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arXiv · 1703.00718

The automorphisms of Petit's algebras

Christian Brown·Susanne Pumpluen
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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.

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Christian Brown, Susanne Pumpluen. 2017-03-02. The automorphisms of Petit's algebras. https://doi.org/10.1080/00927872.2017.1327598

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