$\delta$-derivations of simple Jordan algebras and superalgebras

TL;DR

This paper classifies $rac{1}{2}$-derivations of simple Jordan algebras and superalgebras over algebraically closed fields, showing that non-zero $rac{1}{2}$-derivations are rare and providing a complete description of them.

Contribution

It provides a comprehensive classification of $rac{1}{2}$-derivations for simple Jordan algebras and superalgebras, highlighting the absence of non-zero $rac{1}{2}$-derivations for other $rac{1}{2}$ values.

Findings

Non-zero $rac{1}{2}$-derivations are fully characterized.

No non-zero $rac{1}{2}$-derivations exist for $ eq rac{1}{2}$, 0, 1.

Complete description of $rac{1}{2}$-derivations in these algebra classes.

Abstract

We describe non-trivial $\delta$-derivations of semisimple finite-dimensional Jordan algebras over an algebraically closed field of characteristic not 2, and of simple finite-dimensional Jordan superalgebras over an algebraically closed field of characteristic 0. For these classes of algebras and superalgebras, non-zero $\delta$-derivations are shown to be missing for $\delta\neq 0,{1}{2},1$, and we give a complete account of ${1}{2}$-derivations.