Lie triple derivations of dihedron algebra

Abstract

Let K be a 2-torsion free unital ring and D(K) be dihedron algebra over K. In the present article, we prove that every Lie triple derivation of D(K) can be written as the sum of the Lie triple derivation of K, Jordan triple derivation of K, and some inner derivation of D. We also prove that a generalized Lie triple derivation ϱ:D(K)→D(K) associated with the Lie triple derivation h:D(K)→D(K) exists if ϱ can be represented in the form ϱ(τ) = h(τ) + λτ, where λ lies in the center of D(K). We finally conclude that to obtain the complete algebra of the Lie triple derivation and generalized Lie triple of D(K), we first need to find the Lie triple derivation and Jordan triple derivation of K.