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Annali di Matematica Pura ed Applicata (1923 -)

, Volume 198, Issue 5, pp 1863–1884 | Cite as

Group invariance of integrable Pfaffian systems

  • A. KumperaEmail author
Article
  • 145 Downloads

Abstract

Let \({\mathcal {S}}\) be an integrable Pfaffian system. When it is invariant under a transversally free infinitesimal action of a finite-dimensional real Lie algebra g and consequently invariant under the local action of a Lie group G, we show that the vertical variational cohomology of \({\mathcal {S}}\) is equal to the Lie algebra cohomology of g with values in the space of the horizontal cohomology in maximum dimension. This result, besides giving an effective algorithm for the computation of the variational cohomology of an invariant Pfaffian system, provides a method for detecting obstructions to the existence of finite or infinitesimal actions leaving a given system invariant.

Keywords

Differential systems Partial differential equations Pfaffian systems Standard prolongations Merihedric prolongations Lie groups and algebras Lie groupoids and algebroids Finite and infinitesimal actions Variational cohomology Lie algebra cohomology Local models Integration Foliations 

Mathematics Subject Classification

Primary 53C05 Secondary 53C15 53C17 

Notes

Acknowledgements

Our acknowledgements are due to Ercüment Ortaçgil for his precious help and encouragement and also for suggesting us the use of quaternions. Initially, the author tried to apply Élie Cartan’s beautiful technique of the moving frame, but the quaternions simplified and rendered more concise our argumentation. It was also Prof. Ortaçgil that indicated to us the so practical text [14].

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Copyright information

© Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag GmbH Germany, part of Springer Nature 2019
corrected publication 2019

Authors and Affiliations

  1. 1.Universidade Estadual de Campinas (State University of Campinas)CampinasBrazil

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