Abstract
This overview paper is intended as a quick introduction to Lie algebras of vector fields. Originally introduced in the late nineteenth century by Sophus Lie to capture symmetries of ordinary differential equations, these algebras, or infinitesimal groups, are a recurring theme in twentieth-century research on Lie algebras. I will focus on so-called transitive or even primitive Lie algebras, and explain their theory due to Lie, Morozov, Dynkin, Guillemin, Sternberg, Blattner, and others. This paper gives just one, subjective overview of the subject, without trying to be exhaustive.
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References
Blattner R.J.: Induced and produced representations of Lie algebras. Trans. Am. Math. Soc. 144, 457–474 (1969)
Brion, M.: Variétés sphériques. Notes of the S.M.F. session Opérations Hamiltoniennes et opérations de groupes algébriques. Grenoble 1997. http://www-fourier.ujf-grenoble.fr/~mbrion/notes.html
Brion M., Luna D., Vust T.: Espaces homogènes sphériques. Invent. Math. 84, 617–632 (1986)
Draisma, J.: \({{\tt blattner}}\), an implementation of the realisation formula. \({{\tt GAP}}\) -code: http://www.win.tue.nl/~jdraisma/index.php?location=programs
Draisma, J.: \({{\tt maximal\_subalgebras}}\), an implementation of Dynkin’s classification. \({{\tt LiE}}\) -code: http://www.win.tue.nl/~jdraisma/index.php?location=programs
Draisma J.: Recognizing the symmetry type of o.d.e.’s. J. Pure Appl. Algebra 164(1–2), 109–128 (2001)
Draisma, J.: On a conjecture of Sophus Lie. In: Differential Equations and the Stokes Phenomenon. Proceedings of a Workshop held at Groningen University from May 28–30, 2001. World Scientific, Singapore (2002)
Dynkin E.B.: Maximal subgroups of the classical groups. Am. Math. Soc. Transl. II. Ser. 6, 245–378 (1957)
Dynkin E.B.: Semisimple subalgebras of semisimple Lie algebras. Am. Math. Soc. Transl. II. Ser. 6, 111–244 (1957)
The GAP Group.: GAP—Groups, Algorithms, and Programming, Version 4.3. http://www.gap-system.org (2002)
González-López A., Kamran N., Olver P.J.: Lie algebras of first order differential operators in two complex variables. Am. J. Math. 114(6), 1163–1185 (1992)
González-López A., Kamran N., Olver P.J.: Lie algebras of vector fields in the real plane. Proc. Lond. Math. Soc. III. Ser. 64(2), 339–368 (1992)
Guillemin V.W.: Infinite dimensional primitive Lie algebras. J. Differ. Geom. 4, 257–282 (1970)
Guillemin V.W., Sternberg S.: An algebraic model of transitive differential geometry. Bull. Am. Math. Soc. 70, 16–47 (1964)
Helgason, S.: Differential geometry and symmetric spaces. In: Pure and Applied Mathematics, vol. XII. Academic Press, New York-London (1962)
Jacobson, N.: Lie Algebras. Dover, New York (1969, reprint)
Karpelevich, F.I.: Über nicht-halbeinfache maximale Teilalgebren halbeinfacher Liescher Algebren. Dokl. Akad. Nauk SSSR 76, 775–778 (1951)
van Leeuwen, M.A.A., Cohen, A.M., Lisser, B.: LiE: A Package for Lie Group Computations. Amsterdam (1992). http://wwwmathlabo.univ-poitiers.fr/~maavl/LiE/
Lie, S.: Vorlesungen über Differentialgleichungen mit bekannten infinitesimalen Transformationen. Bearbeitet und herausgegeben von G. Scheffers. B.G. Teubner, Leipzig (1891)
Lie, S.: Gruppenregister. Gesammelte Abhandlungen, vol. 5, pp. 767–773. B.G. Teubner, Leipzig (1924)
Lie, S., Engel, F.: Transformationsgruppen, vol. 3. B.G. Teubner, Leipzig (1893)
Morozov V.V.: Sur les groupes primitifs. Rec. Math. Moscou n. Ser. 5, 355–390 (1939)
Olver, P.J.: Applications of Lie Groups to Differential Equations, 2nd edn. Graduate Texts in Mathematics, vol. 107. Springer, New York (1986)
Oudshoorn W.R., van der Put M.: Lie symmetries and differential Galois groups of linear equations. Math. Comput. 71(237), 349–361 (2002)
Reid G.J.: Algorithms for reducing a system of pdes to standard form, determining the dimension of its solution space and calculating its Taylor series solution. J. Appl. Math. 2(4), 293–318 (1991)
Reid G.J.: Finding abstract Lie symmetry algebras of differential equations without integrating determining equations. Eur. J. Appl. Math. 2(4), 319–340 (1991)
Richter D.A.: \({{\mathbb Z}}\) -gradations of Lie algebras and infinitesimal generators. J. Lie Theory 9(1), 113–123 (1999)
Robertz, D.: Janet bases and applications. In: Rosenkranz, M., et al. (eds.) Gröbner Bases in Symbolic Analysis. Radon Series on Computational and Applied Mathematics, vol. 2, pp. 139–168. Walter de Gruyter, Berlin (2007). Based on talks delivered at the special semester on Gröbner bases and related methods, Linz, Austria, May 2006
Schwarz, F.: Janet bases for symmetry groups. In: Buchberger, B., Winkler, F. (eds.) Gröbner Bases and Applications, pp. 221–234. Cambridge University Press (1998)
Sharpe, R.W.: Differential geometry: Cartan’s generalization of Klein’s Erlangen program. Graduate Texts in Mathematics, vol. 166. Springer, Berlin (1997)
Singer I.M., Sternberg S.: The infinite groups of Lie and Cartan. I: The transitive groups. J. Anal. Math. 15, 1–114 (1965)
Strade, H.: Simple Lie algebras over fields of positive characteristic. I: Structure theory. de Gruyter Expositions in Mathematics, vol. 38. de Gruyter, Berlin (2004)
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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Draisma, J. Transitive Lie Algebras of Vector Fields: An Overview. Qual. Theory Dyn. Syst. 11, 39–60 (2012). https://doi.org/10.1007/s12346-011-0062-9
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DOI: https://doi.org/10.1007/s12346-011-0062-9