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