Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Some proofs of the Poincaré-Birkhoff-Witt theorem and related matters

Gyula Lakos

Abstract

This expository paper focuses on free Lie $K$-algebras and the basic PBW theorem. We argue in various ways that the basic PBW theorem is a quite close consequence of the Magnus-Witt theorems concerning free Lie algebras.

Some proofs of the Poincaré-Birkhoff-Witt theorem and related matters

Abstract

This expository paper focuses on free Lie -algebras and the basic PBW theorem. We argue in various ways that the basic PBW theorem is a quite close consequence of the Magnus-Witt theorems concerning free Lie algebras.

Paper Structure

This paper contains 7 sections, 21 theorems, 72 equations.

Table of Contents

  1. About free Lie algebras. Version 1 (Abstract version)
  2. About free Lie algebras. Version 2 ("Classical" version)
  3. $\mathrm F_{K}^{\mathop{\mathrm{Lie}}\nolimits}$ via $\mathrm F_{\mathbb Q}^{\mathop{\mathrm{Lie}}\nolimits}$
  4. The $\mathrm F_{K}^{\mathop{\mathrm{Lie}}\nolimits}$ case from the MW theorems and the coproduct structure
  5. The $\mathrm F_{K}^{\mathop{\mathrm{Lie}}\nolimits}$ case directly
  6. From $\mathrm F_{K}^{\mathop{\mathrm{Lie}}\nolimits}$ to the basic PBW theorem
  7. The Witt--Lazard proof of the global PBW theorems

Key Result

Lemma 1.1

$\mathrm I^{\mathop{\mathrm{Lie}}\nolimits}_K[X_\lambda :\lambda\in\Lambda]$ is generated by the elements where $Z_1,Z_2,Z_3$ are monomials of the $X_\lambda$, and $M(\ldots)$ is a $\boldsymbol[,\boldsymbol]$-bracketing with $s+1$ many positions (but not necessarily in the indicated order), and $k\in K$.

Theorems & Definitions (48)

  • Lemma 1.1
  • proof
  • Proposition 1.2
  • proof
  • Corollary 1.3
  • proof
  • Proposition 1.4
  • proof
  • Theorem 1.5
  • Theorem 1.6
  • ...and 38 more