The automorphisms of Petit's algebras

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 ${\mathrm{Aut}}_F(K)$ and $n\ge m-1$, 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.

Final version, to appear in Communications in Algebra

References in corpus (1)

• The nonassociative algebras used to build fast-decodable space-time block codes

Cited by in corpus (3)

• Nonassociative cyclic extensions of fields and central simple algebras
• How a nonassociative algebra reflects the properties of a skew polynomial
• Automorphisms and isomorphisms of Jha-Johnson semifields obtained from skew polynomial rings