Non-abelian extensions and automorphisms of post-Lie algebras

TL;DR

This paper develops a framework for understanding non-abelian extensions and automorphisms of post-Lie algebras through new concepts, cohomology, and exact sequences, advancing the algebraic theory of these structures.

Contribution

It introduces crossed modules and cat$^1$-post-Lie algebras, establishes their equivalence, and constructs a non-abelian cohomology and Wells exact sequence for post-Lie algebras.

Findings

Equivalence of crossed modules and cat$^1$-post-Lie algebras

Construction of non-abelian cohomology for classification

Development of Wells exact sequence for automorphism inducibility

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.