Generalized Lie-Type Derivations of Alternative Algebras

Abstract

In this paper we intend to describe generalized Lie-type derivations using, among other things, a generalization for alternative algebras of the result: “If $F:A\to A$ is a generalized Lie n-derivation associated with a Lie n-derivation D, then a linear map $H=F-D$ satisfies $H(p_n(x_1,x_2,\ldots ,x_n)) =p_n(H(x_1),x_2,\ldots ,x_n)$ for all $x_1,x_2,\ldots ,x_n\in A$”. Thus, if A is a unital alternative algebra with a nontrivial idempotent e₁ satisfying certain conditions, then a generalized Lie-type derivation $F : A \rightarrow A$ is of the form $F(x) = \lambda x + \Xi(x)$ for all $x \in A$, where $\lambda \in Z(A)$ and $\Xi : A \rightarrow A$ is a Lie-type derivation.