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