From affine algebraic racks to Leibniz algebras and Yang-Baxter operators
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arXiv
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866918297742606336 |
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| author | Ta, Luc |
| author_facet | Ta, Luc |
| contents | We introduce analogues of algebraic groups called algebraic racks, which are pointed rack objects in the category of schemes over a ground field. Addressing a problem of Loday, we construct functors assigning left and right Leibniz algebras to affine algebraic racks. These functors are compatible with closed subracks and ideals, and they recover the Lie algebras of linear algebraic groups (via conjugation quandles) and the Leibniz algebras of algebraic Lie racks. We also study properties of coordinate algebras and Leibniz algebras of affine algebraic racks. Finally, we use rack schemes to functorially construct (co-)nondegenerate Yang-Baxter operators in various categories. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_15909 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | From affine algebraic racks to Leibniz algebras and Yang-Baxter operators Ta, Luc Algebraic Geometry Group Theory Quantum Algebra Rings and Algebras Primary 14L99, Secondary 16T25, 17A32, 17B45, 18C40, 57K12 We introduce analogues of algebraic groups called algebraic racks, which are pointed rack objects in the category of schemes over a ground field. Addressing a problem of Loday, we construct functors assigning left and right Leibniz algebras to affine algebraic racks. These functors are compatible with closed subracks and ideals, and they recover the Lie algebras of linear algebraic groups (via conjugation quandles) and the Leibniz algebras of algebraic Lie racks. We also study properties of coordinate algebras and Leibniz algebras of affine algebraic racks. Finally, we use rack schemes to functorially construct (co-)nondegenerate Yang-Baxter operators in various categories. |
| title | From affine algebraic racks to Leibniz algebras and Yang-Baxter operators |
| topic | Algebraic Geometry Group Theory Quantum Algebra Rings and Algebras Primary 14L99, Secondary 16T25, 17A32, 17B45, 18C40, 57K12 |
| url | https://arxiv.org/abs/2512.15909 |