The Commutator Subalgebra and Schur Multiplier of aPair of Nilpotent Lie Algebras

摘要

Let (L, N) be a pair of finite dimensional nilpotent Lie algebras, in which N is an ideal in L. In the present article, we prove that if the factor Lie algebras L/N and N/Z(L, N) are of dimensions m and n, respectively, then the commutator subalgebra [L, N] is of dimension at most 1/2n(n + 2m - 1),and also determine when dim([L, N]) =1/2n(n + 2m - 1). In addition, we introduce the notion of the Schur multiplier M(L,N) of an arbitrary pair (L, N) of Lie algebras, and show that if N admits a complement K in L with dim(N) = n and dim(K) = m, then the dimension of M(L,N) is bounded above by 1/2n(n + 2m - 1). In this case, we characterize the pairs (L, N) for which dim(M(L, N)) is either 1/2n(n + 2m - 1) or 1/2n(n + 2m — 1) — 1.