Jordan left {g, h}-derivations over some algebras

TL;DR

This paper introduces and characterizes left {g, h}-derivations and Jordan left {g, h}-derivations on certain algebras, showing their properties and differences through examples and structural results.

Contribution

It defines new derivation concepts, proves their non-existence in some algebras, and characterizes their structure over specific algebra classes.

Findings

No Jordan left {g, h}-derivation over matrix and quaternion algebras for g ≠ h.

Examples show not all Jordan left {g, h}-derivations are left {g, h}-derivations.

Jordan left {g, h}-derivations coincide with left {g, h}-derivations in tensor products and polynomial algebras.

Abstract

In this article, left {g, h}-derivation and Jordan left {g, h}-derivation on algebras are introduced. It is shown that there is no Jordan left {g, h}-derivation over $\mathcal{M}_n(C)$ and $\mathbb{H}_{\mathbb{R}}$, for g not equal to h. Examples are given which show that every Jordan left $\{g, h\}$-derivation over $\mathcal{T}_n(C)$, $\mathcal{M}_n(C)$ and $\mathbb{H}_{\mathbb{R}}$ are not left $\{g, h\}$-derivations. Moreover, we characterize left $\{g, h\}$-derivation and Jordan left $\{g, h\}$-derivation over $\mathcal{T}_n(C)$, $\mathcal{M}_n(C)$ and $\mathbb{H}_{\mathbb{R}}$ respectively. Also, we prove the result of Jordan left $\{g, h\}$-derivation to be a left $\{g, h\}$-derivation over tensor products of algebras as well as for algebra of polynomials.