Automorphisms and derivations of finite-dimensional algebras

TL;DR

This paper characterizes certain linear maps related to automorphisms and derivations of finite-dimensional algebras, showing how they decompose and relate to Jordan automorphisms, with implications for local automorphisms.

Contribution

It provides new structural descriptions of maps satisfying specific algebraic conditions, linking them to Jordan automorphisms and inner derivations in finite-dimensional algebras.

Findings

Maps satisfying $xD(x)x otin [A,A]$ decompose into inner derivations and radical maps.

Characterization of maps $T$ with $T(x)^3 - x^3 otin [A,A]$ as scaled Jordan automorphisms.

Every local Jordan automorphism of a simple algebra is a Jordan automorphism.

Abstract

Let $A$ be a finite-dimensional algebra over a field $F$ with char$(F)\ne 2$. We show that a linear map $D:A\to A$ satisfying $xD(x)x\in [A,A]$ for every $x\in A$ is the sum of an inner derivation and a linear map whose image lies in the radical of $A$. Assuming additionally that $A$ is semisimple and char$(F)\ne 3$, we show that a linear map $T:A\to A$ satisfies $T(x)^3- x^3 \in [A,A]$ for every $x\in A$ if and only if there exist a Jordan automorphism $J$ of $A$ lying in the multiplication algebra of $A$ and a central element $\alpha$ satisfying $\alpha^3=1$ such that $T(x)=\alpha J(x)$ for all $x\in A$. These two results are applied to the study of local derivations and local (Jordan) automorphisms. In particular, the second result is used to prove that every local Jordan automorphism of a finite-dimensional simple algebra $A$ (over a field $F$ with char$(F)\ne 2,3$) is a Jordan automorphism.