A New Proof of the Existence of Free Lie Algebras and an Application. (7th July 2011) - Bonfiglioli, Andrea; Fulci, Roberta
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 setX and a field𝕂 of characteristic zero) the Lie algebraℒ ( 𝕂 〈 X 〉 ) of the Lie polynomials in the letters ofX (over the field𝕂 ) is a free Lie algebra generated byX, all available proofs use the embedding of a Lie algebra𝔤 into its enveloping algebra𝒰 ( 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.