# Derivations on semi-simple Jordan algebras and its applications

## TL;DR

This paper investigates the structure of derivation algebras of semi-simple Jordan algebras over characteristic zero fields, establishing conditions for their simplicity and exploring their automorphism properties.

## Contribution

It provides new criteria for the simplicity of derivation algebras and characterizes derivations and inner derivations in semi-simple Jordan algebras.

## Key findings

- Derivation algebras are simple under certain conditions.
- Equivalence of various derivation-related algebras for semi-simple Jordan algebras.
- Characterization of derivations in Jordan algebras with finite basis.

## Abstract

In this paper, we mainly study the derivation algebras of semi-simple Jordan algebras over a field of characteristic $0$ and give sufficient and necessary conditions that the derivation algebras of them are simple. As an application, we prove that for a semi-simple Jordan algebra $J$, $TDer(Der(J)) = Der(Der(J)) = ad(Der(J))$ under some assumptions. Moreover, we also show that for a semi-simple Jordan algebra $J$ which has a finite basis over a field of characteristic $0$, $TDer(J) = Der(J) = Inn(J)$. This is a corollary about our theorem which concerns Jordan algebras with unit.