Generalized Lie-type derivations of alternative algebras

B. L. M. Ferreira ; G. C. De Moraes

Izvestiâ vysših učebnyh zavedenij. Matematika, no. 9 (2021), pp. 40-48

Résumé

In this paper, we intend to describe generalized Lie-type derivations using, among other things, a generalization for alternative algebras of the following 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 $e_1$ 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.

Détail

Keywords: alternative algebra, generalized Lie derivation.