Unipotent differential algebraic groups as parameterized differential Galois groups
type: Item
instance of: scholarly article
title: Unipotent differential algebraic groups as parameterized differential Galois groups
zbMATH Open document ID: 1364.12005
DOI: 10.1017/S1474748013000200
author: Andrey Minchenko
author: Alexey Ovchinnikov
author: Michael F. Singer
published in: Journal of the Institute of Mathematics of Jussieu
publication date: 2014-09-30T00:00:00Z
full work available at URL: https://arxiv.org/abs/1301.0092
Mathematics Subject Classification ID: 12H05
Mathematics Subject Classification ID: 34M15
Mathematics Subject Classification ID: 12H20
Mathematics Subject Classification ID: 20G05
zbMATH DE Number: 6349676
zbMATH Keywords: differential algebraic groups
zbMATH Keywords: parameterized differential Galois theory
zbMATH Keywords: algorithms
Wikidata QID: Q57440944
MaRDI profile type: Publication
arXiv ID: 1301.0092
cites work: Algebraic Groups and Algebraic Dependence
cites work: Algebraic matric groups and the Picard-Vessiot theory of homogeneous linear ordinary differential equations
cites work: Existence of \(\partial\)-parameterized Picard-Vessiot extensions over fields with algebraically closed constants
cites work: Splitting fields and general differential Galois theory
cites work: Solution of the Inverse Problem of Differential Galois Theory in the Classical Case
cites work: Linear algebraic groups as parameterized Picard-Vessiot Galois groups
cites work: Tannakian categories, linear differential algebraic groups, and parametrized linear differential equations
cites work: Tannakian approach to linear differential algebraic groups.
cites work: Differential Galois theory of linear difference equations
cites work: Complexity of factoring and calculating the GCD of linear ordinary differential operators
cites work: Computing Galois groups of completely reducible differential equations
cites work: Parameterized Picard-Vessiot extensions and Atiyah extensions
cites work: The classification of the semisimple differential algebraic groups and the linear semisimple differential algebraic Lie algebras
cites work: The differential rational representation algebra on a linear differential algebraic group
cites work: Differential Algebraic Groups
cites work: Monodromy groups of parameterized linear differential equations with regular singularities
cites work: Zariski closures of reductive linear differential algebraic groups
cites work: Generalized differential Galois theory
OpenAlex ID: W3099568133
review text: The parametrized Picard-Vessiot (PPV) Galois theory is a Galois theory of linear differential equations, with respect to a derivation \(\partial\), of the form \(\partial Y=A Y\), where A is a squared matrix whose entries belong to a \(\{\partial\}\cup \Delta\)-differential field \(k\), with \(\{\partial\}\cup\Delta\) a finite set of commuting derivations. This theory was presented in [\textit{P. J. Cassidy} and \textit{M. F. Singer}, IRMA Lect. Math. Theor. Phys. 9, 113--155 (2007; Zbl 1356.12004)], where the next example is given: \(\frac{dy}{dx}=\frac{x}{t} y\).NEWLINENEWLINEA PPV extension \(K\) of \(k\) associated with \(\partial Y=A Y\) is a \(\{\partial\}\cup \Delta\)-differential field verifying:NEWLINENEWLINE1. \(K=k\langle Z\rangle\) is generated as a \(\{\partial\}\cup \Delta\)-differential field by the entries of a matrix \(Z\) such that \(\partial Z=AZ\).NEWLINENEWLINE2. The field of constants of \(k\) and \(K\) with respect to \(\partial\) coincide, let as call it \(C\).NEWLINENEWLINEIt was proved in the previous paper that \(C\) is a \(\Delta\)-field and if it is differentially closed the PPV extension always exists and it is unique up to \(\{\partial\}\cup \Delta-k\)-isomorphisms.NEWLINENEWLINENEWLINEThe PPV Galois theory studies the group of differential symmetries of the PPV extension, called a parameterized differential Galois group (PPV group). These are linear differential algebraic groups (LDAGs), groups of matrices whose entries satisfy polynomial differential equations in the parameters.NEWLINENEWLINENEWLINEThe paper under review deals with some aspect of the inverse and direct problems of PPV Galois theory over \(k=\mathcal{U} (x)\), where \(\mathcal{U}\) is a universal differential field with derivations \(\Delta\). It gives a characterization of those LDAGs that occur as PPV groups over \(k=\mathcal{U} (x)\). Namely, a LDAG is a PPV group if and only if it constains a finitely generated Kolchin-dense subgroup. Related with the direct probl…
reviewed by: Sonia Rueda
description: scientific article; zbMATH DE number 6349676