Wells type exact sequence and crossed extensions of algebras with bracket

TL;DR

This paper investigates the structure of algebra extensions with brackets, introduces crossed modules, and establishes an exact sequence relating derivations, cohomology, and algebraic structures.

Contribution

It introduces crossed modules for algebras with bracket and proves their equivalence with internal categories, extending the cohomological framework.

Findings

Derived an exact sequence of Wells type for algebra extensions

Established the equivalence between crossed modules and internal categories

Constructed an eight-term exact sequence in algebra cohomology

Abstract

We study the extensibility problem of a pair of derivations associated with an abelian extension of algebras with bracket, and derive an exact sequence of the Wells type. We introduce crossed modules for algebras with bracket and prove their equivalence with internal categories in the category of algebras with bracket. We interpret the set of equivalence classes of crossed extensions as the second cohomology. Finally, we construct an eight term exact sequence in the cohomology of algebras with bracket.