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Local Integrable Symmetries of Diffieties

François Ollivier, Yirmeyahu J. Kaminski

Abstract

In the framework of diffieties, introduced by Vinogradov, we introduce integrable infinitesimal symmetries and show that they define a one parameter pseudogroup of local diffiety morphisms. We prove some preliminary results allowing to reduce the computation of integrable infinitesimal symmetries of a given order to solving a system of partial differential equations.We provide examples for which we can reduce to a linear system that can be solved by hand computation, and investigate some consequences for the local classification of diffiety, with a special interest for testing if a diffiety is flat.

Local Integrable Symmetries of Diffieties

Abstract

In the framework of diffieties, introduced by Vinogradov, we introduce integrable infinitesimal symmetries and show that they define a one parameter pseudogroup of local diffiety morphisms. We prove some preliminary results allowing to reduce the computation of integrable infinitesimal symmetries of a given order to solving a system of partial differential equations.We provide examples for which we can reduce to a linear system that can be solved by hand computation, and investigate some consequences for the local classification of diffiety, with a special interest for testing if a diffiety is flat.
Paper Structure (39 sections, 33 theorems, 108 equations, 3 figures, 1 algorithm)

This paper contains 39 sections, 33 theorems, 108 equations, 3 figures, 1 algorithm.

Table of Contents

  1. Theoretical framework
  2. Diffieties
  3. Control systems
  4. Diffiety extensions
  5. Strong accessibility
  6. Linearized system
  7. One parameter local automorphisms
  8. Infinitesimal symmetries of the diffiety
  9. Integrable infinitesimal symmetries and flow
  10. The pseudogroup defined by an integrable infinitesimal symmetry
  11. Some theoretical results
  12. Order of an integrable symmetry and an analog of Sluis -- Rouchon theorem
  13. PDE structure defined by an infinitesimal symmetry
  14. Structure of integrable vector fields
  15. Tame symmetries
  16. ...and 24 more sections

Key Result

Proposition 1.18

Consider the system defined by the Cartan field $\tau$. Let $\updelta$ be a derivation. We recursively define $\hat{\uptau}^0\updelta:=\updelta$ and $\hat{\uptau}^{k+1}\updelta:=[\hat{\uptau}^{k}\updelta,\uptau]$. The system is strongly accessible iff the Lie algebra generated by the $\hat{\uptau}^k

Figures (3)

  • Figure 3.1: Chimney and tunnel for derivatives of $x_i$; main derivatives of elements of $\mathcal{A}$ are indicated with $\bullet$.
  • Figure 3.2: Tunnel and stair from ex. \ref{['ex::tunnel1']}
  • Figure 3.3: Chimney and stair from ex. \ref{['ex::chimney']}

Theorems & Definitions (80)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.8
  • Definition 1.11
  • Definition 1.15
  • Definition 1.16
  • Definition 1.17
  • Proposition 1.18
  • proof
  • Definition 1.19
  • ...and 70 more