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 + 1, for some natural number m, and are isomorphism with them. Also, for a nilpotent n-Lie algebra A of dimension d and dim A(2) = m, we find the upper bound dim M(A) <= (GRAPHICS) + ((m - 2) (GRAPHICS) + n - m, where M(A) denotes the multiplier of A. Moreover, in the case where m = 1, the equality holds if and only if A congruent to H(n, 1) circle plus F(d - n - 1), where F(d - n - 1) is an abelian n-Lie algebra of dimension of d - n - 1 and H(n, 1) is the special Heisenberg n-Lie algebra of dimension n + 1. (C) 2015 Elsevier Inc. All rights reserved.