Exterior direct sum n-Lie algebras

For a given n-Lie algebra A, n-Lie algebra structures on the vector space An = { (x1, · · · , xn) | xi ∈ A, 1 ≤ i ≤ n } are constructed. For each s ≥ 2, the n-Lie product [, · · · , ]s is defined on the vector space An , which is called the exterior direct sum n-Lie algebra of the n-Lie algebra A. And it is proved that, n-Lie algebra A can be embedded into its exterior direct sum n-Lie algebras. And some ideals and subalgebras of (An , [, · · · , ]s) are obtained.