On Characterizing Pairs of Non-Abelian Nilpotent and Filiform Lie Algebras by their Schur Multipliers

Let L be an n-dimensional non-abelian nilpotent Lie algebra. Niroomand and Russo (2011) proved that dimM(L) = 1/2(n - 1)(n - 2) + 1 - s(L), where M(L) is the Schur multiplier of L and s(L) is a non-negative integer. They also characterized the structure of L, when s(L) = 0. Assume that (N, L) is a pair of finite dimensional nilpotent Lie algebras, in which L is non-abelian and N is an ideal in L and also M(N, L) is the Schur multiplier of the pair (N, L). If N admits a complement K say, in L such that dimK = m, then dimM(N, L) = 1/2(n(2) + 2nm - 3n - 2m + 2) + 1 - (s(L) - t(K)), where t(K) = 1/2m (m - 1)-dimM(K). In the present paper, we characterize the pairs (N, L), for which 0 <= t(K) <= s(L) <= 3. In particular, we classify the pairs (N, L) such that L is a non-abelian filiform Lie algebra and 0 <= t(K) <= s(L) <= 17.