Homayoon Arabyani - Department of Mathematics, Neyshabur Branch, Islamic Azad University, Neyshabur, Iran.
Elaheh Khamseh - Department of Mathematics, Shahr-e-Qods Branch, Islamic Azad University Tehran, Iran;

Let (N,L) be a pair of finite dimensional nilpotent Lie algebras. If N admits a complement K in L such that dim N = n and dim K = m, then dim M(N,L) = ۱/۲n(n + ۲m - ۱) - t(N,L), where M(N,L) is the Schur multiplier of the pair (N,L) and t(N,L) is a non-negative integer. In this paper, we characterize the pair (N,L) for t(N,L)=۰, ۱, ۲, … , ۲۳, where N is a finite dimensional filiform Lie algebra and N,K are ideals of L such that L = N ⊕ K. Moreover, we classify the pair (N,L) for s′ (N,L) = ۳, where S′ (N,L) = ۱/۲ (n - ۱)(n - ۲) + ۱ + (n - ۱)m – dim M(N,L), L is a finite dimensional nilpotent Lie algebra and N is a non-abelian ideal of L.

Filiform Lie algebra, nilpotent Lie algebra, pair of Lie algebras, Schur multiplier