Poincaré’s proof of the co-called Birkhoff-Witt theorem

Tuong Ton-That; Thai-Duong Tran

Revue d'histoire des mathématiques (1999)

  • Volume: 5, Issue: 2, page 245-280
  • ISSN: 1262-022X

Abstract

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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.

How to cite

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Ton-That, Tuong, and Tran, Thai-Duong. "Poincaré’s proof of the co-called Birkhoff-Witt theorem." Revue d'histoire des mathématiques 5.2 (1999): 245-280. <http://eudml.org/doc/252098>.

@article{Ton1999,
abstract = {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.},
author = {Ton-That, Tuong, Tran, Thai-Duong},
journal = {Revue d'histoire des mathématiques},
keywords = {Henri Poincaré; Lie algebra; universal enveloping algebra; Garrett Birkhoff; Ernst Witt; symmetric algebra},
language = {eng},
number = {2},
pages = {245-280},
publisher = {Société mathématique de France},
title = {Poincaré’s proof of the co-called Birkhoff-Witt theorem},
url = {http://eudml.org/doc/252098},
volume = {5},
year = {1999},
}

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
PB - Société mathématique de France
VL - 5
IS - 2
SP - 245
EP - 280
AB - 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.
LA - eng
KW - Henri Poincaré; Lie algebra; universal enveloping algebra; Garrett Birkhoff; Ernst Witt; symmetric algebra
UR - http://eudml.org/doc/252098
ER -

References

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