Hostname: page-component-76fb5796d-45l2p Total loading time: 0 Render date: 2024-04-27T09:45:31.795Z Has data issue: false hasContentIssue false
  • English
  • Français

Moving Frames for Lie Pseudo–Groups

Published online by Cambridge University Press:  20 November 2018

Peter J. Olver  and
Juha Pohjanpelto
Peter J. Olver
Affiliation:
School of Mathematics, University of Minnesota, Minneapolis, MN 55455 USA e-mail:olver@math.umn.edu
Juha Pohjanpelto
Affiliation:
Department of Mathematics, Oregon State University, Corvallis, OR 97331 USA e-mail:juha@math.oregonstate.edu
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We propose a new, constructive theory of moving frames for Lie pseudo-group actions on submanifolds. The moving frame provides an effective means for determining complete systems of differential invariants and invariant differential forms, classifying their syzygies and recurrence relations, and solving equivalence and symmetry problems arising in a broad range of applications.

Keywords

58A1558A2058H0558J70
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 60 , Issue 6 , 01 December 2008 , pp. 1336 - 1386
Copyright
Copyright © Canadian Mathematical Society 2008

References

[1] Anderson, I.M., The variational bicomplex. Utah State Technical Report, 1989, http://www.math.usu.edu/˜ fg mp/Publications/VB/vb.pdf Google Scholar
[2] Anderson, I.M., The Vessiot handbook. Technical Report, Utah State University, 2000.Google Scholar
[3] Anderson, I.M. and Fels, M. Exterior differential systems with symmetry. Acta Appl. Math. 87(2005), 3–31.Google Scholar
[4] Anderson, I.M. and Pohjanpelto, J., Infinite-dimensional Lie algebra cohomology and the cohomology of invariant Euler-Lagrange complexes: a preliminary report In: Differential geometry and applications,Masaryk Univ., Brno, 1996, pp. 427–448.Google Scholar
[5] Bleecker, D., Gauge theory and variational principles. Addison–Wesley Publ. Co., Reading, Mass., 1981.Google Scholar
[6] Bott, R., On characteristic classes in the framework of Gelfand–Fuks cohomology. Asterisque 32–33(1976) 113–139.Google Scholar
[7] Cantwell, B.J., Introduction to Symmetry Analysis. Cambridge Texts in AppliedMathematics, Cambridge University Press, Cambridge, 2002.Google Scholar
[8] Cartan, É., LaMéthode du Repère Mobile, la Théorie des Groupes Continus, et les Espaces Généralisés. Exposés de Géométrie. 5, Hermann, Paris, 1935.Google Scholar
[9] Cartan, É., Les sous-groupes des groupes continus de transformations. In: OEuvres Complètes Part. II, Vol. 2, Gauthier–Villars, Paris, 1953, pp. 719–856.Google Scholar
[10] Cartan, É., Sur la structure des groupes infinis de transformations. In: OEuvres Complètes; Part. II, Vol. 2, Gauthier–Villars, Paris, 1953, pp. 571–714.Google Scholar
[11] Cartan, É., La structure des groupes infinis. In: OEuvres complètes; part. II, vol. 2, Gauthier–Villars, Paris, 1953, pp. 1335–1384.Google Scholar
[12] Cheh, J., Symmetry pseudogroups of differential equations, Ph.D. thesis, University of Minnesota, 2005.Google Scholar
[13] Cheh, J., Olver, P. J., and Pohjanpelto, J., Maurer–Cartan equations for Lie symmetry pseudogroups of differential equations. J. Math. Phys. 46(2005), no. 2, 023504.Google Scholar
[14] Cheh, J., Olver, P. J., and Pohjanpelto, J., Algorithms for differential invariants of symmetry groups of differential equations. Found. Comput. Math. 8(2008), no. 4, 501–532.Google Scholar
[15] Chern, S. S., and J.K. Moser, Real hypersurfaces in complex manifolds. Acta Math. 133(1974), 219–227.Google Scholar
[16] David, D., Kamran, N., Levi, D., and P.Winternitz, Subalgebras of loop algebras and symmetries of the Kadomtsev–Petviashivili equation. Phys. Rev. Lett. 55(1985), no. 20, 2111–2113.Google Scholar
[17] Di, P. Francesco, Mathieu, P., and Sénéchal, D., Conformal field theory. Graduate Texts in Contemporary Physics, Springer–Verlag, New York, 1997 Google Scholar
[18] Ehresmann, C., Introduction à la théorie des structures infinitésimales et des pseudo-groupes de Lie. In: Géometrie Différentielle; Centre National de la Recherche Scientifique, Paris, 1953, pp. 97–110.Google Scholar
[19] Evtushik, L.E., Pseudo-groups of transformations of geometrical-differential structures and their invariants. J. Math. Sci. 94(1999), no. 5, 1643–1684.Google Scholar
[20] Fefferman, C. and Graham, C.R., Conformal invariants. In: The mathematical heritage ofÉlie Cartan; Numero Hors Série, Astérisque, Lyon, 1984, pp. 95–116.Google Scholar
[21] Fels, M. and Olver, P.J., Moving coframes. II. Regularization and theoretical foundations. Acta Appl. Math. 55(1999), no. 2, 127–208.Google Scholar
[22] Fuks, D.B., Cohomology of infinite-dimensional Lie algebras. Contemporary Soviet Mathematics, Consultants Bureau, New York, 1986.Google Scholar
[23] Gromov, M., Partial Differential Relations. Springer–Verlag, New York, 1986.Google Scholar
[24] Guggenheimer, H.W., Differential geometry. McGraw–Hill, New York, 1963.Google Scholar
[25] Hubert, E., The Aida Maple package, 2006, available at http://www.inria.fr/cafe/Evelyne.Hubert/aida. Google Scholar
[26] Itskov, V., Orbit reduction of contact ideals. The geometrical study of differential equations. Contemp. Math. 285(2001) 171–181.Google Scholar
[27] Itskov, V., Orbit reduction of exterior differential systems. Ph.D. thesis, University of Minnesota, 2002.Google Scholar
[28] Johnson, H. H., Classical differential invariants and applications to partial differential equations. Math. Ann. 148(1962), 308–329.Google Scholar
[29] Kogan, I. A., Inductive construction of moving frames. Contemp. Math. 285(2001), 157–170.Google Scholar
[30] Kogan, I. A., and Olver, P.J., The invariant variational bicomplex. Contemp. Math. 285(2001), 131–144.Google Scholar
[31] Kogan, I. A., and Olver, P.J., Invariant Euler-Lagrange equations and the invariant variational bicomplex. Acta Appl. Math. 76(2003), no. 2, 137–193.Google Scholar
[32] Kumpera, A., Invariants différentiels d’un pseudogroupe de Lie. J. Differential Geometry 10(1975), no. 2, 289–345.Google Scholar
[33] Kuranishi, M., On the local theory of continuous infinite pseudo groups I. Nagoya Math. J. 15(1959), 225–260.Google Scholar
[34] Kuranishi, M., On the local theory of continuous infinite pseudo groups II. Nagoya Math. J. 19(1961), 55–91.Google Scholar
[35] Lie, S., Gruppenregister. In: Gesammelte Abhandlungen 5,B.G. Teubner, Leipzig, 1924, pp. 767–773.Google Scholar
[36] Lie, S., Zur allgemeinen Theorie der partiellen Differentialgleichungen beliebeger Ordnung. Leipz. Berich. 47(1895), 53–128; also In: Gesammelte Abhandlungen 4, B.G. Teubner, Leipzig, 1929, pp. 320–384.Google Scholar
[37] Lisle, I.G., and Reid, G.J., Geometry and structure of Lie pseudogroups from infinitesimal defining systems, J. Symb. Comp. 26 (1998), 355–379 .Google Scholar
[38] Lisle, I. G. and Reid, G.J., Cartan structure of infinite Lie pseudogroups. In: Geometric Approaches to Differential Equations, Austral. Math. Soc. Lect. Ser. 15, Cambridge University Press, Cambridge, 2000, pp. 116–145.Google Scholar
[39] Lisle, I. G. and Reid, G.J., Symmetry classification using non-commutative invariant differential operators. Found. Comput. Math. 6(2006), no. 3, 353–386.Google Scholar
[40] Mackenzie, K. C.H., General theory of Lie groupoids and Lie algebroids. LondonMathematical Society Lecture Notes 213, Cambridge University Press, Cambridge, 2005.Google Scholar
[41] Mansfield, E. L., Algorithms for symmetric differential systems. Found. Comput. Math. 1(2001), no. 4, 335–383.Google Scholar
[42] Martina, L., Sheftel, M.B., and Winternitz, P., Group foliation and non-invariant solutions of the heavenly equation. J. Phys. A 34(2001), no. 43, 9243–9263.Google Scholar
[43] McLachlan, R.I. and Quispel, G.R.W.,What kinds of dynamics are there? Lie pseudogroups, dynamical systems and geometric integration. Nonlinearity 14(2001), no. 6, 1689–1705.Google Scholar
[44] Morozov, O.I., Moving coframes and symmetries of differential equations. J. Phys. A 35(2002), no. 12, 2965–2977.Google Scholar
[45] Morozov, O.I., Structure of symmetry groups via Cartan's method: survey of four approaches. SIGMA Symmetry Integrability Geom. Methods Appl. 1(2005), Paper 006.Google Scholar
[46] Nutku, Y. and M.B. Sheftel, Differential invariants and group foliation for the complex Monge–Ampère equation. J. Phys. A 34(2001), 137–156.Google Scholar
[47] Olver, P.J., Applications of Lie groups to differential equations. Second Edition, Graduate Texts in Mathematics 107, Springer–Verlag, New York, 1993.Google Scholar
[48] Olver, P.J., Equivalence, invariants, and symmetry. Cambridge University Press, Cambridge, 1995.Google Scholar
[49] Olver, P.J., Moving frames and singularities of prolonged group actions. SelectaMath. 6(2000), no. 1, 41–77.Google Scholar
[50] Olver, P.J., Joint invariant signatures. Found. Comput. Math. 1(2001), no. 1, 3–67.Google Scholar
[51] Olver, P.J., Geometric foundations of numerical algorithms and symmetry. Appl. Algebra Engrg. Comm. Comput. 11(2001), no. 5, 417–436.Google Scholar
[52] Olver, P.J. and Pohjanpelto, J., Regularity of pseudogroup orbits. In: Symmetry and perturbation theory,World Scientific Publ., Hackensack, NJ, 2005, pp. 244–254.Google Scholar
[53] Olver, P.J. and Pohjanpelto, J., Maurer–Cartan forms and the structure of Lie pseudo-groups. SelectaMath. 11(2005), no. 1, 99–126.Google Scholar
[54] Olver, P.J. and Pohjanpelto, J., emphDifferential invariant algebras of Lie pseudo-groups. preprint, University of Minnesota, 2007, http://www.math.umn.edu/˜ olver/mf /psc.pdf. Google Scholar
[55] Ovsiannikov, L.V., Group Analysis of Differential Equations. Academic Press, New York, 1982.Google Scholar
[56] Pommaret, J.-F., Systems of partial differential equations and Lie pseudogroups. Mathematics and its Applications 14, Gordon and Breach Science Publishers, New York, 1978.Google Scholar
[57] Reid, G.J., Finding abstract Lie symmetry algebras of differential equations without integating determining equations. European J. Appl. Math. 2(1991), no. 4, 319–340.Google Scholar
[58] Reid, G.J. and Wittkopf, A.D., Determination of maximal symmetry groups of classes of differential equations. Proceedings of the 2000 International Symposium on Symbolic and Algebraic Computation, ACM, New York, 2000, pp. 272–280.Google Scholar
[59] Salmon, R., Lectures on geophysical fluid dynamics. Oxford University Press, New York, 1998.Google Scholar
[60] Sussmann, H.J., Orbits of families of vector fields and integrability of distributions. Trans. Amer. Math. Soc. 180(1973), 171–188.Google Scholar
[61] Tsujishita, T., On variational bicomplexes associated to differential equations. Osaka J. Math. 19(1982), no. 2, 311–363.Google Scholar
[62] Vessiot, E., Sur l’intégration des systèmes différentiels qui admettent des groupes continues de transformations. Acta. Math. 28(1904), no. 1, 307–349.Google Scholar
[63] Wells, R.O., Jr., Differential analysis on complex manifolds. Graduate Texts inMathematics 65, Springer–Verlag, New York, 1980.Google Scholar