On the Local Theory of Continuous Infinite Pseudo Groups I*

Masatake Kuranishi

doi:10.1017/S0027763000006747

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The local theory of continuous (infinite) pseudo-groups of transformations was originated by S. Lie, and developed by himself, F. Engel, E. Vessiot, E. Cartan, etc. In the beginning, the definition was not clear and we can find several different definitions in the papers of pioneers. In 1902, E. Cartan introduced a definition using his theory of exterior differential systems and made an extensive study in his series of papers [1], [2], and [31 The writer will adopt his definition in this series of papers. A continuous pseudo-group of transformations is, roughly speaking, a collection of real (or complex) analytic homeo-morphisms of domains in a real (or complex) euclidean space, which is closed under the operations of composition and inverse, and which forms the general solutions of a system of partial differential equations.

Footnotes

A part of this series of papers was done at the Institute for Advanced Study, Princeton, in 1955-56, and at the University of Chicago in 1956-57. At the University of Chicago, the writer was partially supported by the United States Air Force under Contract No. AF 18 (600)-1383 monitered by the Office of Scientific Research.

References

[1] Cartan, E., Sur la structure des groupes infinis de transformations, Ann. Ec. Norm. Sup. 21 (1904), pp. 153–206 and 22 (1905), pp. 219–308.Google Scholar

[2] Cartan, E., Les sous-groupes des groupes continus de transformations, ibid. 25 (1908), pp. 57–194.Google Scholar

[3] Cartan, E., Les groupes de transformations continus, infinis, simples, ibid. 26 (1909), pp. 93–161.Google Scholar

[4] Cartan, E., Seminaire Ec. Norm. Sup. 1950–51 (multilithed).Google Scholar

Crossref Citations

This article has been cited by the following publications. This list is generated based on data provided by Crossref.

Johnson, H. H. 1963. Lie groups associated with pseudo groups. Proceedings of the American Mathematical Society, Vol. 14, Issue. 1, p. 84.

Guillemin, Victor W. and Sternberg, Shlomo 1964. An algebraic model of transitive differential geometry. Bulletin of the American Mathematical Society, Vol. 70, Issue. 1, p. 16.

Johnson, H. H. 1964. Bracket and exponential for a new type of vector field. Proceedings of the American Mathematical Society, Vol. 15, Issue. 3, p. 432.

D’atri, Joseph E. 1965. Homomorphisms of Continuous Pseudogroups. Nagoya Mathematical Journal, Vol. 25, Issue. , p. 143.

Hermann, Robert 1965. E. Cartan's geometric theory of partial differential equations. Advances in Mathematics, Vol. 1, Issue. 3, p. 265.

Singer, I. M. and Sternberg, Shlomo 1965. The infinite groups of Lie and Cartan Part I, (The transitive groups). Journal d'Analyse Mathématique, Vol. 15, Issue. 1, p. 1.

Weisfeld, Morris 1968. Tensor products of 𝐹-vector spaces. Proceedings of the American Mathematical Society, Vol. 19, Issue. 6, p. 1373.

Guillemin, Victor 1968. A Jordan-Hölder decomposition for a certain class of infinite dimensional Lie algebras. Journal of Differential Geometry, Vol. 2, Issue. 3,

Ibragimov, Nail H and Anderson, Robert L 1977. Lie-Bäcklund tangent transformations. Journal of Mathematical Analysis and Applications, Vol. 59, Issue. 1, p. 145.

Boyer, Charles P. and Plebański, Jerzy F. 1978. General relativity and G-structures I. general theory and algebraically degenerate spaces. Reports on Mathematical Physics, Vol. 14, Issue. 1, p. 111.

Hazewinkel, Michiel 1991. Encyclopaedia of Mathematics. p. 55.

Strunkov, S. P. Parshin, A. N. Sprindzhuk, V. G. Dolgachev, I. V. Gorchakov, Yu. M. Malyshev, A. V. Rakhmanov, E. A. Lavrik, A. F. Rozov, N. Kh. Solomentsev, E. D. Shtern, A. I. Fedoryuk, M. V. Kim, G. D. Brychkov, Yu. A. Prudnikov, A. P. Babich, V. M. Gulin, A. V. Nikulin, M. S. Sokolov, D. D. Popov, V. L. Arkhangelskiĭ, A. V. Lumiste, Ü. Alekseevskiĭ, D. V. Kotov, V. E. Kuptsov, L. P. Il’in, V. A. Bogdanov, Yu. S. Davidenko, D. F. Goluzina, E. G. Korbut, A. A. Popov, V. I. Starzhinskiĭ, V. M. Alimov, Sh. A. Proknorov, A. V. Kudryavtsev, L. D. Prokhorov, A. V. Afanas’ev, V. V. Ikramov, Kh. D. Plisko, V. E. Kopytov, V. M. Skornyakov, L. A. Davydov, Yu. M. Sidorov, L. A. Modenov, P. S. Parkhomenko, A. S. Nechaev, V. I. Bogatyĭ, S. A. Kol’tsov, P. P. Krechet, V. G. Mints, G. E. Voĭtsekhovskiĭ, M. I. Bukhshtab, A. A. Karmanov, V. G. Il’yashenko, Yu. S. Kuz’min, L. V. Arkhangel’skiĭ, A. V. Sazonov, V. V. Stepanov, S. A. Zalgaller, V. A. Ovseevich, A. I. Millionshchikov, V. M. Shevrin, L. N. Zhurbenko, I. G. Sklyarenko, E. G. Tarakanov, V. E. Mikheev, V. M. Kaluzhnin, L. A. Supruneko, D. A. Minlos, R. A. Khvedelidze, B. V. Suprunenko, U. A. Lukashenko, T. P. Reĭzin’, L. E. Bogolyubov, N. N. Onishchik, A. L. Ivanov, A. B. Sobolev, V. I. Kvasnikov, I. A. Latyshev, V. I. Danilov, V. I. Shokurov, V. V. Chernavskiĭ, A. V. Golubov, B. I. Zarkhin, Yu. G. Lizorkin, P. I. van de Craats, J. Lenskiĭ, V. S. Sabitov, I. Kh. Fomenko, A. T. Korneva, L. A. Anosov, D. V. Kulikov, Vik. S. Rumyantsev, V. V. Rudyak, Yu. B. Soldatov, A. P. D’yakonov, E. G. Sevast’yanov, B. A. Volkov, I. I. Lomonosov, V. I. Shul’man, V. S. Kulikov, V. S. Shmel’kin, A. L. Merzlyakov, Yu. I. Matveev, S. V. Vinberg, E. B. Markushevich, A. I. Korneĭchuk, N. P. Motornyĭ, V. P. Kharshiladze, A. F. Shtan’ko, M. A. Kurzhanskiĭ, A. B. Nikol’skiĭ, N. K. Pavlov, B. S. Malygin, S. N. Postnikov, M. M. Aleksandrov, P. S. Dolzhenko, E. P. Konyushkov, A. A. Vorob’ev, N. N. Lyapunov, A. N. Schul’man, V. S. Sobolev, S. K. Kudryavtsev, B. V. Maslov, S. Yu. Rozenberg, G. Salomaa, A. Adyan, S. I. Solomentsev, E. V. Prilenko, A. I. Fabrikant, V. I. Fofanova, T. S. Kokorin, A. I. Dragalin, A. G. Grishin, V. N. Shiryaev, A. N. Chirka, E. M. Markov, V. T. Ivanova, O. A. Zhevlakov, K. A. Bredikhin, B. M. Artemov, S. N. Chudakov, N. G. Shikin, E. V. Shchekut’ev, A. R. Voskresenskiĭ, V. E. Remeslennikov, V. N. Bokut’, L. A. Dezin, A. A. Smirnov, D. M. Ereshko, F. I. Fedorov, V. V. Dushskiĭ, V. A. Palamodov, V. P. Kolmogorov, A. N. Prokhorov, Yu. V. Petrov, V. V. Yurinskiĭ, V. V. Ershov, A. P. Bol’shev, L. N. Kreĭn, S. G. Tsalenko, M. Sh. Kas’yanov, V. N. Levner, E. V. Vaĭnikko, G. M. Ponomarev, V. I. Norden, A. P. Shirokov, A. P. Govorov, V. E. Kirillov, A. A. El’kin, A. G. Mazurov, V. D. Kuzichev, A. S. Mal’tsev, A. A. Polyakov, V. Z. Arknangel’skiĭ, A. V. Shubin, M. A. and Petrosyan, L. A. 1995. Encyclopaedia of Mathematics. p. 307.

Kamran, Niky and Robart, Thierry 1997. Perspectives sur la théorie des pseudogroupes de transformations analytiques de type infini. Journal of Geometry and Physics, Vol. 23, Issue. 3-4, p. 308.

Olver, Peter J. 2003. Moving frames. Journal of Symbolic Computation, Vol. 36, Issue. 3-4, p. 501.

Olver, Peter J. 2005. Computer Algebra and Geometric Algebra with Applications. Vol. 3519, Issue. , p. 105.

Cheh, Jeongoo Olver, Peter J. and Pohjanpelto, Juha 2005. Maurer–Cartan equations for Lie symmetry pseudogroups of differential equations. Journal of Mathematical Physics, Vol. 46, Issue. 2,

Neeb, Karl-Hermann 2006. Towards a Lie theory of locally convex groups. Japanese Journal of Mathematics, Vol. 1, Issue. 2, p. 291.

Pohjanpelto, J. 2008. Reduction of exterior differential systems with infinite dimensional symmetry groups. BIT Numerical Mathematics, Vol. 48, Issue. 2, p. 337.

Olver, Peter J. and Pohjanpelto, Juha 2008. Symmetries and Overdetermined Systems of Partial Differential Equations. Vol. 144, Issue. , p. 127.

Kruglikov, Boris and Lychagin, Valentin 2008. Handbook of Global Analysis. p. 725.

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