Cohomology of Lie Algebra Morphism Triples and Some Applications

A Lie algebra morphism triple is a triple \((\mathfrak {g}, \mathfrak {h}, \phi )\) consisting of two Lie algebras \(\mathfrak {g}, \mathfrak {h}\) and a Lie algebra homomorphism \(\phi : \mathfrak {g} \rightarrow \mathfrak {h}\). We define representations and cohomology of Lie algebra morphism triples. As applications of our cohomology, we study some aspects of deformations, abelian extensions of Lie algebra morphism triples and classify skeletal sh Lie algebra morphism triples. Finally, we consider the cohomology of Lie group morphism triples and find a relation with the cohomology of Lie algebra morphism triples.