Poincaré's Proof of the so-called Birkhoff-Witt Theorem

Tuong Ton-That; Thai-Duong Tran

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Poincaré's Proof of the so-called Birkhoff-Witt Theorem

Tuong Ton-That·Thai-Duong Tran

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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 precedence of his discovery of many important concepts, especially that of the \textit{universal enveloping algebra} of a Lie algebra over a field of characteristic zero, the \textit{% canonical map} (\textit{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.

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Tuong Ton-That, Thai-Duong Tran. 2000-04-27. Poincaré's Proof of the so-called Birkhoff-Witt Theorem. https://arxiv.org/abs/math/9908139

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