On set-theoretic solutions of pentagon equation and positive basis Hopf algebras
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2026
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| _version_ | 1866912860019359744 |
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| author | Colazzo, Ilaria Janssens, Geoffrey |
| author_facet | Colazzo, Ilaria Janssens, Geoffrey |
| contents | We investigate the connection between bijective, not necessarily finite, set-theoretic solutions of the pentagon equation and Hopf algebras. Firstly, we prove that finite solutions correspond to Hopf algebras with the positive basis property. As a corollary we generalise Lu-Yan-Zhu classification to arbitrary characteristic $0$ fields $k$. Secondly, we study the general problem of when a Hopf algebra has a basis yielding a set-theoretic solution. Finally, we classify all (co)commutative bijective solutions. This result requires to obtain a description of all bases of a group algebra $k[G]$ yielding a set-theoretic solution. We namely show that such bases correspond, through a Fourier transform, to splittings $A \rtimes N$ of $G$ with $A$ a finite abelian group. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_22089 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On set-theoretic solutions of pentagon equation and positive basis Hopf algebras Colazzo, Ilaria Janssens, Geoffrey Rings and Algebras Quantum Algebra We investigate the connection between bijective, not necessarily finite, set-theoretic solutions of the pentagon equation and Hopf algebras. Firstly, we prove that finite solutions correspond to Hopf algebras with the positive basis property. As a corollary we generalise Lu-Yan-Zhu classification to arbitrary characteristic $0$ fields $k$. Secondly, we study the general problem of when a Hopf algebra has a basis yielding a set-theoretic solution. Finally, we classify all (co)commutative bijective solutions. This result requires to obtain a description of all bases of a group algebra $k[G]$ yielding a set-theoretic solution. We namely show that such bases correspond, through a Fourier transform, to splittings $A \rtimes N$ of $G$ with $A$ a finite abelian group. |
| title | On set-theoretic solutions of pentagon equation and positive basis Hopf algebras |
| topic | Rings and Algebras Quantum Algebra |
| url | https://arxiv.org/abs/2601.22089 |