Non-abelian extensions and Wells exact sequences of Hom-Lie algebras with derivations

Abstract

In this paper, we consider the non-abelian extensions of HLieDer triples,
in which an HLieDer triple includes a Hom-Lie algebra and a derivation.
First we introduce the non-abelian cohomology for HLieDer triples by which we classify non-abelian extensions of HLieDer triples.
Then given a non-abelian extension of
HLieDer triples, we show that the obstruction for a pair of HLieDer triple automorphisms to
be inducible can be seen as the image of a suitable Wells map.
As a byproduct, we obtain the Wells short-exact sequence in the context of HLieDer triples.
Finally, we discuss these results in the case of abelian extensions of HLieDer triples.