Parametric set-theoretic Yang-Baxter equation: p-racks, solutions & quantum algebras
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arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866918326225076224 |
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| author | Doikou, Anastasia |
| author_facet | Doikou, Anastasia |
| contents | The theory of the parametric set-theoretic Yang-Baxter equation is established from a purely algebraic point of view. The first step towards this objective is the introduction of certain generalizations of the familiar shelves and racks called parametric (p)-shelves and racks. These objects satisfy a parametric self-distributivity condition and lead to solutions of the Yang-Baxter equation. Novel, non-reversible solutions are obtained from p-shelf/rack solutions by a suitable parametric twist, whereas all reversible set-theoretic solutions are reduced to the identity map via a parametric twist. The universal algebras associated to both p-rack and generic parametric, set-theoretic solutions are next presented and the corresponding universal R-matrices are derived. The admissible universal Drinfel'd twist is constructed allowing the derivation of the general set-theoretic universal R-matrix. By introducing the concept of a parametric coproduct we prove the existence of a parametric co-associativity. We show that the parametric coproduct is an algebra homomorphism and the universal R-matrices satisfy intertwining relations with the algebra coproducts |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_04088 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Parametric set-theoretic Yang-Baxter equation: p-racks, solutions & quantum algebras Doikou, Anastasia Mathematical Physics Quantum Algebra Rings and Algebras The theory of the parametric set-theoretic Yang-Baxter equation is established from a purely algebraic point of view. The first step towards this objective is the introduction of certain generalizations of the familiar shelves and racks called parametric (p)-shelves and racks. These objects satisfy a parametric self-distributivity condition and lead to solutions of the Yang-Baxter equation. Novel, non-reversible solutions are obtained from p-shelf/rack solutions by a suitable parametric twist, whereas all reversible set-theoretic solutions are reduced to the identity map via a parametric twist. The universal algebras associated to both p-rack and generic parametric, set-theoretic solutions are next presented and the corresponding universal R-matrices are derived. The admissible universal Drinfel'd twist is constructed allowing the derivation of the general set-theoretic universal R-matrix. By introducing the concept of a parametric coproduct we prove the existence of a parametric co-associativity. We show that the parametric coproduct is an algebra homomorphism and the universal R-matrices satisfy intertwining relations with the algebra coproducts |
| title | Parametric set-theoretic Yang-Baxter equation: p-racks, solutions & quantum algebras |
| topic | Mathematical Physics Quantum Algebra Rings and Algebras |
| url | https://arxiv.org/abs/2405.04088 |