on schur multiplier of a pair of lie algebras

Nilpotent lie algebras have played an important role in mathematics in the classification theory oflie algebras. let (n, l) be a pair of finite dimensional lie algebras. let k be an ideal of l suchthat l = n ⊕ k and n be a filiform ideal of l. also, let dim n = n and dim k = m. thens′(n, l) = 1/2(n − 1)(n − 2) + 1 + (n − 1)m − dim m(n, l). in this paper, we characterize the pair(n, l) for s′(n; l) = 0, 1, 2,..., 15.