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ISRN Algebra
Volume 2011 (2011), Article ID 247403, 11 pages
http://dx.doi.org/10.5402/2011/247403
Research Article

A New Proof of the Existence of Free Lie Algebras and an Application

Dipartimento di Matematica, Università degli Studi di Bologna, Piazza di Porta San Donato 5, 40126 Bologna, Italy

Received 23 March 2011; Accepted 20 April 2011

Academic Editors: K. Dekimpe and J. Kakol

Copyright © 2011 Andrea Bonfiglioli and Roberta Fulci. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The existence of free Lie algebras is usually derived as a consequence of the Poincaré-Birkhoff-Witt theorem. Moreover, in order to prove that (given a set 𝑋 and a field 𝕂 of characteristic zero) the Lie algebra ( 𝕂 𝑋 ) of the Lie polynomials in the letters of 𝑋 (over the field 𝕂 ) is a free Lie algebra generated by 𝑋 , all available proofs use the embedding of a Lie algebra 𝔤 into its enveloping algebra 𝒰 ( 𝔤 ) . The aim of this paper is to give a much simpler proof of the latter fact without the aid of the cited embedding nor of the Poincaré-Birkhoff-Witt theorem. As an application of our result and of a theorem due to Cartier (1956), we show the relationships existing between the theorem of Poincaré-Birkhoff-Witt, the theorem of Campbell-Baker-Hausdorff, and the existence of free Lie algebras.