Multiplicative Lie higher derivations of unital algebras with idempotents

Let ℛ be a commutative ring with identity and A be a unital algebra with nontrivial idempotent e over ℛ. Motivated by Benkovič’s systematic and powerful work [2, 3, 4, 5, 6, 7, 8], we will study multiplicative Lie higher derivations (i.e. those Lie higher derivations without additivity assumption) on A in this article. Let D = {L_k}_k∈ℕ be a multiplicative Lie higher derivation on A. It is shown that under suitable assumptions, D = {L_k}_k∈ℕ is of standard form; i.e. each component L_k (k ≥ 1) can be expressed through an additive higher derivation and a central mapping vanishing on all commutators of A.