In 1994, Moneyhun introduced and studied the concept of isoclinism in Lie algebras. Moghaddam and parvaneh in 2009 introduced and gave some properties of isoclinism of a pair of Lie algebras. In this paper we introduce the concept of relative n-isoclinism between two pairs of Lie algebras. They form equivalence classes and we show that each equivalent class contains an n-stem pair of Lie algebras. We also show that for a relative n-isoclinism family of Lie algebras C say, consists of (M, L) such that L is finitely generated and [M,n L] is finite dimension, then there exists an n-stem pair of Lie algebras (R, S) ∈ C such that dim([R,n−1 S] = min{dim([M,n−1 L]); f or all (M, L) ∈ C}. Mathematics Subject Classification (2010): Primary 17B30, 17B60, 17B99; Secondary 20E99, 20D15.Key words: Lie algebra, n-isoclinism, relative n-isoclinism, n-stem Lie algebra.
Introduction. We consider all Lie algebras over a fixed fieldF and assume M to be an ideal of a Lie algebra L, with the Lie product [ , ]. Then (M, L) is said to be a pair of Lie algebras. The lower and the upper central series of L are defined inductively by L 1 = L , L n+1 = [L n , L], for all n ≥ 1, and Z 0 (L) = {0}, Z 1 (L) = Z(L), and Z n+1 (L)/Z n (L) = Z(L/Z n (L)), for all n ≥ 0. Also one may consider Quaestiones Mathematicae is co-published by NISC (Pty) Ltd and Taylor & Francis 28 M.R.R. Moghaddam, F. Saeedi, S. Tajnia and B. Veisi [M, L] = ⟨[m, l]; m ∈ M, l ∈ L⟩, Z(M, L) = {m ∈ M ; [m, l] = 0, ∀l ∈ L} = Z(L) ∩ M , to be the commutator and the centre of the pair of Lie algebras (M, L), respectively. We also denote [M, L, . . . , L n−times
