On the right nucleus of Petit algebras
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 associat...
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| Format: | Thesis (University of Nottingham only) |
| Language: | English |
| Published: |
2022
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| Online Access: | https://eprints.nottingham.ac.uk/68418/ |
| _version_ | 1848800485417418752 |
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| author | Owen, Adam |
| author_facet | Owen, Adam |
| author_sort | Owen, Adam |
| building | Nottingham Research Data Repository |
| collection | Online Access |
| description | 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). |
| first_indexed | 2025-11-14T20:52:19Z |
| format | Thesis (University of Nottingham only) |
| id | nottingham-68418 |
| institution | University of Nottingham Malaysia Campus |
| institution_category | Local University |
| language | English |
| last_indexed | 2025-11-14T20:52:19Z |
| publishDate | 2022 |
| recordtype | eprints |
| repository_type | Digital Repository |
| spelling | nottingham-684182022-08-02T04:40:10Z https://eprints.nottingham.ac.uk/68418/ On the right nucleus of Petit algebras Owen, Adam 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). 2022-08-02 Thesis (University of Nottingham only) NonPeerReviewed application/pdf en cc_by https://eprints.nottingham.ac.uk/68418/1/AdamOwenThesisFinal.pdf Owen, Adam (2022) On the right nucleus of Petit algebras. PhD thesis, University of Nottingham. Nonassociative Algebras Skew Polynomial Rings A-polynomials Central Simple Algebras Division Rings Space-time Block Codes |
| spellingShingle | Nonassociative Algebras Skew Polynomial Rings A-polynomials Central Simple Algebras Division Rings Space-time Block Codes Owen, Adam On the right nucleus of Petit algebras |
| title | On the right nucleus of Petit algebras |
| title_full | On the right nucleus of Petit algebras |
| title_fullStr | On the right nucleus of Petit algebras |
| title_full_unstemmed | On the right nucleus of Petit algebras |
| title_short | On the right nucleus of Petit algebras |
| title_sort | on the right nucleus of petit algebras |
| topic | Nonassociative Algebras Skew Polynomial Rings A-polynomials Central Simple Algebras Division Rings Space-time Block Codes |
| url | https://eprints.nottingham.ac.uk/68418/ |