Quandles as pre-Lie skew braces, set-theoretic Hopf algebras & universal R-matrices
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866910613041577984 |
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| author | Doikou, Anastasia Rybolowicz, Bernard Stefanelli, Paola |
| author_facet | Doikou, Anastasia Rybolowicz, Bernard Stefanelli, Paola |
| contents | We present connections between left non-degenerate solutions of the set-theoretic braid equation and left shelves using Drinfel'd homomorphisms. We generalize the notion of affine quandle, by using heap endomorphisms and metahomomorphisms, and identify the underlying Yang-Baxter algebra for solutions of the braid equation associated to a given quandle. We introduce the notion of the pre-Lie skew brace and identify certain affine quandles that give rise to pre-Lie skew braces. Generalisations of the braiding of a group, associated to set-theoretic solutions of the braid equation are also presented. These generalized structures encode part of the underlying Hopf algebra. We then introduce the quasi-triangular (quasi) Hopf algebras and universal R-matrices for rack and set-theoretic algebras. Generic set-theoretic solutions coming from heap endomorphisms are also identified. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_12704 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Quandles as pre-Lie skew braces, set-theoretic Hopf algebras & universal R-matrices Doikou, Anastasia Rybolowicz, Bernard Stefanelli, Paola Quantum Algebra Mathematical Physics Rings and Algebras We present connections between left non-degenerate solutions of the set-theoretic braid equation and left shelves using Drinfel'd homomorphisms. We generalize the notion of affine quandle, by using heap endomorphisms and metahomomorphisms, and identify the underlying Yang-Baxter algebra for solutions of the braid equation associated to a given quandle. We introduce the notion of the pre-Lie skew brace and identify certain affine quandles that give rise to pre-Lie skew braces. Generalisations of the braiding of a group, associated to set-theoretic solutions of the braid equation are also presented. These generalized structures encode part of the underlying Hopf algebra. We then introduce the quasi-triangular (quasi) Hopf algebras and universal R-matrices for rack and set-theoretic algebras. Generic set-theoretic solutions coming from heap endomorphisms are also identified. |
| title | Quandles as pre-Lie skew braces, set-theoretic Hopf algebras & universal R-matrices |
| topic | Quantum Algebra Mathematical Physics Rings and Algebras |
| url | https://arxiv.org/abs/2401.12704 |