, MA, 1990, pp. 111–195].

Tannakian Categories, Linear Differential Algebraic Groups, and Parametrized Linear Differential Equations

We provide conditions for a category with a fiber functor to be equivalent to the category of representations of a linear differential algebraic group. This generalizes the notion of a neutral Tannakian category used to characterize the category of representations of a linear algebraic group.

Tannakian Approach to Linear Differential Algebraic Groups

Tannaka's Theorem states that a linear algebraic group G is determined by the category of finite dimensional G-modules and the forgetful functor. We extend this result to linear differential algebraic groups by introducing a category corresponding to their representations and show how this category determines such a group.

Generalized differential Galois theory

A Galois theory of differential fields with parameters is developed in a manner that generalizes Kolchin's theory. It is shown that all connected differential algebraic groups are Galois groups of some appropriate differential field extension.

Quantifier elimination for algebraic $D$-groups

We prove that if G is an algebraic D-group (in the sense of Buium over a differentially closed field (K,∂) of characteristic 0, then the first order structure consisting of G together with the algebraic D-subvarieties of G, G x G,..., has quantifier-elimination. In other words, the projection on G of a D-constructible subset of G n+1 is D-constructible. Among the consequences is that any finite-dimensional differential algebraic group is interpretable in an algebraically closed field.

Lazard’s Theorem for differential algebraic groups and proalgebraic groups

We prove that a differential group whose underlying variety is an affine space is unipotent. The problem is reduced to an infinite-dimensional version of Lazard's Theorem.

Pro-algebraic and differential algebraic group structures on affine spaces

We prove the following result, solving a problem raised in an article by A. Buium and Ph. J. Cassidy, to appear in the collected works of E. Kolchin. If G is a differential algebraic group whose underlying set is some affine n -space, then G is unipotent. A key result, possibly of independent interest, concerns infinite-dimensional group schemes: a group scheme whose underlying scheme is Spec ( K [ X 1 , X 2 ,...]) ( K an algebraically closed field of characteristic 0) is a projective limit of unipotent algebraic groups.

Galois Theory of Differential Equations, Algebraic Groups and Lie Algebras

The Galois theory of linear differential equations is presented, including full proofs. The connection with algebraic groups and their Lie algebras is given. As an application the inverse problem of differential Galois theory is discussed. There are many exercises in the text.

David Marker, Introduction to the model theory of fields. Model theory of fields, Lecture notes in logic, no. 5, Springer, Berlin, Heidelberg, New York, etc., 1996, pp. 1–37. - David Marker. Model theory of differential fields. Model theory of fields, Lecture notes in logic, no. 5, Springer, Berlin, Heidelberg, New York, etc., 1996, pp. 38–113. - Anand Pillay. Differential algebraic

David Marker, Introduction to the model theory of fields. Model theory of fields, Lecture notes in logic, no. 5, Springer, Berlin, Heidelberg, New York, etc., 1996, pp. 1–37. - David Marker. Model theory of differential fields. Model theory of fields, Lecture notes in logic, no. 5, Springer, Berlin, Heidelberg, New York, etc., 1996, pp. 38–113. - Anand Pillay. Differential algebraic groups and the number of countable differentially closed fields. Model theory of fields, Lecture notes in logic, no. 5, Springer, Berlin, Heidelberg, New York, etc., 1996, pp. 114–134. - Margit Messmer. Some model theory of separably closed fields. Model theory of fields, Lecture notes in logic, no. 5, Springer, Berlin, Heidelberg, New York, etc., 1996, pp. 135–152. - Volume 63 Issue 2

Some foundational questions concerning differential algebraic groups

In this paper we solve some problems posed by Kolchin about differential algebraic groups. The main result (from which the others follow) is the embeddability of any differential algebraic group in an algebraic group. A crucial intermediate result, and one of independent interest, is a generalisation of Weil’s theorem on recovering an algebraic group from birational data, to pro-algebraic groups.

A gap theorem for abelian varieties over differential fields

In [B1] the first author proved a theorem about differential algebraic groups (in the sense of Cassidy-Kolchin [C], [K2]) that implied the geometric LangMordell conjecture [L]; cf. Theorem 1 below. The proof in [B1] used an analytic argument, based on “Big Picard”. Hrushovski [H] was able to replace this analytic argument by a remarkable model theoretic argument, based in its turn on the difficult theory developed in [HZ]. Now Hrushovski’s model theoretic methods further lead to striking new results on differential algebraic groups [HS]; cf. Theorem 3 below. So a natural challenge presents itself: to find a proof for these new results that is free from model theory. In this note we shall give in particular a quick analytic proof of Theorem 3, based, again, on “Big Picard”. The main result of this note is the “Gap” Theorem 2 below, which should be viewed as a significant complement to Theorem 1; Theorems 1 and 2 will then easily imply Theorem 3. Some comments are in order concerning the role of the authors in this paper. The second author asked whether analytic methods as in [B1] could yield strong minimality and local modularity of the “Manin kernel” of a simple abelian variety which does not descend to the constants (Theorem 3 below). The first author then proved strong minimality (assertion 1 of Theorem 3) by proving a weaker version of Theorem 2 (namely for simple abelian varieties). The second author suggested some general ideas, analogous to those in [P1], for obtaining local modularity. In trying to carry out details, the first author came up with the proof of the present Theorem 2, from which everything follows. Recall some basic terminology of differential algebra [K1], [C], [B2]. (The definitions below will suffice to understand the statements of the Theorems below, without assuming any previous knowledge of differential algebra; for the proofs, however, familiarity with [B1], [B2] is required.) Let F be a δ−field (i.e. a field of characteristic zero equipped with a derivation δ.) One defines the ring of δ−polynomials F{y1, ..., yN} as the ring of usual polynomials with F−coefficients in the variables δyj , 1 ≤ j ≤ N, i ≥ 0. There is an obvious notion of order for δ−polynomials. F is said to be δ−closed if for any A, B ∈ F{y}, B = 0, such that the order of A is strictly bigger than the order of B there

A remark on differential algebraic groups.

Let U be a universal differential field of characteristic 0 with a set of commuting derivations. In [C] it is shown that if G is an almost simple differential algebraic group then G is isogeneous (as a differential algebraic group) to a linear algebraic group relative to a δclosed subfield of U. Here we show, using some model-theoretic facts, that isogeneous can be replaced by isomorphic. In differential algebra the underlying universe is taken to be a “universal” differential field U with a set of commuting derivations. Starting with subsets of Un defined by differential polynomial equations (theδ-closed subsets of Un), and maps defined by differential polynomials, a category of differential algebraic sets (varieties) is constructed [K]. A differential algebraic group is a group object in this category. If G is such a group then G is said to be A-connected if G has no proper δ-closed subgroup of finite index. G is said to be almost - simple (or, in the language of [C], δ-simple) if G has no proper nont...

Splitting differential algebraic groups

Splitting differential algebraic groups

Differential and difference algebra

In a survey the main results in differential and difference algebra (differential rings and modules; differential fields; differential-algebraic geometry; differential and difference dimension; differential and difference Galois theory; integration in finite terms; differential algebraic groups), compiled from the reviews published inReferativnyl Zhurnal Matematika for the years 1975–1986, are reflected.

Differential Algebraic Group Structures on the Plane

Differential Algebraic Group Structures on the Plane

Differential algebraic group structures on the plane

The differential algebraic group structures on the affine line and plane are classified. The additive group G a {G_a} of the coefficient field is the only differential algebraic group structure on the line. Every differential algebraic group with underlying set in the plane is unipotent and is isomorphic to a group whose law of composition is defined by the formula \[ ( u 1 , u 2 ) ( v 1 , v 2 ) = ( u 1 + v 1 , u 2 + v 2 + f ( u 1 , v 1 ) ) , ({u_1},{u_2})({v_1},{v_2}) = ({u_1} + {v_1},{u_2} + {v_2} + f({u_1},{v_1})), \] where f is a 2-cocycle of G a {G_a} into G a {G_a} .

The differential rational representation algebra on a linear differential algebraic group

In [2], the author began the study of affine differential algebraic groups. The study of such groups is somewhat analogous to the study of algebraic groups. However, the non-Noetherian character of the differential rings associated with differential algebraic sets has made it necessary to develop new techniques for the study of algebraic differential equations. The reader is referred to Kolchin [7]. It is well-known, easy to prove, and central to the study of algebraic groups, that the ring of everywhere defined rational functions on an affine algebraic set is the coordinate ring. The differential ring of everywhere defined differential rational functions on an affine differential algebraic set is not the differential coordinate ring, and is not, in general, finitely generated as a differential algebra. For this reason, an affine differential algebraic group need not be linear, i.e., it need not be differentially rationally isomorphic to a differential algebraic matric group. If G is an affine differential algebraic group, the differential algebra, generated over the coefficient field by the coordinate functions of all the linear differential rational representations of G, is called the dz@rentiaZ rational representation algebra on G, and is denoted by W(G). The purpose of this note is to prove the following theorem and its corollaries:

Typical differential dimension of the intersection of linear differential algebraic groups

Let % be a universal differential field of characteristic zero. Let “y; and “yz be two irreducible closed (in the differential Zariski topology) subsets of %!n of differential dimension rr and rz , respectively. In general, it is not true that every irreducible component of the intersection Y1 n $$, has a differential dimension 3 rr + ~a - n. Indeed Ritt [3, p. 1331 has an example of an irreducible closed subset of &s of differential dimension 2 that intersects the hyperplane defined by the equation y3 = 0 at the single point (0, 0,O). In this paper (Theorem 4.1 and corollaries) we show that such abnormality does not arise in two special cases: namely, when Vr and Vs are differential algebraic subgroups of either G,” or GL(n). Here G,” denotes the differential algebraic group whose underlying set is 9P and GL(n) denotes the general linear group of n