Centroid structures of n-Lie algebras

We concern the centroid Γ(L) of an n-Lie algebra L over a field F of characteristic p⩾0. Let C(L) be the central derivations of L. We obtain the following results.(1)If p is not a factor of n-1, then C(L) is the intersection of Γ(L) and the derivation algebra of L.(2)Let B be a nonzero ideal of L and invariant under Γ(L). Then the vanishing ideal of B is isomorphic to a subspace of Hom(L/B,ZL(B), where ZL(B) is the centralizer of B.(3)Suppose L=L1⊕L2 with L1,L2 ideals of L. Then Γ(L1) and Γ(L2) are components of Γ(L).(4)If L is a Heisenberg n-Lie algebra over an algebraically closed field of characteristic 0, then Γ(L) is generated by central derivations and scalars, and C(L) is made up of all the inner derivations of L.(5)If dimL⩾2 and Γ(L) consists of scalars, then the centroid of the tensor product of an associative algebra and L is the same as the tensor product of the centroids of the two algebras.