On the multiplier of nilpotent n-Lie algebras

In this article, we show that all finite dimensional special Heisenberg n  -Lie algebras have same dimension, say mn+1mn+1, for some natural number m, and are isomorphism with them. Also, for a nilpotent n-Lie algebra A of dimension d   and dim⁡A2=mdim⁡A2=m, we find the upper bound dim⁡M(A)≤(d−m+1n)+(m−2)(d−mn−1)+n−m, where M(A)M(A) denotes the multiplier of A  . Moreover, in the case where m=1m=1, the equality holds if and only if A≅H(n,1)⊕F(d−n−1)A≅H(n,1)⊕F(d−n−1), where F(d−n−1)F(d−n−1) is an abelian n  -Lie algebra of dimension of d−n−1d−n−1 and H(n,1)H(n,1) is the special Heisenberg n  -Lie algebra of dimension n+1n+1.