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The early proofs of the theorem of Campbell, Baker, Hausdorff, and Dynkin. (English) Zbl 1245.01002

“One of the most versatile results originating from the early theory of groups of transformations \((\dots)\) states that, in the algebra of formal series in two non-commuting indeterminates \(x\) and \(y\), the series naturally associated to \(\log(e^xe^y)\) is a series of Lie polynomials in \(x\) and \(y\).” The result, sometimes called the Exponential Theorem, has found important applications to physics, group theory, linear PDEs, Lie groups and Lie algebras, numerical analysis. The aim of this paper is to recall nearly forgotten contributions given by the forerunners of the Theorem, in particular by Italian mathematician E. Pascal, whose work had been “of decisive importance”. With the intention to ease the access to those contributions the authors furnish the mathematical details and offer an explanation in modern language.
The paper is divided into sections. After an introduction there follow sections on contributions by F. Schur, J. E. Campbell, H. Poincaré, E. Pascal, H. F. Baker, F. Hausdorff, and E. B. Dynkin, with the final one on commentaries by F. Hausdorff and N. Bourbaki which, in the authors’ opinion, were rather “cold” and thus played a major role in subsequent neglecting those early contributions which still seem to be of some value. – The paper is completed with an extensive bibliography.

MSC:

01-02 Research exposition (monographs, survey articles) pertaining to history and biography
01A55 History of mathematics in the 19th century
01A60 History of mathematics in the 20th century
22-03 History of topological groups
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[1] Abbaspour H., Moskowitz M. (2007) Basic Lie theory. World Scientific, Hackensack, NJ · Zbl 1137.17001
[2] Alexandrov, P.S., et al. 1979. Die Hilbertschen Probleme. Ostwalds Klassiker der exakten Wissenschaften. Leipzig: Akademische Verlagsgesellschaft Geest & Portig K.-G.
[3] Baker H.F. (1901) On the exponential theorem for a simply transitive continuous group, and the calculation of the finite equations from the constants of structure. Proceedings of the London Mathematical Society 34: 91–127 · JFM 32.0159.01 · doi:10.1112/plms/s1-34.1.91
[4] Baker H.F. (1902) Further applications of matrix notation to integration problems. Proceedings of the London Mathematical Society 34: 347–360 · JFM 33.0352.03
[5] Baker H.F. (1903) On the calculation of the finite equations of a continuous group. Proceedings of the London Mathematical Society 35: 332–333 · JFM 34.0181.03
[6] Baker H.F. (1905) Alternants and continuous groups. Proceedings of the London Mathematical Society (2) 3: 24–47 · JFM 36.0225.01 · doi:10.1112/plms/s2-3.1.24
[7] Białynicki-Birula I., Mielnik B., Plebański J. (1969) Explicit solution of the continuous Baker–Campbell–Hausdorff problem and a new expression for the phase operator. Annals of Physics 51: 187–200 · Zbl 0172.56601 · doi:10.1016/0003-4916(69)90351-0
[8] Biermann K.-R. (1988) Die Mathematik und ihre Dozenten an der Berliner Universität 1810–1933. Stationen auf dem Wege eines mathematischen Zentrums von Weltgeltung, 2nd improved edn. Akademie-Verlag, Berlin · Zbl 0261.01016
[9] Birkhoff G. (1936) Continuous groups and linear spaces. Recueil Mathématique [Matematicheskii Sbornik] N.S. 1(43) 5: 635–642 · JFM 62.1231.03
[10] Birkhoff G. (1938) Analytic groups. Transactions of the American Mathematical Society 43: 61–101 · JFM 64.1092.02 · doi:10.1090/S0002-9947-1938-1501934-4
[11] Blanes S., Casas F. (2004) On the convergence and optimization of the Baker-Campbell-Hausdorff formula. Linear Algebra and its Applications 378: 135–158 · Zbl 1054.17005 · doi:10.1016/j.laa.2003.09.010
[12] Blanes S., Casas F., Oteo J.A., Ros J. (1998) Magnus and Fer expansions for matrix differential equations: The convergence problem. Journal of Physics A: Mathematical and General 31: 259–268 · Zbl 0946.34014 · doi:10.1088/0305-4470/31/1/023
[13] Blanes S., Casas F., Oteo J.A., Ros J. (2009) The Magnus expansion and some of its applications. Physics Reports 470: 151–238 · doi:10.1016/j.physrep.2008.11.001
[14] Bonfiglioli, A., and R. Fulci. 2012. Topics in noncommutative algebra. The theorem of Campbell, Baker, Hausdorff and Dynkin. Lecture Notes in Mathematics 2034. Berlin: Springer-Verlag. · Zbl 1231.17001
[15] Borel, A. 2001. Essays in the history of Lie groups and algebraic groups. History of Mathematics 21. Providence/Cambridge: American Mathematical Society/London Mathematical Society. · Zbl 1087.01011
[16] Bose A. (1989) Dynkin’s method of computing the terms of the Baker–Campbell–Hausdorff series. Journal of Mathematical Physics 30: 2035–2037 · Zbl 0676.17004 · doi:10.1063/1.528242
[17] Boseck, H., Czichowski, G., and K.-P. Rudolph. (1981) Analysis on topological groups–general Lie theory. Teubner-Texte zur Mathematik 37. Leipzig: BSB B. G. Teubner Verlagsgesellschaft. · Zbl 0558.22012
[18] Bourbaki, N. 1972. Éléments de Mathématique. Fasc. XXXVII: Groupes et algèbres de Lie. Chap. II: Algèbres de Lie libres. Chap. III: Groupes de Lie. Actualités scientifiques et industrielles, 1349. Paris: Hermann. · Zbl 0244.22007
[19] Campbell J.E. (1897a) On a law of combination of operators bearing on the theory of continuous transformation groups. Proceedings of the London Mathematical Society 28: 381–390 · JFM 28.0321.01
[20] Campbell J.E. (1897b) Note on the theory of continuous groups. Bulletin of the American Mathematical Society 4: 407–408 · JFM 29.0120.03 · doi:10.1090/S0002-9904-1898-00524-4
[21] Campbell J.E. (1898) On a law of combination of operators (second paper). Proceedings of the London Mathematical Society 29: 14–32 · JFM 29.0324.01
[22] Campbell J.E. (1903) Introductory treatise on Lie’s theory of finite continuous transformation groups. Clarendon Press, Oxford · JFM 34.0390.04
[23] Cartier P. (1956) Demonstration algébrique de la formule de Hausdorff. Bulletin de la Société Mathématique de France 84: 241–249 · Zbl 0072.01605
[24] Casas F. (2007) Sufficient conditions for the convergence of the Magnus expansion. Journal of Physics A: Mathematical and Theoretical 40: 15001–15017 · Zbl 1131.34008 · doi:10.1088/1751-8113/40/50/006
[25] Casas F., Murua A. (2009) An efficient algorithm for computing the Baker–Campbell–Hausdorff series and some of its applications. Journal of Mathematical Physics 50: 033513–103351323 · Zbl 1202.17004 · doi:10.1063/1.3078418
[26] Cohen A. (1931) An introduction to the Lie theory of one-parameter groups. G.E. Stechert & Co., New York
[27] Czy\.z J. (1989) On Lie supergroups and superbundles defined via the Baker–Campbell–Hausdorff formula. Journal of Geometry and Physics 6: 595–626 · Zbl 0716.17005 · doi:10.1016/0393-0440(89)90028-4
[28] Czy\.z J. (1994) Paradoxes of measures and dimensions originating in Felix Hausdorff’s ideas. World Scientific Publishing Co. Inc., Singapore · Zbl 0819.54001
[29] Day J., So W., Thompson R.C. (1991) Some properties of the Campbell–Baker–Hausdorff series. Linear and Multilinear Algebra 29: 207–224 · Zbl 0728.22008 · doi:10.1080/03081089108818072
[30] Dieudonné, J.A. 1974. Treatise on analysis, Vol. IV. Pure and Applied Mathematics 10-IV. New York: Academic Press. · Zbl 0292.58001
[31] Dieudonné, J.A. 1982. A panorama of pure mathematics. As seen by N. Bourbaki. Pure and Applied Mathematics 97. New York: Academic Press. · Zbl 0482.00003
[32] Djoković D.Ž. (1975) An elementary proof of the Baker–Campbell–Hausdorff–Dynkin formula. Mathematische Zeitschrift 143: 209–211 · Zbl 0298.22010 · doi:10.1007/BF01214376
[33] Dragt A.J., Finn J.M. (1976) Lie series and invariant functions for analytic symplectic maps. Journal of Mathematical Physics 17: 2215–2217 · Zbl 0343.70011 · doi:10.1063/1.522868
[34] Duistermaat J.J., Kolk J.A.C. (2000) Lie groups. Springer, Universitext. Berlin · Zbl 0955.22001
[35] Dynkin E.B. (1947) Calculation of the coefficients in the Campbell–Hausdorff formula (Russian). Doklady Akademii Nauk SSSR (N. S.) 57: 323–326 · Zbl 0029.24507
[36] Dynkin E.B. (1949) On the representation by means of commutators of the series log (e x e y ) for noncommutative x and y (Russian). Matematicheskii Sbornik (N.S.) 25(67(1)): 155–162 · Zbl 0041.16102
[37] Dynkin E.B. (1950) Normed Lie algebras and analytic groups. Uspekhi Matematicheskikh Nauk 5:1(35): 135–186. · Zbl 0052.26202
[38] Dynkin, E.B. 2000. Selected papers of E. B. Dynkin with commentary. Eds. A.A. Yushkevich, G.M. Seitz, and A.L. Onishchik. Providence: American Mathematical Society. · Zbl 1056.01014
[39] Eichler M. (1968) A new proof of the Baker–Campbell–Hausdorff formula. Journal of the Mathematical Society of Japan 20: 23–25 · Zbl 0157.07601 · doi:10.2969/jmsj/02010023
[40] Eisenhart, L.P. 1933. Continuous groups of transformations. Princeton: University Press. (Reprinted 1961, New York: Dover Publications.) · JFM 59.0430.01
[41] Eriksen E. (1968) Properties of higher-order commutator products and the Baker–Hausdorff formula. Journal of Mathematical Physics 9: 790–796 · Zbl 0169.04703 · doi:10.1063/1.1664643
[42] Folland G.B. (1975) Subelliptic estimates and function spaces on nilpotent Lie groups. Arkiv for Matematik 13: 161–207 · Zbl 0312.35026 · doi:10.1007/BF02386204
[43] Folland, G.B., and E.M. Stein. 1982. Hardy spaces on homogeneous groups. Mathematical Notes 28. Princeton/Tokyo: Princeton University Press/University of Tokyo Press. · Zbl 0508.42025
[44] Friedrichs K.O. (1953) Mathematical aspects of the quantum theory of fields. V. Fields modified by linear homogeneous forces. Communications on Pure and Applied Mathematics 6: 1–72 · Zbl 0052.44504 · doi:10.1002/cpa.3160060101
[45] Gilmore R. (1974) Baker–Campbell–Hausdorff formulas. Journal of Mathematical Physics 15: 2090–2092 · Zbl 0288.17005 · doi:10.1063/1.1666587
[46] Glöckner H. (2002a) Algebras whose groups of units are Lie groups. Studia Mathematica 153: 147–177 · Zbl 1009.22021 · doi:10.4064/sm153-2-4
[47] Glöckner, H. 2002b. Infinite-dimensional Lie groups without completeness restrictions. In Geometry and analysis on finite and infinite-dimensional Lie Groups, vol. 55, ed. A. Strasburger, W. Wojtynski, J. Hilgert, and K.-H. Neeb, 43–59. Warsaw: Banach Center Publications. · Zbl 1020.58009
[48] Glöckner H. (2002c) Lie group structures on quotient groups and universal complexifications for infinite-dimensional Lie groups. Journal of Functional Analysis 194: 347–409 · Zbl 1022.22021 · doi:10.1006/jfan.2002.3942
[49] Glöckner H., Neeb K.-H. (2003) Banach–Lie quotients, enlargibility, and universal complexifications. Journal für die reine und angewandte Mathematik (Crelle’s Journal) 560: 1–28 · Zbl 1029.22029 · doi:10.1515/crll.2003.056
[50] Godement, R. 1982. Introduction à la théorie des groupes de Lie. Tome 2. Publications Mathématiques de l’Université Paris VII. Paris: Université de Paris VII, U.E.R. de Mathématiques.
[51] Gorbatsevich, V.V., A.L. Onishchik, and E.B. Vinberg. 1997. Foundations of Lie theory and Lie transformation groups. New York: Springer. · Zbl 0999.17500
[52] Gordina M. (2005) Hilbert–Schmidt groups as infinite-dimensional Lie groups and their Riemannian geometry. Journal of Functional Analysis 227: 245–272 · Zbl 1078.22015 · doi:10.1016/j.jfa.2005.05.011
[53] Hairer E., Lubich Ch., Wanner G. (2006) Geometric numerical integration. Structure-preserving algorithms for ordinary differential equations. Springer, New York · Zbl 1094.65125
[54] Hall, B.C. 2003. Lie groups, Lie algebras, and representations: an elementary introduction. Graduate texts in mathematics. New York: Springer.
[55] Hausdorff F. (1906) Die symbolische Exponentialformel in der Gruppentheorie. Berichte der Königlich-Sächsischen Gesellschaft der Wissenschaften zu Leipzig (Leipziger Berichte), Math. Phys. Cl. 58: 19–48 · JFM 37.0176.02
[56] Hausdorff, F. 2001. Gesammelte Werke. Band IV: Analysis, Algebra und Zahlentheorie. Herausgegeben von S.D. Chatterji, R. Remmert und W. Scharlau. (Collected works. Vol. IV: Analysis, algebra, number theory.). Eds. S.D. Chatterji, R. Remmert, and W. Scharlau. Berlin: Springer.
[57] Hausner, M., and J.T. Schwartz. 1968. Lie Groups. Lie algebras. Notes on mathematics and its applications. New York: Gordon and Breach. · Zbl 0192.35902
[58] Hawkins, Th. 1989. Line geometry, differential equations and the birth of Lie’s theory of groups. In The history of modern mathematics, vol. I, pp. 275–327. Boston: Academic Press.
[59] Hawkins Th. (1991) Jacobi and the birth of Lie’s theory of groups. Archive for History of Exact Sciences 42: 187–278 · Zbl 0767.01019 · doi:10.1007/BF00375135
[60] Hawkins, Th. 2000. Emergence of the Theory of Lie groups. An essay in the history of mathematics 1869–1926. Sources and studies in the history of mathematics and physical sciences. New York: Springer. · Zbl 0965.01001
[61] Hilgert J., Hofmann K.H. (1986) On Sophus Lie’s fundamental theorem. Journal of Functional Analysis 67: 239–319 · Zbl 0597.22005 · doi:10.1016/0022-1236(86)90028-5
[62] Hilgert J., Neeb K.-H. (1991) Lie-Gruppen und Lie-Algebren. Vieweg, Braunschweig · Zbl 0760.22005
[63] Hochschild G.P. (1965) The structure of Lie groups. San Holden-Day Inc., Francisco · Zbl 0131.02702
[64] Hofmann, K.H. 1972. Die Formel von Campbell, Hausdorff und Dynkin und die Definition Liescher Gruppen. In Theory of sets and topology, ed. W. Rinow, 251–264. Berlin: VEB Deutscher Verlag der Wissenschaften. · Zbl 0264.22008
[65] Hofmann, K.H. 1975. Théorie directe des groupes de Lie. I, II, III, IV. Séminaire Dubreil, Algèbre, tome 27, n. 1 (1973/1974), Exp. 1 (1–24), Exp. 2 (1–16), Exp. 3 (1–39), Exp. 4 (1–15).
[66] Hofmann K.H., Morris S.A. (2005) Sophus Lie’s third fundamental theorem and the adjoint functor theorem. Journal of Group Theory 8: 115–133 · Zbl 1061.22001 · doi:10.1515/jgth.2005.8.1.115
[67] Hofmann, K.H., and S.A. Morris. 2006. The structure of compact groups. A primer for the student–A handbook for the expert, 2nd revised edn., de Gruyter Studies in Mathematics 25. Berlin: Walter de Gruyter. · Zbl 1139.22001
[68] Hofmann, K.H., and S.A. Morris. 2007. The Lie theory of connected pro-Lie goups. A structure theory for pro-Lie algebras, pro-Lie groups, and connected locally compact groups. EMS tracts in mathematics 2. Zürich: European Mathematical Society. · Zbl 1153.22006
[69] Hofmann K.H., Neeb K.-H. (2009) Pro-Lie groups which are infinite-dimensional Lie groups. Mathematical Proceedings of the Cambridge Philosophical Society 146: 351–378 · Zbl 1165.22017 · doi:10.1017/S030500410800128X
[70] Hörmander L. (1967) Hypoelliptic second order differential equations. Acta Mathematica 119: 147–171 · Zbl 0156.10701 · doi:10.1007/BF02392081
[71] Ince, E.L. 1927. Ordinary differential equations. London: Longmans, Green & Co. (Reprinted 1956, New York: Dover Publications Inc.) · JFM 53.0399.07
[72] Iserles A., Munthe-Kaas H.Z., Nørsett S.P., Zanna A. (2000) Lie-group methods. Acta Numerica 9: 215–365 · Zbl 1064.65147 · doi:10.1017/S0962492900002154
[73] Iserles A., Nørsett S.P. (1999) On the solution of linear differential equations in Lie groups. Philosophical Transactions of the Royal Society A 357: 983–1019 · Zbl 0958.65080 · doi:10.1098/rsta.1999.0362
[74] Jacobson, N. 1962. Lie algebras. Interscience tracts in pure and applied mathematics 10. New York: Interscience Publishers/Wiley.
[75] Klarsfeld S., Oteo J.A. (1989a) Recursive generation of higher-order terms in the Magnus expansion. Physical Review A 39: 3270–3273 · Zbl 0725.58039 · doi:10.1103/PhysRevA.39.3270
[76] Klarsfeld S., Oteo J.A. (1989b) The Baker–Campbell–Hausdorff formula and the convergence of Magnus expansion. Journal of Physics A: Mathematical and General 22: 4565–4572 · Zbl 0725.58039 · doi:10.1088/0305-4470/22/21/018
[77] Kobayashi H., Hatano N., Suzuki M. (1998) Goldberg’s theorem and the Baker–Campbell–Hausdorff formula. Physica A 250: 535–548 · Zbl 0924.22006 · doi:10.1016/S0378-4371(97)00557-8
[78] Kolsrud M. (1993) Maximal reductions in the Baker–Hausdorff formula. Journal of Mathematical Physics 34: 270–286 · Zbl 0772.17004 · doi:10.1063/1.530381
[79] Kumar K. (1965) On expanding the exponential. Journal of Mathematical Physics 6: 1928–1934 · Zbl 0137.23904 · doi:10.1063/1.1704742
[80] Magnus W. (1950) A connection between the Baker-Hausdorff formula and a problem of Burnside. Annals of Mathematics 52: 111–126 · Zbl 0037.30401 · doi:10.2307/1969512
[81] Magnus W., Karrass A., Solitar D. (1966) Combinatorial group theory. Interscience, New York · Zbl 0138.25604
[82] McLachlan R.I., Quispel R. (2002) Splitting methods. Acta Numerica 11: 341–434 · Zbl 1105.65341 · doi:10.1017/S0962492902000053
[83] Michel J. (1976) Calculs dans les algèbres de Lie libre: la série de Hausdorff et le problème de Burnside. Astérisque 38: 139–148 · Zbl 0351.17012
[84] Mielnik B., Plebański J. (1970) Combinatorial approach to Baker–Campbell–Hausdorff exponents. Annales de l’Institut Henri Poincaré 12: 215–254 · Zbl 0206.13602
[85] Moan, P.C. 1998. Efficient approximation of Sturm-Liouville problems using Lie-group methods. Technical Report 1998/NA11, Department of Applied Mathematics and Theoretical Physics, University of Cambridge, England.
[86] Moan P.C., Niesen J. (2008) Convergence of the Magnus series. Foundations of Computational Mathematics 8: 291–301 · Zbl 1154.34307 · doi:10.1007/s10208-007-9010-0
[87] Moan P.C., Oteo J.A. (2001) Convergence of the exponential Lie series. Journal of Mathematical Physics 42: 501–508 · Zbl 1016.34008 · doi:10.1063/1.1330198
[88] Montgomery D., Zippin L. (1955) Topological transformation groups. Interscience Publishers, New York · Zbl 0068.01904
[89] Murray F.J. (1962) Perturbation theory and Lie algebras. Journal of Mathematical Physics 3: 451–468 · Zbl 0149.46601 · doi:10.1063/1.1724245
[90] Nagel A., Stein E.M., Wainger S. (1985) Balls and metrics defined by vector fields I, basic properties. Acta Mathematica 155: 103–147 · Zbl 0578.32044 · doi:10.1007/BF02392539
[91] Neeb K.-H. (2006) Towards a Lie theory of locally convex groups. Japanese Journal of Mathematics (3) 1: 291–468 · Zbl 1161.22012 · doi:10.1007/s11537-006-0606-y
[92] Newman M., So W., Thompson R.C. (1989) Convergence domains for the Campbell–Baker–Hausdorff formula. Linear Multilinear Algebra 24: 301–310 · Zbl 0713.22007 · doi:10.1080/03081088908817923
[93] Omori, H. 1997. Infinite-dimensional Lie groups. Translations of Mathematical Monographs 158. Providence: American Mathematical Society.
[94] Oteo J.A. (1991) The Baker-Campbell-Hausdorff formula and nested commutator identities. Journal of Mathematical Physics 32: 419–424 · Zbl 0725.47052 · doi:10.1063/1.529428
[95] Pascal E. (1901a) Sopra alcune indentità fra i simboli operativi rappresentanti trasformazioni infinitesime. Rendiconti/Istituto Lombardo di Scienze e Lettere, Milano (2) 34: 1062–1079 · JFM 32.0376.01
[96] Pascal E. (1901b) Sulla formola del prodotto di due trasformazioni finite e sulla dimostrazione del cosidetto secondo teorema fondamentale di Lie nella teoria dei gruppi. Rendiconti/Istituto Lombardo di Scienze e Lettere, Milano (2) 34: 1118–1130 · JFM 32.0376.02
[97] Pascal E. (1902a) Sopra i numeri bernoulliani. Rendiconti/Istituto Lombardo di Scienze e Lettere, Milano (2) 35: 377–389 · JFM 33.0457.02
[98] Pascal E. (1902b) Del terzo teorema di Lie sull’esistenza dei gruppi di data struttura. Rendiconti/Istituto Lombardo di Scienze e Lettere, Milano (2) 35: 419–431 · JFM 33.0373.01
[99] Pascal E. (1902c) Altre ricerche sulla formola del prodotto di due trasformazioni finite e sul gruppo parametrico di un dato. Rendiconti / Istituto Lombardo di Scienze e Lettere, Milano (2) 35: 555–567 · JFM 33.0373.02
[100] Pascal, E. 1903. I Gruppi Continui di Trasformazioni (Parte generale della teoria). Manuali Hoepli, Nr. 327 bis 328; Milano: Hoepli. · JFM 34.0181.02
[101] Poincaré H. (1899) Sur les groupes continus. Comptes Rendus de l’Académie des Sciences. Série I. Mathématique 128: 1065–1069 · JFM 30.0334.01
[102] Poincaré H. (1900) Sur les groupes continus. Transactions of the Cambridge Philosophical Society 18: 220–255 · JFM 31.0386.01
[103] Poincaré H. (1901) Quelques remarques sur les groupes continus. Rendiconti del Circolo Matematico di Palermo 15: 321–368 · JFM 32.0373.01 · doi:10.1007/BF03013569
[104] Reinsch M.W. (2000) A simple expression for the terms in the Baker–Campbell–Hausdorff series. Journal of Mathematical Physics 41: 2434–2442 · Zbl 0974.22015 · doi:10.1063/1.533250
[105] Reutenauer, C. 1993. Free Lie algebras. London Mathematical Society Monographs (New Series) 7. Oxford: Clarendon Press. · Zbl 0798.17001
[106] Richtmyer R.D., Greenspan S. (1965) Expansion of the Campbell–Baker–Hausdorff formula by computer. Communications on Pure and Applied Mathematics 18: 107–108 · Zbl 0156.26303 · doi:10.1002/cpa.3160180111
[107] Robart Th. (1997) Sur l’intégrabilité des sous-algèbres de Lie en dimension infinie. Canadian Journal of Mathematics 49: 820–839 · Zbl 0888.22016 · doi:10.4153/CJM-1997-042-7
[108] Robart Th. (2004) On Milnor’s regularity and the path-functor for the class of infinite dimensional Lie algebras of CBH type. Algebras, Groups and Geometries 21: 367–386 · Zbl 1130.17306
[109] Rossmann, W. 2002. Lie Groups. An Introduction Through Linear Groups. Oxford graduate texts in mathematics 5. Oxford: Oxford University Press. · Zbl 0989.22001
[110] Rothschild L.P., Stein E.M. (1976) Hypoelliptic differential operators and nilpotent groups. Acta Mathematica 137: 247–320 · Zbl 0346.35030 · doi:10.1007/BF02392419
[111] Sagle, A.A., R.E. Walde. 1973. Introduction to Lie groups and Lie algebras. Pure and applied mathematics 51. New York: Academic Press. · Zbl 0252.22001
[112] Schmid W. (1982) Poincaré and Lie groups. Bulletin of the American Mathematical Society (New Series) 6: 175–186 · Zbl 0493.22001 · doi:10.1090/S0273-0979-1982-14972-2
[113] Schmid, R. 2010. Infinite-dimensional Lie groups and algebras in Mathematical Physics. Advances in Mathematical Physics: Article ID 280362. doi: 10.1155/2010/280362 .
[114] Schur F. (1890a) Neue Begründung der Theorie der endlichen Transformationsgruppen. Mathematische Annalen 35: 161–197 · JFM 21.0371.01 · doi:10.1007/BF01443876
[115] Schur F. (1890b) Beweis für die Darstellbarkeit der infinitesimalen Transformationen aller transitiven endlichen Gruppen durch Quotienten beständig convergenter Potenzreihen. Berichte der Königlich-Sächsischen Gesellschaft der Wissenschaften zu Leipzig (Leipziger Berichte), Math. phys. Cl. 42: 1–7 · JFM 22.0375.01
[116] Schur F. (1891) Zur Theorie der endlichen Transformationsgruppen. Mathematische Annalen 38: 263–286 · JFM 23.0381.01 · doi:10.1007/BF01199254
[117] Schur F. (1893) Ueber den analytischen Charakter der eine endliche continuirliche Transformationsgruppe darstellenden Functionen. Mathematische Annalen 41: 509–538 · JFM 25.0632.01 · doi:10.1007/BF01443741
[118] Sepanski, M.R. 2007. Compact Lie groups. Graduate texts in mathematics 235. New York: Springer.
[119] Serre, J.P. 1965. Lie algebras and Lie groups: 1964 lectures given at Harvard University, 1st edn. 1965, New York: W. A. Benjamin, Inc., 2nd ed. 1992. Lecture notes in mathematics 1500. Berlin: Springer. · Zbl 0132.27803
[120] Specht W. (1948) Die linearen Beziehungen zwischen höheren Kommutatoren. Mathematische Zeitschrift 51: 367–376 · Zbl 0031.00602 · doi:10.1007/BF01181601
[121] Suzuki M. (1977) On the convergence of exponential operators–the Zassenhaus formula, BCH formula and systematic approximants. Communications in Mathematical Physics 57: 193–200 · Zbl 0366.47016 · doi:10.1007/BF01614161
[122] Thompson R.C. (1982) Cyclic relations and the Goldberg coefficients in the Campbell–Baker–Hausdorff formula. Proceedings of the American Mathematical Society 86: 12–14 · Zbl 0497.17002
[123] Thompson R.C. (1989) Convergence proof for Goldberg’s exponential series. Linear Algebra and its Applications 121: 3–7 · Zbl 0678.22003 · doi:10.1016/0024-3795(89)90688-5
[124] Ton-That T., Tran T.D. (1999) Poincaré’s proof of the so-called Birkhoff–Witt theorem. Revue d’Histoire des Mathématiques 5: 249–284 · Zbl 0958.01012
[125] Tu L.W. (2004) Une courte démonstration de la formule de Campbell–Hausdorff. Journal of Lie Theory 14: 501–508 · Zbl 1061.22007
[126] Van Est W.T., Korthagen J. (1964) Non-enlargible Lie algebras. Proceedings of the Koninklijke Nederlandse Akademie van Wetenschappen. Series A 67 = Indagationes Mathematicae 26: 15–31 · Zbl 0121.27503
[127] Varadarajan, V.S. 1984. Lie groups, Lie algebras, and their representations. (Reprint of the 1974 edition.) Graduate texts in mathematics 102. New York: Springer.
[128] Vasilescu F.-H. (1972) Normed Lie algebras. Canadian Journal of Mathematics 24: 580–591 · Zbl 0237.17006 · doi:10.4153/CJM-1972-052-7
[129] Veldkamp F.D. (1980) A note on the Campbell–Hausdorff formula. Journal of Algebra 62: 477–478 · Zbl 0423.22009 · doi:10.1016/0021-8693(80)90197-0
[130] Wei J. (1963) Note on the global validity of the Baker–Hausdorff and Magnus theorems. Journal of Mathematical Physics 4: 1337–1341 · Zbl 0152.02203 · doi:10.1063/1.1703910
[131] Weiss G.H., Maradudin T.D. (1962) The Baker Hausdorff formula and a problem in Crystal Physics. Journal of Mathematical Physics 3: 771–777 · Zbl 0108.44804 · doi:10.1063/1.1724280
[132] Wever F. (1949) Operatoren in Lieschen Ringen. Journal für die reine und angewandte Mathematik (Crelle’s Journal) 187: 44–55 · Zbl 0036.29901
[133] Wichmann E.H. (1961) Note on the algebraic aspect of the integration of a system of ordinary linear differential equations. Journal of Mathematical Physics 2: 876–880 · Zbl 0106.29003 · doi:10.1063/1.1724235
[134] Wilcox R.M. (1967) Exponential operators and parameter differentiation in quantum physics. Journal of Mathematical Physics 8: 962–982 · Zbl 0173.29604 · doi:10.1063/1.1705306
[135] Wojtyński W. (1998) Quasinilpotent Banach-Lie algebras are Baker–Campbell–Hausdorff. Journal of Functional Analysis 153: 405–413 · Zbl 0934.46053 · doi:10.1006/jfan.1997.3202
[136] Yosida K. (1937) On the exponential-formula in the metrical complete ring. Proceedings of the Imperial Academy Tokyo 13: 301–304. doi: 10.3792/pia/1195579861 · JFM 63.1000.02 · doi:10.3792/pia/1195579861
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