Capability, Schur Multiplier and Some Exact Sequences of a Pair of Lie Algebras

The aim of this work is to find some criteria for detecting the capability of a pair of Lie algebras. We exactly characterize the structure of all pairs of capable Lie algebras in the class of abelian and Heisenberg Lie algebras. It could generalize the definition, notation and investigation on groups and Lie algebras. Among the other results, we also give some exact sequences on the Schur multiplier and exterior product of Lie algebras.