nLab
Picard-Vessiot theory

Contents

Context

Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

tangent cohesion

differential cohesion

graded differential cohesion

singular cohesion

id id fermionic bosonic bosonic Rh rheonomic reduced infinitesimal infinitesimal & étale cohesive ʃ discrete discrete continuous * \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}{}& ʃ &\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,

  • Henri Gillet, Sergey Gorchinskiy?, Alexey Ovchinnikov, Parameterized Picard–Vessiot extensions and Atiyah extensions, Adv. Math. 238 (2013) 322-411 doi

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 15:59:15. See the history of this page for a list of all contributions to it.