Poincaré's proof of the co-called Birkhoff-Witt theorem
[La démonstration de Poincaré du théorème dit de Birkhoff-Witt]
Revue d'histoire des mathématiques, Tome 5 (1999) no. 2, pp. 249-284.

Une analyse méthodique des travaux faits en connexion avec l'article, “Sur les groupes continus”, de Henri Poincaré révèle des erreurs historiques et des jugements injustes en ce qui concerne sa contribution à la théorie de Lie. Une étude approfondie de cet article confirme l'antériorité de sa découverte de plusieurs concepts importants ; notamment de l'algèbre enveloppante universelle d'une algèbre de Lie sur le corps réel ou le corps complexe, et de l'application canonique (la symétrisation) de l'algèbre symétrique sur l'algèbre enveloppante universelle. L'essentiel de cet article consiste en un examen approfondi de sa démonstration rigoureuse et complète du théorème de Birkhoff-Witt.

A methodical analysis of the research related to the article, “Sur les groupes continus”, of Henri Poincaré reveals many historical misconceptions and inaccuracies regarding his contribution to Lie theory. A thorough reading of this article confirms the priority of his discovery of many important concepts, especially that of the universal enveloping algebra of a Lie algebra over the real or complex field, and the canonical map (symmetrization) of the symmetric algebra onto the universal enveloping algebra. The essential part of this article consists of a detailed discussion of his rigorous, complete, and enlightening proof of the so-called Birkhoff-Witt theorem.

@article{RHM_1999__5_2_249_0,
     author = {Ton-That, Tuong and Tran, Thai-Duong},
     title = {Poincar\'e's proof of the co-called {Birkhoff-Witt} theorem},
     journal = {Revue d'histoire des math\'ematiques},
     pages = {249--284},
     publisher = {Soci\'et\'e math\'ematique de France},
     volume = {5},
     number = {2},
     year = {1999},
     zbl = {0958.01012},
     language = {en},
     url = {http://archive.numdam.org/item/RHM_1999__5_2_249_0/}
}
TY  - JOUR
AU  - Ton-That, Tuong
AU  - Tran, Thai-Duong
TI  - Poincaré's proof of the co-called Birkhoff-Witt theorem
JO  - Revue d'histoire des mathématiques
PY  - 1999
SP  - 249
EP  - 284
VL  - 5
IS  - 2
PB  - Société mathématique de France
UR  - http://archive.numdam.org/item/RHM_1999__5_2_249_0/
LA  - en
ID  - RHM_1999__5_2_249_0
ER  - 
%0 Journal Article
%A Ton-That, Tuong
%A Tran, Thai-Duong
%T Poincaré's proof of the co-called Birkhoff-Witt theorem
%J Revue d'histoire des mathématiques
%D 1999
%P 249-284
%V 5
%N 2
%I Société mathématique de France
%U http://archive.numdam.org/item/RHM_1999__5_2_249_0/
%G en
%F RHM_1999__5_2_249_0
Ton-That, Tuong; Tran, Thai-Duong. Poincaré's proof of the co-called Birkhoff-Witt theorem. Revue d'histoire des mathématiques, Tome 5 (1999) no. 2, pp. 249-284. http://archive.numdam.org/item/RHM_1999__5_2_249_0/

[1] Encyclopaedia [1988-1994] Encyclopaedia of Mathematics, ed. by I.M.Vinogradov, Dordrecht-Boston: Kluwer Academic Publishers, 1988-1994, translated from Математическая энциклопедия, главный редактор И.М. Виноградов 495, Москва́: Советская энциклопедия, 1977. (the rubric “Birkhoff-Witt theorem” is in Tome1, p.495).

[2] Barrow-Green (June) [1997] Poincaré and the Three-Body Problem, Providence: American Mathematical Society, 1997. | MR | Zbl

[3] Bell (Eric T.) [1937] Men of Mathematics, New York: Simon & Schuster, 1937. | JFM

[4] Birkhoff (Garrett) [1937] Representability of Lie algebras and Lie groups by matrices, Annals of Mathematics, 38 (April 1937), pp. 526-532. | JFM | MR | Zbl

[5] Bourbaki (Nicolas) [1960] Éléments de mathématiques, Fascicule XXVI. Groupes et algèbres de Lie, Chap. 1, Paris: Hermann, 1960. | MR

[6] Bourbaki (Nicolas) [1969] Éléments d'histoire des mathématiques, Paris: Hermann, 1969. | MR | Zbl

[7] Bourbaki (Nicolas) [1972] Éléments de mathématiques, Fascicule XXXVII. Groupes et algèbres de Lie, Paris: Hermann, 1972. | MR

[8] Bourbaki (Nicolas) [1975] Elements of Mathematics. Lie Groups and Lie Algebras, Part I, Chapter I-3, Reading, MA: Addison-Wesley, 1975; Paris: Hermann, 1971-73. | MR

[9] Boyer (Carl) [1968] A History of Mathematics, Princeton: Princeton University Press, 1968. | MR | Zbl

[10] Cartan (Henri) & Eilenberg (Samuel) [1956] Homological Algebra, Princeton: Princeton University Press, 1956. | MR | Zbl

[11] Chevalley (Claude) [1955] Théorie des groupes de Lie, vol. III, Paris: Hermann, 1955. | MR | Zbl

[12] Cohn (Paul M.) [1981] Universal Algebra, Dordrecht: Reidel, 1981. | MR

[13] Dixmier (Jacques) [1974] Algèbres enveloppantes, Paris: Gauthier-Villars, 1974. | MR | Zbl

[14] Gittleman (Arthur) [1975] History of Mathematics, Columbus, OH: C.E. Merrill Publishing Company, 1975.

[15] Godement (Roger) [1982] Introduction à la théorie des groupes de Lie, 2 vols., Publications mathématiques de l'Université de Paris VII, Paris: Univ. Paris VII, 1982. | Zbl

[16] Harish-Chandra [1949] On Representations of Lie Algebras, Annals of Mathematics, 50 (October 1949), pp. 900-915. | MR | Zbl

[17] Harish-Chandra [1951] On Some Applications of the Universal Enveloping Algebra of a Semisimple Lie Algebra, Transactions of the American Mathematical Society, 70-71 (1951), pp. 28-96. | MR | Zbl

[18] Hoffman (Kenneth) & Kunze (Ray) [1971] Linear Algebra, second ed., Englewood Cliffs, NJ: Prentice-Hall, 1971. | MR

[19] Humphreys (James E.) [1972] Introduction to Lie Algebras and Representation Theory, New York: Springer-Verlag, 1972. | MR | Zbl

[20] Jacobson (Nathan) [1962] Lie Algebras, New York: John Wiley & Sons, 1962. | MR | Zbl

[21] Knapp (Anthony) [1986] Representation Theory of Semisimple Groups, Princeton: Princeton University Press, 1986. | MR | Zbl

[22] Kuroś (Aleksandrevich Gennadievich) [1963] Lectures on General Algebras, New York: Chelsea, 1963. | MR

[23] Lang (Serge) [1965] Algebra, Reading, MA: Addison-Wesley, 1965. | MR | Zbl

[24] Lazard (Michel) [1952] Sur les algèbres enveloppantes universelles de certaines algèbres de Lie, Comptes rendus hebdomadaires des séances de l'Académie des sciences de Paris Sér. I (18 Feb. 1952), pp.788-792. | MR | Zbl

[25] Lazard (Michel) [1954] Sur les algèbres enveloppantes de certaines algèbres de Lie, Publications scientifiques de l'Université d'Alger, Sér. A, 1 (1954), pp. 281-294. | MR | Zbl

[26] Lorentz (Hendrik Antoon) [1921] Deux mémoires de Henri Poincaré sur la physique mathématique, Acta mathematica, 38 (1921), pp. 293-308. | MR

[27] Painlevé (Paul) [1921] Henri Poincaré, Acta Math., 38 (1921), pp. 309-402.

[28] Poincaré (Henri) [Œuvres] Œuvres de Henri Poincaré, 11 vols., Paris: Gauthier-Villars, 1916- . | Zbl

[29] Poincaré (Henri) [1881] Formes cubiques ternaires et quaternaires, Journal de l'École polytechnique Paris, XXXI (1881), pp. 199-253. | JFM

[30] Poincaré (Henri) [1883] Sur la reproduction des formes, C.R. Acad. sci. Paris (29 oct. 1883), pp.949-951. | JFM

[31] Poincaré (Henri) [1899] Sur les groupes continus, C.R. Acad. sci. Paris, 128 (1899), pp. 1065-1069. | JFM

[32] Poincaré (Henri) [1900] Sur les groupes continus, Transactions of the Cambridge Philosophical Society, 18 (1900), pp. 220-255 = Œuvres de Henri Poincaré, vol., III, Paris: Gauthier-Villars (1934), pp.173-212. | JFM

[33] Poincaré (Henri) [1901] Quelques remarques sur les groupes continus, Rendiconti del Circolo matematico di Palermo, 15 (1901), pp.213-260. | JFM

[34] Poincaré (Henri) [1906] Sur la dynamique de l'électron, Rend. Circ. Mat. Palermo, 21 (1906), pp.129-176. | JFM

[35] Poincaré (Henri) [1908] Nouvelles remarques sur les groupes continus, Rend. Circ. Mat. Palermo, 25 (1908), pp.261-321. | JFM

[36] Poincaré (Henri) [1912] Sur la théorie des quanta, Journal de physique théorique et appliquée 5e sér., 2 (1912), pp.5-34. | JFM

[37] Poincaré (Henri) [In memoriam 1921] Henri Poincaré in memoriam, Acta Math., 38 (1921). | JFM

[38] Schmid (Wilfried) [1982] Poincaré and Lie groups, Bulletin of the American Mathematical Society (N.S.), 6 (1982), pp. 175-186. This article is also reprinted in Felix E.Browder (ed.), The Mathematical Heritage of Henri Poincaré, Proceedings of Symposia in Pure Mathematics, Vol. 39, Providence, R.I.: American Mathematical Society, 1983, Part 1, pp.157-168. | MR | Zbl

[39] Schwartz (Laurent) [1998 (1975)] Les tenseurs, Paris: Hermann, 1998; 1e éd. 1975. | MR

[40] Varadarajan (V.S.) [1984 (1974)] Lie Groups, Lie Algebras and Their Representations, New York: Springer-Verlag, 1984; Englewood Cliffs, NJ: Prentice-Hall, 1974. | MR | Zbl

[41] Weyl (Hermann) [1946] The Classical Groups, Their Invariants and Representations, second ed., Princeton: Princeton Univ. Press, 1946. | JFM | MR | Zbl

[42] Witt (Ernst) [1937] Treue Darstellung Liescher Ringe, Journal für die reine und angewandte Mathematik, 177 (1937), pp. 152-160. | JFM | Zbl