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 $\delta :J \times J \rightarrow J$ 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.