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Free post-groups, post-groups from group actions, and post-Lie algebras

Mahdi Jasim Hasan Al-Kaabi, Kurusch Ebrahimi-Fard, Dominique Manchon

Abstract

After providing a short review on the recently introduced notion of post-group by Bai, Guo, Sheng and Tang, we exhibit post-group counterparts of important post-Lie algebras in the literature, including the infinite-dimensional post-Lie algebra of Lie group integrators. The notion of free post-group is examined, and a group isomorphism between the two group structures associated to a free post-group is explicitly constructed.

Free post-groups, post-groups from group actions, and post-Lie algebras

Abstract

After providing a short review on the recently introduced notion of post-group by Bai, Guo, Sheng and Tang, we exhibit post-group counterparts of important post-Lie algebras in the literature, including the infinite-dimensional post-Lie algebra of Lie group integrators. The notion of free post-group is examined, and a group isomorphism between the two group structures associated to a free post-group is explicitly constructed.
Paper Structure (14 sections, 10 theorems, 74 equations)

This paper contains 14 sections, 10 theorems, 74 equations.

Table of Contents

  1. Introduction
  2. Basic definitions and properties
  3. Quick review of the main results of reference BGST2023
  4. Post-groups, braided groups and skew-braces
  5. Post-Lie groups and post-Lie algebras
  6. Post-Hopf algebras vs. $D$-algebras
  7. Weak post-groups
  8. Examples
  9. Two post-groups naturally associated to a group
  10. Weak post-groups from group actions
  11. Weak post-Lie groups from Lie group actions
  12. The free post-group generated by a diagonal left-regular magma
  13. A group isomorphism
  14. An analogy with Gavrilov's $K$-map

Key Result

Proposition 1

BGST2023 The binary map $*:G\times G\to G$ defined by $a*b:=a.(a\rhd b)$ provides $G$ with a second group structure. Both groups $(G,.)$ and $(G,*)$ share the same unit $e$, and the inverse for $*$ is given by The binary map $*$ is an action of the group $(G,*)$ on the group $(G,.)$ by automorphisms.

Theorems & Definitions (39)

  • Definition 1
  • Proposition 1
  • proof
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Theorem 2
  • Proposition 3
  • proof
  • ...and 29 more