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The Cartier-Quillen-Milnor-Moore theorem in the Post-Hopf case

Pierre Catoire

Abstract

We give the definition of left/right Post-Lie algebras and left/right Post-Hopf algebras and establish a link between those objects. We get a Cartier-Quillen-Milnor-Moore theorem for Post-Hopf algebras. We give another description for free Post-Lie algebras and a description for Post-Lie algebras obtained from an associative product.

The Cartier-Quillen-Milnor-Moore theorem in the Post-Hopf case

Abstract

We give the definition of left/right Post-Lie algebras and left/right Post-Hopf algebras and establish a link between those objects. We get a Cartier-Quillen-Milnor-Moore theorem for Post-Hopf algebras. We give another description for free Post-Lie algebras and a description for Post-Lie algebras obtained from an associative product.
Paper Structure (23 sections, 47 theorems, 166 equations, 2 figures)

This paper contains 23 sections, 47 theorems, 166 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Document structure
  3. Acknowledgments
  4. Left and Right Post-Lie and Post-Hopf
  5. Notations and reminders
  6. Classical reminders
  7. A short introduction to multisets
  8. Definitions
  9. Some properties
  10. The Cartier-Quillen-Milnor-Moore theorem for Post-Hopf algebras
  11. Preliminary lemmas and definitions
  12. The Cartier-Quillen-Milnor-Moore theorem in the Post-Hopf case
  13. A graded version
  14. Preliminary lemmas and definitions
  15. The theorem
  16. ...and 8 more sections

Key Result

Lemma \ref{Lem:linkLR}

Let $(\mathfrak{h},\left[ , \right],\lhd)$ be a left Post-Lie algebra. Then $(\mathfrak{h},\left[ , \right]^{\mathrm{op}},\lhd^{\mathrm{op}})$ is a right Post-Lie algebra. This development of Post-Lie algebras motivates us to do an algebraic study of these objects like it was done with Pre-Lie algeb

Figures (2)

  • Figure 1: Some diagrams
  • Figure 2: Construction of $\Phi$ and $\Phi'$

Theorems & Definitions (146)

  • Definition \ref{defi:LPL}: left Post-Lie algebras
  • Lemma \ref{Lem:linkLR}
  • Proposition \ref{Prop:OptiLRPH}
  • Theorem \ref{thm:PrimPL}
  • Theorem \ref{thm:extendUg}
  • Corollary \ref{Cor:CQQM}: Cartier-Quillen-Milnor-Moore for Post-Hopf algebras
  • Corollary \ref{Cor:FreePLMag}
  • Lemma \ref{lem:Forestact}
  • Definition \ref{defi:levelledtrees}: levelled planar trees
  • Definition \ref{defi:wordtree}
  • ...and 136 more