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

Bo Hou; Jun Zhao

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arXiv · 2211.10842

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

Bo Hou·Jun Zhao

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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.

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Bo Hou, Jun Zhao. 2022-11-20. Crossed modules, non-abelian extensions of associative conformal algebras and Wells exact sequences. https://arxiv.org/abs/2211.10842

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