Linear Differential Algebraic Groups Research Articles

Picard-Vessiot extensions, linear differential algebraic groups, and their torsors

Let K be a differential field with algebraically closed field of constants CK . Let K P V ∞ be the (iterated) Picard-Vessiot closure of K. Let G be a linear differential algebraic group over K, and X a differential algebraic torsor for G over K. We prove that X ( K P V ∞ ) is Kolchin dense in X. In the special case that G is finite-dimensional we prove that X ( K P V ∞ ) = X ( K diff ) (where Kdiff is the differential closure of K). We also give close relationships between Picard-Vessiot extensions of K and the category of torsors for suitable finite-dimensional linear differential algebraic groups over K.

Triviality of differential Galois cohomology of linear differential algebraic groups

For a partial differential field K, we show that the triviality of the first differential Galois cohomology of every linear differential algebraic group over K is equivalent to K being algebraically, Picard–Vessiot, and linearly differentially closed. This cohomological triviality condition is also known to be equivalent to the uniqueness up to an isomorphism of a Picard–Vessiot extension of a linear differential equation with parameters.

New classes of parameterized differential Galois groups

This paper is on the inverse parameterized differential Galois problem. We show that surprisingly many groups do not occur as parameterized differential Galois groups over K(x) even when K is algebraically closed. We then combine the method of patching over fields with a suitable version of Galois descent to prove that certain groups do occur as parameterized differential Galois groups over k((t))(x). This class includes linear differential algebraic groups that are generated by finitely many unipotent elements and also semisimple connected linear algebraic groups.

The Picard–Vessiot theory, constrained cohomology, and linear differential algebraic groups

We prove that a differential field K is algebraically closed and Picard–Vessiot closed if and only if the differential Galois cohomology group, H∂1(K,G), is trivial for any linear differential algebraic group G over K. We give applications to the parameterized Picard–Vessiot theory and pose several problems.

Calculating differential Galois groups of parametrized differential equations, with applications to hypertranscendence

The main motivation of our work is to create an efficient algorithm that decides hypertranscendence of solutions of linear differential equations, via the parameterized differential and Galois theories. To achieve this, we expand the representation theory of linear differential algebraic groups and develop new algorithms that calculate unipotent radicals of parameterized differential Galois groups for differential equations whose coefficients are rational functions. P. Berman and M.F. Singer presented an algorithm calculating the differential Galois group for differential equations without parameters whose differential operator is a composition of two completely reducible differential operators. We use their algorithm as a part of our algorithm. As a result, we find an effective criterion for the algebraic independence of the solutions of parameterized differential equations and all of their derivatives with respect to the parameter.

On central extensions of simple differential algebraic groups

We consider central extensions Z ↪→ E G in the category of linear differential algebraic groups. We show that if G is simple non-commutative and Z is unipotent with the differential type smaller than that of G, then such an extension splits. We also give a construction of central extensions illustrating that the condition on differential types is important for splitting. Our results imply that non-commutative almost simple linear differential algebraic groups, introduced by Cassidy and Singer, are simple.

Reductive Linear Differential Algebraic Groups and the Galois Groups of Parameterized Linear Differential Equations

We develop the representation theory for reductive linear differential algebraic groups (LDAGs). In particular, we exhibit an explicit sharp upper bound for orders of derivatives in differential representations of reductive LDAGs, extending existing results, which were obtained for SL(2) in the case of just one derivation. As an application of the above bound, we develop an algorithm that tests whether the parameterized differential Galois group of a system of linear differential equations is reductive and, if it is, calculates it.

Isomonodromic differential equations and differential categories

We study isomonodromicity of systems of parameterized linear differential equations and related conjugacy properties of linear differential algebraic groups by means of differential categories. We prove that isomonodromicity is equivalent to isomonodromicity with respect to each parameter separately under a filtered-linearly closed assumption on the field of functions of parameters. Our result implies that one does not need to solve any non-linear differential equations to test isomonodromicity anymore. This result cannot be further strengthened by weakening the requirement on the parameters as we show by giving a counterexample. Also, we show that isomonodromicity is equivalent to conjugacy to constants of the associated parameterized differential Galois group, extending a result of P. Cassidy and M. Singer, which we also prove categorically. We illustrate our main results by a series of examples, using, in particular, a relation between the Gauss–Manin connection and parameterized differential Galois groups.

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.

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.

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