Poincar\'e's Proof of the so-called Birkhoff-Witt Theorem
Abstract
A methodical analysis of the research related to the article, ``Sur les groupes continus'', of Henri Poincar\'e 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 universal enveloping algebra of a Lie algebra over a field of characteristic zero, 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.
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