# The automorphisms of Petit's algebras

## TL;DR

This paper investigates the automorphisms of a class of nonassociative algebras generalizing certain quotient algebras, providing explicit automorphism group descriptions under specific conditions and exploring isomorphism criteria.

## Contribution

It computes automorphism groups of Petit algebras when the automorphism commutes with all automorphisms of the base field and extends results to finite Galois extensions.

## Key findings

- Automorphism groups are fully characterized when $\sigma$ commutes with all automorphisms.
- Partial results are obtained for cases where the order of $\sigma$ is less than the degree of $f$.
- In finite Galois extensions, detailed structure of automorphism groups is described.

## Abstract

Let $\sigma$ be an automorphism of a field $K$ with fixed field $F$. We study the automorphisms of nonassociative unital algebras which are canonical generalizations of the associative quotient algebras $K[t;\sigma]/fK[t;\sigma]$ obtained when the twisted polynomial $f\in K[t;\sigma]$ is invariant, and were first defined by Petit. We compute all their automorphisms if $\sigma$ commutes with all automorphisms in ${\rm Aut}_F(K)$ and $n\geq m-1$, where $n$ is the order of $\sigma$ and $m$ the degree of $f$,and obtain partial results for $n<m-1$. In the case where $K/F$ is a finite Galois field extension, we obtain more detailed information on the structure of the automorphism groups of these nonassociative unital algebras over $F$. We also briefly investigate when two such algebras are isomorphic.