Non-abelian cohomology of Nijenhuis Lie algebras and the inducibility of automorphisms and derivations

TL;DR

This paper develops a non-abelian cohomology framework for Nijenhuis Lie algebras, classifies their extensions, and studies the inducibility of automorphisms and derivations within this context.

Contribution

It introduces a non-abelian cohomology theory for Nijenhuis Lie algebras, providing classification of extensions and criteria for inducibility of automorphisms and derivations.

Findings

Classifies abelian extensions of Nijenhuis Lie algebras.

Identifies the obstruction to inducibility in non-abelian cohomology.

Provides conditions for the inducibility of automorphisms and derivations.

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.