Poincaré's proof of the co-called Birkhoff-Witt theorem
Revue d'histoire des mathématiques, Volume 5 (1999) no. 2, pp. 249-284.

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.

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.

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Ton-That, Tuong; Tran, Thai-Duong. Poincaré's proof of the co-called Birkhoff-Witt theorem. Revue d'histoire des mathématiques, Volume 5 (1999) no. 2, pp. 249-284. http://archive.numdam.org/item/RHM_1999__5_2_249_0/

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