The non-abelian extension and Wells map of Leibniz conformal algebra
Jun Zhao, Bo Hou, Xin Zhou
TL;DR
This work addresses the problem of classifying non-abelian extensions of a Leibniz conformal algebra $R$ by a Leibniz conformal algebra $H$ through a non-abelian cohomology theory $H^2_{nab}(R,H)$. By constructing a differential graded Lie algebra $\mathfrak{L}$, the authors show that Maurer-Cartan elements parametrize extensions, linking cohomology with a Deligne groupoid viewpoint. They derive a Wells map that governs the inducibility of automorphisms and establish exact sequences relating automorphism groups to $H^2_{nab}(R,H)$, including abelian cases. The extensibility problem for derivations is treated via a Wells obstruction map $\mathcal{W}:{\rm g}(R,H)\to H^2(R,H)$, with precise conditions for extending derivations and a corresponding exact sequence. Overall, the paper provides a unified, DG-Lie-algebraic framework for non-abelian extensions, automorphism inducibility, and derivation extensibility in the Leibniz conformal setting.
Abstract
In this paper, we study the theory of non-abelian extensions of a Leibniz conformal algebra $R$ by a Leibniz conformal algebra $H$ and prove that all the non-abelian extensions are classified by non-abelian $2$nd cohomology $H^2_{nab}(R,H)$ in the sense of equivalence. Then we introduce a differential graded Lie algebra $\mathfrak{L}$ and show that the set of its Maurer-Cartan elements in bijection with the set of non-abelian extensions. Finally, as an application of non-abelian extension, we consider the inducibility of a pair of automorphisms about a non-abelian extension, and give the fundamental sequence of Wells of Leibniz conformal algebra $R$. Especially, we discuss the extensibility problem of derivations about an abelian extension of $R$.