Relationship between the CBHD theorem, the PBW theorem and the free lie algebras

Abstract

THE aim of this chapter is to unravel the close relationship existing = between the Theorems of CBHD and of Poincaré-Birkhoff-Witt (“PBW” for short) and to show how the existence of free Lie algebras intervenes.We have analyzed, in Chap. 3, how the PBWTheoremintervenes in the classical approach to the proof of CBHD, in that PBWcan be used to prove in a simple way Friedrichs’s characterization of L(V ) (see Theorem 3.13 on page 133). Also, as for the proofs of CBHD in Chap. 4, the rÔle of the free Lie algebras was broadly manifest.