Document Type : Research Paper
Department of Mathematics, Neyshabur Branch, Islamic Azad University, Neyshabur, Iran.
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) = 1/2n(n + 2m - 1) - 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)=0, 1, 2, … , 23, 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) = 3, where S′ (N,L) = 1/2 (n - 1)(n - 2) + 1 + (n - 1)m – dim M(N,L), L is a finite dimensional nilpotent Lie algebra and N is a non-abelian ideal of L.