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Non-abelian extensions and automorphisms of post-Lie algebras

Lisi Bai, Tao Zhang

TL;DR

The paper develops a non-abelian extension theory for post-Lie algebras by introducing non-abelian 2-cocycles and proving their cohomology classifies extensions. It establishes an equivalence between crossed modules and cat^1-post-Lie algebras, and studies automorphism inducibility through a Wells-type framework that yields a short exact sequence connecting automorphism groups with the second non-abelian cohomology. The results extend classical non-abelian extension theory to the post-Lie setting and provide explicit obstruction-theoretic criteria for inducibility of automorphisms, including a specialization to abelian extensions. These tools enable systematic classification and analysis of extensions and automorphisms in post-Lie algebras, with potential implications for related algebraic structures and their applications.

Abstract

In this paper, we introduce the concepts of crossed modules of post-Lie algebras and cat$^1$-post-Lie algebras. It is proved that these two concepts are equivalent to each other. Secondly, we construct a non-abelian cohomology for post-Lie algebras to classify their non-abelian extensions. At last, we investigate the inducibility problem of a pair of automorphisms for post-Lie algebras and construct a Wells exact sequence to solve it.

Non-abelian extensions and automorphisms of post-Lie algebras

TL;DR

The paper develops a non-abelian extension theory for post-Lie algebras by introducing non-abelian 2-cocycles and proving their cohomology classifies extensions. It establishes an equivalence between crossed modules and cat^1-post-Lie algebras, and studies automorphism inducibility through a Wells-type framework that yields a short exact sequence connecting automorphism groups with the second non-abelian cohomology. The results extend classical non-abelian extension theory to the post-Lie setting and provide explicit obstruction-theoretic criteria for inducibility of automorphisms, including a specialization to abelian extensions. These tools enable systematic classification and analysis of extensions and automorphisms in post-Lie algebras, with potential implications for related algebraic structures and their applications.

Abstract

In this paper, we introduce the concepts of crossed modules of post-Lie algebras and cat-post-Lie algebras. It is proved that these two concepts are equivalent to each other. Secondly, we construct a non-abelian cohomology for post-Lie algebras to classify their non-abelian extensions. At last, we investigate the inducibility problem of a pair of automorphisms for post-Lie algebras and construct a Wells exact sequence to solve it.
Paper Structure (8 sections, 12 theorems, 107 equations)

This paper contains 8 sections, 12 theorems, 107 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Crossed modules of post-Lie algebras
  4. Non-abelian extension of post-Lie algebras
  5. Automorphisms
  6. Inducibility for automorphisms
  7. Wells exact sequence
  8. The spacial case of abelian extension

Key Result

Proposition 3.2

Let $A$ and $H$ be two post-Lie algebras with an action $(\rho,\phi,\psi)$ of $A$ on $H$. A homomorphism $\partial:H\rightarrow A$ is a crossed module of post-Lie algebras if and only if the maps $(\partial,\mathrm{id}_ H) : H \ltimes H \to {A}\ltimes H$ and $(\mathrm{id}_A, \partial) : {A}\ltimes H

Theorems & Definitions (34)

  • Definition 2.1: Vallette
  • Definition 2.2
  • Definition 2.3
  • Definition 3.1
  • Proposition 3.2
  • proof
  • Definition 3.3
  • Definition 3.4
  • Definition 3.5
  • Theorem 3.6
  • ...and 24 more