Biderivations and triple homomorphisms on perfect Jordan algebras

Abstract

In this paper, we mainly study a class of biderivations and triple homomorphisms on perfect Jordan algebras. Let $J$ be a Jordan algebra and $\delta :J\times J\to J$ a symmetric biderivation satisfying $\delta (w,u\circ v)=w\cdot \delta (u,v),\forall u,v,w\in J$. If $J$ is perfect and satisfies $Z(J)=\{0\}$, then $\delta$ is of the form $\delta (x,y)=\gamma (x\circ y)$ for all $x,y\in J$, where $\gamma \in Cent(J)$ satisfying $z\cdot \gamma (x\circ y)=x\cdot \gamma (y\circ z)+y\cdot \gamma (x\circ z),\forall x,y,z\in J$. This is the special case of our main theorem which concerns biderivations having their range in a $J$-module. What's more, we give an algorithm which can be applied to find biderivations satisfying $\delta (w,u\circ v)=w\cdot \delta (u,v),\forall u,v,w\in J$ on any Jordan algebra. We also show that for a triple homomorphism between perfect Jordan algebras, $f(x^2)=(f(x){)}^2$ or $f(x^2)=-(f(x){)}^2$. As an application, such $f$ is a homomorphism if and only if $f(x^2)=(f(x){)}^2$. Moreover, we give an algorithm which can be applied to any Jordan algebra.

15pages

References in corpus (1)

• Biderivations and commuting linear maps on Lie algebras