On the Right Nucleus of Petit Algebras

TL;DR

This paper investigates the structure of nonassociative algebras derived from skew polynomial rings, focusing on their right nuclei, and introduces generalized A-polynomials with applications in differential polynomial rings.

Contribution

It provides a detailed analysis of the right nucleus of certain nonassociative algebras, generalizes Petit's results, and introduces semi-invariant elements and generalized A-polynomials.

Findings

Determined the structure of the right nucleus for specific nonassociative algebras.

Established the equivalence of semi-invariant elements and the nucleus when $f$ is not right-invariant.

Constructed examples of generalized A-polynomials with central simple eigenrings.

Abstract

Let $D$ be division algebra over its center $C$, let $\sigma$ be an endormorphism of $D$, let $\delta$ be a left $\sigma$-derivation of $D$, and let $R=D[t;\sigma,\delta]$ be a skew polynomial ring. We study the structure of a class of nonassociative algebras, denoted by $S_f$, whose construction canonically generalises that of the associative quotient algebras $R/Rf$ where $f\in R$ is right-invariant. We determine the structure of the right nucleus of $S_f$ when the polynomial $f$ is bounded and not right invariant and either $\delta = 0$, or $\sigma = {\rm id}_D$. As a by-product, we obtain a new proof on the size of the right nuclei of the cyclic (Petit) semifields $\mathbb{S}_f$. We look at subalgebras of the right nucleus of $S_f$, generalising several of Petit's results \cite{petit1966certains} and introduce the notion of semi-invariant elements of the coefficient ring $D$. The set of semi-invariant elements is shown to be equal to the nucleus of $S_f$ when $f$ is not right-invariant. Moreover, we compute the right nucleus of $S_f$ for certain $f$. In the final chapter of this thesis we introduce and study a special class of polynomials in $R$ called generalised A-polynomials. In a differential polynomial ring over a field of characteristic zero, A-polynomials were originally introduced by Amitsur \cite{amitsur1954differential}. We find examples of polynomials whose eigenring is a central simple algebra over the field $C \cap {\rm Fix}(\sigma) \cap {\rm Const}(\delta)$.