Non-abelian cohomology of Nijenhuis Lie algebras and the inducibility of automorphisms and derivations
Apurba Das
TL;DR
This work develops a non-abelian cohomology framework for Nijenhuis Lie algebras and uses it to classify non-abelian extensions via $H^2_{nab}((\mathfrak{g}, N); (\mathfrak{h}, S))$, with abelian extensions tied to $H^2((\mathfrak{g}, N); (V, S))$. It then investigates inducibility problems for automorphisms and derivations within (non-)abelian extensions, formulating Wells-type obstructions that live in cohomology groups and proving they vanish exactly when inducibility holds. The paper constructs Wells maps and exact sequences relating automorphism/derivation groups to cohomology, providing a coherent obstruction-theory framework for Nijenhuis Lie algebras. It lays groundwork for a full cochain complex and potential applications to deformation theory and homotopy algebras, while aligning with analogous results in Lie and Rota-Baxter contexts.
Abstract
In this paper, we first introduce the non-abelian cohomology group of a Nijenhuis Lie algebra with values in another Nijenhuis Lie algebra and show that it parametrizes the isomorphism classes of all non-abelian extensions. In particular, we obtain a classification result for abelian extensions of a Nijenhuis Lie algebra by a given Nijenhuis representation. Next, given a non-abelian extension of Nijenhuis Lie algebras, we investigate the inducibility of a pair of Nijenhuis Lie algebra automorphisms and show that the corresponding obstruction lies in the non-abelian cohomology group. Subsequently, we also consider the inducibility of a pair of Nijenhuis Lie algebra derivations in a given abelian extension.