Almost classical skew bracoids
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866913623284121600 |
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| author | Martin-Lyons, Isabel |
| author_facet | Martin-Lyons, Isabel |
| contents | We investigate two sub-classes of skew bracoids, the first consists of those we term almost a brace, meaning the multiplicative group decomposes as a certain semi-direct product, and then those that are almost classical, which additionally specifies the relationship between the multiplicative group and the additive. Skew bracoids with these properties have applications in Hopf-Galois theory, in particular for questions concerning the Hopf-Galois correspondence, and can also yield solutions to the set-theoretic Yang-Baxter equation. We use this skew bracoid perspective to give a new construction building on the induced Hopf-Galois structures of Crespo, Rio and Vela, recover a result of Greither and Pareigis on the Hopf-Galois correspondence, and examine the solutions that arise from skew bracoids, in particular where more than one solution may be drawn from a single skew bracoid. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2412_10268 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Almost classical skew bracoids Martin-Lyons, Isabel Rings and Algebras Group Theory 12F10, 16T05, 16T25, 20N99 We investigate two sub-classes of skew bracoids, the first consists of those we term almost a brace, meaning the multiplicative group decomposes as a certain semi-direct product, and then those that are almost classical, which additionally specifies the relationship between the multiplicative group and the additive. Skew bracoids with these properties have applications in Hopf-Galois theory, in particular for questions concerning the Hopf-Galois correspondence, and can also yield solutions to the set-theoretic Yang-Baxter equation. We use this skew bracoid perspective to give a new construction building on the induced Hopf-Galois structures of Crespo, Rio and Vela, recover a result of Greither and Pareigis on the Hopf-Galois correspondence, and examine the solutions that arise from skew bracoids, in particular where more than one solution may be drawn from a single skew bracoid. |
| title | Almost classical skew bracoids |
| topic | Rings and Algebras Group Theory 12F10, 16T05, 16T25, 20N99 |
| url | https://arxiv.org/abs/2412.10268 |