TY - JOUR
A2 - Dekimpe, K.
A2 - Kakol, J.
AU - Bonfiglioli, Andrea
AU - Fulci, Roberta
PY - 2011
DA - 2012/03/07
TI - A New Proof of the Existence of Free Lie Algebras and an Application
SP - 247403
VL - 2011
AB - 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.
SN - null
UR - https://doi.org/10.5402/2011/247403
DO - 10.5402/2011/247403
JF - ISRN Algebra
PB - International Scholarly Research Network
KW -
ER -