Crossed modules, non-abelian extensions of associative conformal algebras and Wells exact sequences

TL;DR

This paper develops a cohomological framework for classifying non-abelian extensions of associative conformal algebras using crossed modules, Maurer-Cartan elements, and Wells exact sequences.

Contribution

It introduces new notions of crossed modules and strongly homotopy algebras for associative conformal algebras and links them to Hochschild cohomology and non-abelian cohomology.

Findings

Classification of non-abelian extensions via non-abelian cohomology.

Representation of extensions as Maurer-Cartan elements in a dg Lie algebra.

Establishment of Wells exact sequences in the context of associative conformal algebras.

Abstract

In this paper, we introduce the notions of crossed module of associative conformal algebras, 2-term strongly homotopy associative conformal algebras, and discuss the relationship between them and the 3-th Hochschild cohomology of associative conformal algebras. We classify the non-abelian extensions by introducing the non-abelian cohomology. We show that non-abelian extensions of an associative conformal algebra can be viewed as Maurer-Cartan elements of a suitable differential graded Lie algebra, and prove that the Deligne groupoid of this differential graded Lie algebra corresponds one to one with the non-abelian cohomology. Based on this classification, we study the inducibility of a pair of automorphisms about a non-abelian extension of associative conformal algebras, and give the fundamental sequence of Wells in the context of associative conformal algebras. Finally, we consider the extensibility of a pair of derivations about an abelian extension of associative conformal algebras, and give an exact sequence of Wells type.