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. Bonfiglioli, R. Fulci (2011). A new proof of the existence of free Lie algebras and an application. ISRN ALGEBRA, 2011, 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.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
