A categorical approach to Picard-Vessiot theory

Infinitesimal group schemes as iterative differential Galois groups

This article is concerned with Galois theory for iterative differential fields (ID-fields) in positive characteristic. More precisely, we consider purely inseparable Picard-Vessiot extensions, because these are the ones having an infinitesimal group scheme as iterative differential Galois group. In this article we prove a necessary and sufficient condition to decide whether an infinitesimal group scheme occurs as Galois group scheme of a Picard-Vessiot extension over a given ID-field or not. In particular, this solves the inverse ID-Galois problem for infinitesimal group schemes.

On General Extension Fields for the Classical Groups in Differential Galois Theory

Let $G$ be one of the classical groups of Lie rank $l$. We make a similar construction of a general extension field in differential Galois theory for $G$ as E. Noether did in classical Galois theory for finite groups. More precisely, we build a differential field $E$ of differential transcendence degree $l$ over the constants on which the group $G$ acts and show that it is a Picard-Vessiot extension of the field of invariants $E^G$. The field $E^G$ is differentially generated by $l$ differential polynomials which are differentially algebraically independent over the constants. They are the coe…

Galois theory for iterative connections and nonreduced Galois groups

This article presents a theory of modules with iterative connection. This theory is a generalisation of the theory of modules with connection in characteristic zero to modules over rings of arbitrary characteristic. We show that these modules with iterative connection (and also the modules with integrable iterative connection) form a Tannakian category, assuming some nice properties for the underlying ring, and we show how this generalises to modules over schemes. We also relate these notions to stratifications on modules, as introduced by A. Grothendieck in order to extend integrable (ordinar…

Real Liouvillian Extensions of Partial Differential Fields

In this paper, we establish Galois theory for partial differential systems defined over formally real differential fields with a real closed field of constants and over formally $p$-adic differential fields with a $p$-adically closed field of constants. For an integrable partial differential system defined over such a field, we prove that there exists a formally real (resp. formally $p$-adic) Picard-Vessiot extension. Moreover, we obtain a uniqueness result for this Picard-Vessiot extension. We give an adequate definition of the Galois differential group and obtain a Galois fundamental theorem…

Galois theory of Artinian simple module algebras

This main purpose of this article is the unification of the Galois theory of algebraic differential equations by Umemura and the Galois theory of algebraic difference equations by Morikawa-Umemura in a common framework using Artinian simple D-module algebras, where D is a bialgebra. We construct the Galois hull of an extension of Artinian simple D-module algebras and define its Galois group, which consists of infinitesimal coordinate transformations fulfilling certain partial differential equations and which we call Umemura functor. We eliminate the restriction to characteristic 0 from the abo…

关于常微分算子中心化子的分类：一种有效的微分代数方法

On the classification of centralizers of ODOs: An effective differential algebra approach

本文概述了利用微分代数和符号计算使常微分算子交换环与代数曲线对应关系计算有效的最新进展，描述了谱曲线定义理想及谱层计算，并旨在发展谱问题的Picard-Vessiot理论。

微分中心单代数的微分伽罗瓦群及其射影表示

Differential Galois Groups of Differential Central Simple Algebras and their Projective Representations

本文研究微分中心单代数的分裂性与微分伽罗瓦群表示，证明分裂等价于 Picard-Vessiot 扩张的嵌入，并建立分裂代数同构类与代数群在 PGL_n 中表示类之间的双射，进而用 δ-右理想刻画群的约化性与可解性。

经典群微分伽罗瓦理论中的正问题

On the Direct Problem in Differential Galois Theory for the Classical Groups

本文研究经典群微分伽罗瓦理论的正问题，通过分析一般Picard-Vessiot扩张的结构，结合已知算法，确定特殊化线性微分方程的微分伽罗瓦群。