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On Hom-associative ...
On Hom-associative Ore Extensions
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- Bäck, Per, 1987- (author)
- Mälardalens universitet,Utbildningsvetenskap och Matematik,MAM
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- Silvestrov, Sergei, Professor, 1970- (thesis advisor)
- Mälardalens universitet,Utbildningsvetenskap och Matematik
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- Richter, Johan, Docent (thesis advisor)
- Blekinge tekniska högskola, Institutionen för matematik och naturvetenskap
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- Hellström, Lars, Doktor, 1974- (thesis advisor)
- Mälardalens universitet,Utbildningsvetenskap och Matematik
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- Malyarenko, Anatoliy, Professor, 1957- (thesis advisor)
- Mälardalens universitet,Utbildningsvetenskap och Matematik
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- Arnlind, Joakim, Professor (opponent)
- Linköpings universitet
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(creator_code:org_t)
- ISBN 9789174855531
- Västerås : Mälardalens universitet, 2022
- English.
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Series: Mälardalen University Press Dissertations, 1651-4238 ; 360
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Abstract
Subject headings
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- In this thesis, we introduce and study hom-associative Ore extensions. These are non-unital, non-associative, non-commutative polynomial rings in which the associativity condition is “twisted” by an additive group homomorphism. In particular, they are examples of hom-associative algebras, and they generalize the classical non-commutative polynomial rings introduced by Ore known as Ore extensions to the non-unital, hom-associative setting. At the same time, when the twisted associativity condition is null, they also generalize to the general non-unital, non-associative setting. We deduce necessary and sufficient conditions for hom-associative Ore extensions to exist, and construct concrete examples thereof. These include hom-associative generalizations of the quantum plane, the universal enveloping algebra of the two-dimensional non-abelian Lie algebra, and the first Weyl algebra, to name a few. The aforementioned algebras turn out to be formal hom-associative deformations of their associative counterparts, the latter two which cannot be formally deformed in the associative setting. Moreover, these are all weakly unital algebras, and we provide a way of embedding any multiplicative, non-unital hom-associative algebra into a multiplicative, weakly unital hom-associative algebra. This generalizes the classical unitalization of non-unital, associative algebras. We then study the hom-associative Weyl algebras in arbitrary characteristic, classify them up to isomorphism, and in the zero characteristic case, we prove that an analogue of the Dixmier conjecture is true. We also study hom-modules over hom-associative rings, and by doing so, we are able to prove a Hilbert's basis theorem for hom-associative Ore extensions. Our theorem includes as special cases both the classical Hilbert's basis theorem for Ore extensions and a Hilbert's basis theorem for unital, non-associative Ore extensions. Last, we construct examples of both hom-associative and non-associative Ore extensions which are all Noetherian by our theorem.
Subject headings
- NATURVETENSKAP -- Matematik -- Algebra och logik (hsv//swe)
- NATURAL SCIENCES -- Mathematics -- Algebra and Logic (hsv//eng)
Keyword
- Mathematics/Applied Mathematics
- matematik/tillämpad matematik
Publication and Content Type
- vet (subject category)
- dok (subject category)
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Bäck, Per, 1987-
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Silvestrov, Serg ...
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Richter, Johan, ...
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Hellström, Lars, ...
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Malyarenko, Anat ...
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Arnlind, Joakim, ...
- About the subject
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- NATURAL SCIENCES
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NATURAL SCIENCES
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and Mathematics
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and Algebra and Logi ...
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Mälardalen Unive ...
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Mälardalen University