The Lie group in infinite dimension. (English) Zbl 1223.22018

Tryhuk, V.; Chrastinová, V.; Dlouhý, O.

Abstr. Appl. Anal. 2011, Article ID 919538, 35 p. (2011).

Summary: A Lie group acting on finite-dimensional space is generated by its infinitesimal transformations and conversely, any Lie algebra of vector fields in finite dimension generates a Lie group (the first fundamental theorem). This classical result is adjusted for the infinite-dimensional case. We prove that the (local, \(C^{\infty}\) smooth) action of a Lie group on infinite-dimensional space (a manifold modelled on \(\mathbb R^{\infty}\)) may be regarded as a limit of finite-dimensional approximations and the corresponding Lie algebra of vector fields may be characterized by certain finiteness requirements. The result is applied to the theory of generalized (or higher-order) infinitesimal symmetries of differential equations.

Reviewer: Liviu Popescu (Craiova)

MSC:

22E65	Infinite-dimensional Lie groups and their Lie algebras: general properties
34C14	Symmetries, invariants of ordinary differential equations

Keywords:

Lie groups

References:

[1]	R. L. Anderson and N. H. Ibragimov, Lie-Bäcklund Transformations in Applications, vol. 1 of SIAM Studies in Applied Mathematics, SIAM, Philadelphia, Pa, USA, 1979. · Zbl 1063.35528 · doi:10.1088/0031-8949/20/3-4/024
[2]	I. M. Anderson, N. Kamran, and P. J. Olver, “Internal, external, and generalized symmetries,” Advances in Mathematics, vol. 100, no. 1, pp. 53-100, 1993. · Zbl 0809.58044 · doi:10.1006/aima.1993.1029
[3]	N. Kamran, Selected Topics in the Geometrical Study of Differential Equations, vol. 96 of CBMS Regional Conference Series in Mathematics, American Mathematical Society, Washington, DC, USA, 2002. · Zbl 1134.58001
[4]	P. J. Olver, Applications of Lie Groups to Differential Equations, vol. 107 of Graduate Texts in Mathematics, Springer, New York, NY, USA, 1986. · Zbl 0588.22001
[5]	P. J. Olver, Equivalence, Invariants, and Symmetry, Cambridge University Press, Cambridge, UK, 1995. · Zbl 0837.58001
[6]	A. M. Vinogradov, I. S. Krasil’shchik, and V. V. Lychagin, Introduction to the Theory of Nonlinear Differential Equations, “Nauka”, Moscow, Russia, 1986. · Zbl 0592.35002
[7]	A. M. Vinogradov, Cohomological Analysis of Partial Differential Equations and Secondary Calculus, vol. 204 of Translations of Mathematical Monographs, American Mathematical Society, Providence, RI, USA, 2001. · Zbl 1152.58308
[8]	J. Chrastina, “From elementary algebra to Bäcklund transformations,” Czechoslovak Mathematical Journal, vol. 40(115), no. 2, pp. 239-257, 1990. · Zbl 0726.58041
[9]	J. Chrastina, The Formal Theory of Differential Equations, vol. 6 of Folia Facultatis Scientiarium Naturalium Universitatis Masarykianae Brunensis. Mathematica, Masaryk University, Brno, Czech Republic, 1998. · Zbl 0906.35002
[10]	V. Tryhuk and V. Chrastinová, “Automorphisms of curves,” Journal of Nonlinear Mathematical Physics, vol. 16, no. 3, pp. 259-281, 2009. · Zbl 1184.58003 · doi:10.1142/S1402925109000224
[11]	V. Chrastinová and V. Tryhuk, “Generalized contact transformations,” Journal of Applied Mathematics, Statistics and Informatics, vol. 3, no. 1, pp. 47-62, 2007.
[12]	V. Tryhuk and V. Chrastinová, “On the mapping of jet spaces,” Journal of Nonlinear Mathematical Physics, vol. 17, no. 3, pp. 293-310, 2010. · Zbl 1207.58004 · doi:10.1142/S140292511000091X
[13]	V. Chrastinová and V. Tryhuk, “Automorphisms of submanifolds,” Advances in Difference Equations, vol. 2010, Article ID 202731, 26 pages, 2010. · Zbl 1197.53009 · doi:10.1155/2010/202731
[14]	N. H. Ibragimov, Transformation Groups Applied to Mathematical Physics, Mathematics and Its Applications, D. Reidel, Dordrecht, The Netherlands, 1985. · Zbl 0558.53040
[15]	J. Mike\vs, A. Van, and I. Hinterleitner, Geodesic Mappings and Some Generalizations, Palacký University Olomouc, Faculty of Science, Olomouc, Czech Republic, 2009.
[16]	V. N. Chetverikov, “On the structure of integrable \?-fields,” Differential Geometry and Its Applications, vol. 1, no. 4, pp. 309-325, 1991. · Zbl 0782.58054 · doi:10.1016/0926-2245(91)90011-W
[17]	L. P. Eisenhart, Continuous Groups of Transformations, Princeton University Press, 1933. · Zbl 0008.10801
[18]	É. Cartan, Les systèmes extérieurs et leurs applications géométriques, Hermann, Paris, France, 1946. · Zbl 0211.12701
[19]	É. Cartan, Les systèmes extérieurs et leurs applications géométriques, Actualits scientifiques et industrielles. No. 994, Hermann, Paris, France, 2nd edition, 1971. · Zbl 0211.12701
[20]	V. Guillemin and S. Sternberg, Geometric Asymptotics, Mathematical Surveys, No. 1, American Mathematical Society, Providence, RI, USA, 1977. · Zbl 0364.53011
[21]	A. Kushner, V. Lychagin, and V. Rubtsov, Contact Geometry and Non-Linear Differential Equations, vol. 101 of Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge, UK, 2007. · Zbl 1122.53044
[22]	É. Goursat, Le\ccons sur l’intégration des équations aux dérivées partielles premier ordre, Zweite Auflage, Paris, France, 1891. · JFM 23.0401.02
[23]	É. Goursat, Le\ccons sur l’intégration des équations aux dérivées partielles premier ordre, J. Hermann, Paris, France, 1920. · JFM 48.0537.05
[24]	J. Chrastina, “What the differential equations should be,” in Proceedings of the Conference on Differential Geometry and Its Applications, Part 2, pp. 41-50, Univ. J. E. Purkyn\ve, Brno, Czech Republic, 1984. · Zbl 0571.58030
[25]	J. Chrastina, “On formal theory of differential equations. I,” \vCasopis Pro P Matematiky, vol. 111, no. 4, pp. 353-383, 1986. · Zbl 0638.35008
[26]	J. Chrastina, “On formal theory of differential equations. II,” \vCasopis Pro P Matematiky, vol. 114, no. 1, pp. 60-105, 1989. · Zbl 0703.34001
[27]	J. Chrastina, “On formal theory of differential equations. III,” Mathematica Bohemica, vol. 116, no. 1, pp. 60-90, 1991. · Zbl 0728.58041