# nLab Picard-Vessiot theory

Contents

### Context

#### Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

• (shape modality $\dashv$ flat modality $\dashv$ sharp modality)

$(\esh \dashv \flat \dashv \sharp )$

• dR-shape modality$\dashv$ dR-flat modality

$\esh_{dR} \dashv \flat_{dR}$

infinitesimal cohesion

tangent cohesion

differential cohesion

singular cohesion

$\array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& \esh &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }$

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

# Contents

## Idea

Picard-Vessiot theory is the Galois theory of homogeneous linear ordinary differential equations. A version for homogeneous linear difference equations has also been developed. Alternatively, terms differential Galois theory and difference Galois theory have also been used.

## Literature

#### Classical references

• E. R. Kolchin, Algebraic matric groups and the Picard-Vessiot theory of homogeneous linear ordinary differential equations, Ann. Math. (2nd ser) 49:1 (1948) 1-42 jstor doi
• Charles H. Franke, Picard-Vessiot theory of linear homogeneous difference equations, Trans. Amer. Math. Soc. 108:3 (1963) 491-515 jstor doi

#### In categorical setup

Tannakian approach has been developed in

In the setup of differential abelian tensor categories over differential rings,

Galois theory for differential equations is one of the examples in

Difference equations are viewed from topos perspective (with sporadic recourse to Galois theory) in

• Ivan Tomašić, A topos-theoretic view of difference algebra, arxiv/2001.09075

#### Connections to and variants in model theory

• D. Marker, Model theory of differential fields, In Model theory of fields, edited by Marker, Messmer, Pillay. Lecture Notes in Logic 5. ASL-Peters, 2006
• Moshe Kamensky, Anand Pillay, Interpretations and differential Galois extensions, IMRN 24 (2016) 7390–7413 doi
• Z. Chatzidakis, Anand Pillay, Generalized Picard-Vessiot extensions and differential Galois cohomology, 2017, pdf

#### Hopf algebraic approach

• Mitsuhiro Takeuchi, A Hopf algebraic approach to the Picard-Vessiot theory, J. Algebra 122:2 (1989) 481-509 doi

…Hopf algebraic approach simplifies and generalizes the theory. Among other things the generalization gives a Picard-Vessiot theory for positive characteristic, for fields with not necessarily commuting derivations, and even for fields with (a set of) higher derivations.

• K. Amano, A. Masuoka, M. Takeuchi, Hopf algebraic approach to Picard-Vessiot theory, Handbook of Algebra, Volume 6, (2009) 127-171 doi

Last revised on January 22, 2021 at 20:59:15. See the history of this page for a list of all contributions to it.