Cohomology and automorphisms for a matched pair of 3-Lie algebras

TL;DR

This paper develops the cohomology theory for matched pairs of 3-Lie algebras, exploring their representations, deformations, extensions, and automorphisms, with a focus on algebraic structures and their symmetries.

Contribution

It introduces the cohomology groups, deformation theory, and automorphism inducibility for matched pairs of 3-Lie algebras, extending the understanding of their algebraic and symmetry properties.

Findings

Defined low-dimensional cohomology groups for matched pairs.

Analyzed infinitesimal deformations and abelian extensions.

Presented results on automorphism inducibility via Wells exact sequence.

Abstract

We begin by reviewing the definition of 3-Lie algebras and the fundamental concepts of matched pairs. Subsequently, we introduce the representation theory of matched pairs and define the semidirect product. Building on this foundation, we define the low-dimensional cohomology groups of matched pairs. In addition, we explore the infinitesimal deformations and abelian extensions of matched pairs. Finally, we examine the inducibility of automorphisms of matched pairs and present related results through the Wells exact sequence.