Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 5 (2009), 103, 40 pages      arXiv:0911.3344      http://dx.doi.org/10.3842/SIGMA.2009.103
Contribution to the Special Issue “Élie Cartan and Differential Geometry”

Isomorphism of Intransitive Linear Lie Equations

Jose Miguel Martins Veloso
Faculdade de Matematica, UFPA, Belem, PA, CEP 66075-110, Brasil

Received February 09, 2009, in final form November 11, 2009; Published online November 17, 2009

Abstract
We show that formal isomorphism of intransitive linear Lie equations along transversal to the orbits can be extended to neighborhoods of these transversal. In analytic cases, the word formal is dropped from theorems. Also, we associate an intransitive Lie algebra with each intransitive linear Lie equation, and from the intransitive Lie algebra we recover the linear Lie equation, unless of formal isomorphism. The intransitive Lie algebra gives the structure functions introduced by É. Cartan.

Key words: Lie equations; Lie groupoids; intransitive; isomorphism.

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References

  1. Cartan É., Sur le structure des groupes infinis de transformations, Ann. Sci. École Norm. Sup. (3) 21 (1904), 153-206.
    Cartan É., Sur la structure des groupes infinis de transformation (suite), Ann. Sci. École Norm. Sup. (3) 22 (1905), 219-308.
  2. Conn J.F., A new class of counterexamples to the integrability problem, Proc. Nat. Acad. Sci. USA 74 (1977), 2655-2658.
  3. Frölicher A., Nijenhuis A., Theory of vector valued differential forms. I. Derivations in the graded ring of differential forms, Nederl. Akad. Wet. Proc. Ser. A 59 (1956), 338-359.
  4. Goldschmidt H., Existence theorems for analytic linear partial differential equations, Ann. of Math. (2) 86 (1967), 246-270.
  5. Goldschmidt H., Integrability criteria for systems of non-linear partial differential equations, J. Differential Geom. 1 (1967), 269-307.
  6. Goldschmidt H., Sur la structure des équations de Lie. I. Le troisième théorème fondamental, J. Differential Geom. 6 (1972), 357-373.
  7. Goldschmidt H., Sur la structure des équations de Lie. II. Équations formellement transitives, J. Differential Geom. 7 (1972). 67-95.
  8. Goldschmidt H., Sur la structure des équations de Lie. III. La cohomologie de Spencer, J. Differential Geom. 11 (1976). 167-223.
  9. Goldschmidt H., Spencer D., On the non-linear cohomology of Lie equations. I, Acta Math. 136 (1976), 103-170.
    Goldschmidt H., Spencer D., On the non-linear cohomology of Lie equations. II, Acta Math. 136 (1976), 171-239.
  10. Guillemin V.W., Sternberg S., An algebraic model of transitive differential geometry, Bull. Amer. Math. Soc. 70 (1964), 16-47.
  11. Guillemin V.W., Sternberg S., The Lewy counterexample and the local equivalence problem for G-structures, J. Differential Geom. 1 (1967), 127-131.
  12. Kiso K., Local properties of intransitive infinite Lie algebra sheaves, Japan. J. Math. (N.S.) 5 (1979), 101-155.
  13. Kiso K., Infinitesimal automorphisms of G-structures and certain intransitive infinite Lie algebra sheaves, Hokkaido Math. J. 11 (1982), 301-327.
  14. Kumpera A., Spencer D.C., Lie equations, Vol. I, General theory, Annals of Mathematics Studies, Vol. 73, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1972.
  15. Kamber F.W., Tondeur P., Foliated bundles and characteristic classes, Lecture Notes in Mathematics, Vol. 493, Springer-Verlag, Berlin - New York, 1975.
  16. Malgrange B., Equations de Lie. I, J. Differential Geom. 6 (1972), 503-522.
  17. Malgrange B., Equations de Lie. II, J. Differential Geom. 7 (1972), 117-141.
  18. Morimoto T., On the intransitive Lie algebras whose transitive parts are infinite and primitive, J. Math. Soc. of Japan 29 (1977), 35-65.
  19. Ngô Van Quê V.V.T., Définition des pseudogroupes infinitesimaux de Lie intransitifs. Théorème fondamental de réalization en dimension deux, Nagoya Math. J. 86 (1982), 211-228.
  20. Petitjean A., Rodrigues A.M., Correspondance entre algébres de Lie abstraites et pseudo-groupes de Lie transitifs, Ann. of Math. (2) 101 (1975), 268-279.
  21. Ruiz C., Prolongement formel des systèmes différentiels extérieur d'ordre supérieur, C. R. Acad. Sci. Paris Sér. A-B 285 (1977), A1077-A1080.
  22. Ruiz C., Complexe de Koszul du symbole d'un système différentiel extérieur, C. R. Acad. Sci. Paris Sér. A-B 286 (1978), A55-A58.
  23. Ruiz C., Propriétès de dualité du prolongement formel des systèmes différentiels extérieurs, C. R. Acad. Sci. Paris Sér. A-B 286 (1978), A99-A101.
  24. Singer I.M., Sternberg S., The infinite groups of Lie and Cartan. I. The transitive groups, J. Analyse Math. 15 (1965), 1-114.
  25. Veloso J.M.M., Lie's third theorem for intransitive Lie equations, J. Differential Geom. 32 (1990), 185-198.
  26. Veloso J.M., Prolongation projection commutativity theorem, in CR-Geometry and Overdetermined Systems (Osaka, 1994), Adv. Stud. Pure Math., Vol. 25, Math. Soc. Japan, Tokyo, 1997, 386-405.
  27. Veloso J.M.M., New classes of intransitive simple Lie pseudogroups, Bull. Soc. Sci. Lett. Lodz 36 (1986), no. 19, 7 pages.

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