Quandles as pre-Lie skew braces, set-theoretic Hopf algebras & universal R-matrices
| Autor: | Doikou, Anastasia, Rybolowicz, Bernard, Stefanelli, Paola |
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| Rok vydání: | 2024 |
| Předmět: | |
| Zdroj: | J. Phys. A: Math. Theor. 57 405203, 2024 |
| Druh dokumentu: | Working Paper |
| DOI: | 10.1088/1751-8121/ad7769 |
| Popis: | 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. Comment: 36 pages LaTex. Some general comments added in the introduction section. Minor typos corrected |
| Databáze: | arXiv |
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