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

Let L bean n-dimensional non-abelian nilpotent Lie algebra. Niroomad and Russo(2011)proved that dim M(L) = 1 2 (n − 1)(n − 2) +1 − s(L), where M(L) is the Schurmultiplier of L and s(L) isanon-negativeinteger.Theyalsocharacterizedthestructureof L, when s(L) =0.Assumethat(N,L) isapairoffinitedimensional nilpotentLiealgebras,inwhich L is non-abelianand N isanideal in L and also M(N,L) istheSchurmultiplierofthepair(N,L). If N admitsacomplement K say,in L suchthatdimK = m, then dimM(N,L) = 1 2 (n2 + 2nm − 3n − 2m + 2)+1 − (s(L) − t(K)),where t(K) = 1 2 m(m − 1)−dimM(K). In the present paper,we characterize the pairs(N,L), forwhich0  t(K)  s(L)  3. Inparticular,weclas- sifythepairs(N,L) such that L is anon-abelian filiform Lie algebra and 0  t(K)  s(L)  17.