On the right nucleus of Petit algebras

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Owen, Adam (2022) On the right nucleus of Petit algebras. PhD thesis, University of Nottingham.

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Abstract

Let D be division algebra over its center C, let σ be an endormorphism of D, let δ be a left σ-derivation of D, and let R=D[t; σ, δ] be a skew polynomial ring. We study the structure of a class of nonassociative algebras, denoted by Sf, whose construction canonically generalises that of the associative quotient algebras R/Rf where f ∈ R is right-invariant.

We determine the structure of the right nucleus of Sf when the polynomial f is bounded and not right invariant and either δ = 0, or σ = idD. As a by-product, we obtain a new proof on the size of the right nuclei of the cyclic (Petit) semifields Sf.

We look at subalgebras of the right nucleus of Sf, generalising several of Petit's results [Pet66] 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 Sf when f is not right-invariant. Moreover, we compute the right nucleus of Sf 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 [Ami54]. We find examples of polynomials whose eigenring is a central simple algebra over the field C ∩ Fix(σ) ∩ Const(δ).

Item Type: Thesis (University of Nottingham only) (PhD)
Supervisors: Pumpluen, Susanne
Keywords: Nonassociative Algebras, Skew Polynomial Rings, A-polynomials, Central Simple Algebras, Division Rings, Space-time Block Codes
Subjects: Q Science > QA Mathematics > QA150 Algebra
Faculties/Schools: UK Campuses > Faculty of Science > School of Mathematical Sciences
Item ID: 68418
Depositing User: Owen, Adam
Date Deposited: 02 Aug 2022 04:40
Last Modified: 02 Aug 2022 04:40
URI: https://eprints.nottingham.ac.uk/id/eprint/68418

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