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

Abstract EN

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 L(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.