Self-distributive structures, braces & the Yang-Baxter equation
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866917284500471808 |
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| author | Doikou, Anastasia |
| author_facet | Doikou, Anastasia |
| contents | The theory of the set-theoretic Yang-Baxter equation is reviewed from a purely algebraic point of view. We recall certain algebraic structures called shelves, racks and quandles. These objects satisfy a self-distributivity condition and lead to solutions of the Yang-Baxter equation. The quantum algebra as well as the integrability associated to Baxterized involutive set-theoretic solutions is briefly discussed. We then present the theory of the universal algebras associated to rack and general set-theoretic solutions. We show that these are quasi-triangular Hopf algebras and we derive the universal set-theoretic Drinfel'd twist. It is shown that this is an admissible twist allowing the derivation of the universal set-theoretic R-matrix. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_20479 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Self-distributive structures, braces & the Yang-Baxter equation Doikou, Anastasia Mathematical Physics Quantum Algebra Quantum Physics The theory of the set-theoretic Yang-Baxter equation is reviewed from a purely algebraic point of view. We recall certain algebraic structures called shelves, racks and quandles. These objects satisfy a self-distributivity condition and lead to solutions of the Yang-Baxter equation. The quantum algebra as well as the integrability associated to Baxterized involutive set-theoretic solutions is briefly discussed. We then present the theory of the universal algebras associated to rack and general set-theoretic solutions. We show that these are quasi-triangular Hopf algebras and we derive the universal set-theoretic Drinfel'd twist. It is shown that this is an admissible twist allowing the derivation of the universal set-theoretic R-matrix. |
| title | Self-distributive structures, braces & the Yang-Baxter equation |
| topic | Mathematical Physics Quantum Algebra Quantum Physics |
| url | https://arxiv.org/abs/2409.20479 |