Generalized derivations, quasiderivations and centroids of ternary Jordan algebras

TL;DR

This paper explores the structure of ternary Jordan algebras, focusing on the relationships between generalized derivations, quasiderivations, and centroids, and how these concepts interact within the algebraic framework.

Contribution

It establishes the decomposition of generalized derivation algebras into quasiderivations and centroids, and shows how quasiderivations can be embedded as derivations in larger algebras.

Findings

Generalized derivation algebras are sums of quasiderivation algebras and centroids.

Centroids are ideals within generalized derivation algebras.

Quasiderivations can be embedded as derivations in larger ternary Jordan algebras.

Abstract

In this paper, we give some construction about ternary Jordan algebras at first. Next we study relationships between generalized derivations, quasiderivations and centroids of ternary Jordan algebras. We show that for ternary Jordan algebras, generalized derivation algebras are the sum of quasiderivation algebras and centroids where centroids are ideals of generalized derivation algebras. We also prove that quasiderivations can be embedded into larger ternary Jordan algebras as derivations. In particular, we also determine dimensions of ternary Jordan algebras in the case of all linear transformations are quasiderivations. Some properties about centroids of ternary Jordan algebras are also displayed.