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 and a field of characteristic zero) the Lie algebra ℒ(⟨⟩) of the Lie polynomials in the letters of (over the field ) is a free Lie algebra generated by , all available proofs use the embedding of a Lie algebra into its enveloping algebra (). 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.

A new proof of the existence of free Lie algebras and an application / A. Bonfiglioli; R. Fulci. - In: ISRN ALGEBRA. - ISSN 2090-6285. - ELETTRONICO. - 2011:(2011), pp. 1-11. [10.5402/2011/247403]

A new proof of the existence of free Lie algebras and an application

BONFIGLIOLI, ANDREA;FULCI, ROBERTA
2011

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 and a field of characteristic zero) the Lie algebra ℒ(⟨⟩) of the Lie polynomials in the letters of (over the field ) is a free Lie algebra generated by , all available proofs use the embedding of a Lie algebra into its enveloping algebra (). 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.
2011
A new proof of the existence of free Lie algebras and an application / A. Bonfiglioli; R. Fulci. - In: ISRN ALGEBRA. - ISSN 2090-6285. - ELETTRONICO. - 2011:(2011), pp. 1-11. [10.5402/2011/247403]
A. Bonfiglioli; R. Fulci
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11585/110485
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