The symmetric Poincar\'e-Birkhoff-Witt theorem and Dynkin-Magnus commutators

TL;DR

This paper provides alternative proofs for the symmetric Poincaré-Birkhoff-Witt theorem using Magnus recursion and Dynkin's polynomial methods, also deriving a related Nouazé-Revoy type theorem.

Contribution

It introduces new proof techniques for the symmetric PBW theorem and applies universal algebraic principles to derive additional results.

Findings

Alternative proofs of the symmetric PBW theorem

Application of Magnus recursion and Dynkin's methods

Derivation of a Nouazé-Revoy type theorem

Abstract

The objective of this paper is to give alternative proofs for the symmetric Poincar\'e-Birkhoff-Witt theorem utilizing the Magnus recursion formulae or Dynkin's non-commutative polynomial comparison method and simple universal algebraic principles. As an application of these principles, a theorem of Nouaz\'e-Revoy type is also obtained.