Nonlinear Lie higher derivations on triangular algebras

Abstract

Let ℛ be a commutative ring with identity, A, B be unital algebras over ℛ and M be a unital (A, B)-bimodule, which is faithful as a left A-module and also as a right B-module. Let be the triangular algebra consisting of A, B and M. Motivated by the powerful works of Brešar [M. Brešar, Commuting traces of biadditive mappings, commutativity-preserving mappings and Lie mappings, Trans. Amer. Math. Soc. 335 (1993), pp. 525–546] and Yu et al. [W.-Y. Yu and J.-H. Zhang, Nonlinear Lie derivations of triangular algebras, Linear Algebra Appl. 432 (2010), pp. 2953–2960], we will study nonlinear Lie higher derivations on 𝒯 in this article. Let D = {L n } n∈ℕ be a Lie higher derivation on 𝒯 without additive condition. Under mild assumptions, it is shown that D = {L n } n∈ℕ is of standard form, i.e. each component L n (n ≥ 1) can be expressed through an additive higher derivation and a nonlinear functional vanishing on all commutators of 𝒯. As applications, nonlinear Lie higher derivations on some classical triangular algebras are characterized.