Matched pairs and Yetter-Drinfeld braces
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866913746302009344 |
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| author | Ferri, Davide Sciandra, Andrea |
| author_facet | Ferri, Davide Sciandra, Andrea |
| contents | It is proven that a matched pair of actions on a Hopf algebra $H$ is equivalent to the datum of a Yetter-Drinfeld brace, which is a novel structure generalising Hopf braces. This improves a theorem by Angiono, Galindo and Vendramin, originally stated for cocommutative Hopf braces. These Yetter-Drinfeld braces produce Hopf algebras in the category of Yetter-Drinfeld modules over $H$, through an operation that generalises Majid's transmutation. A characterisation of Yetter-Drinfeld braces via 1-cocycles, in analogy to the one for Hopf braces, is given.
Every coquasitriangular Hopf algebra $H$ will be seen to yield a Yetter-Drinfeld brace, where the additional structure on $H$ is given by the transmutation. We compute explicit examples of Yetter-Drinfeld braces on the Sweedler's Hopf algebra, on the algebras $E(n)$, on $\mathrm{SL}_{q}(2)$, and an example in the class of Suzuki algebras. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_10009 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Matched pairs and Yetter-Drinfeld braces Ferri, Davide Sciandra, Andrea Quantum Algebra Category Theory Rings and Algebras Representation Theory Primary 16T05, Secondary 16T25, 18M15 It is proven that a matched pair of actions on a Hopf algebra $H$ is equivalent to the datum of a Yetter-Drinfeld brace, which is a novel structure generalising Hopf braces. This improves a theorem by Angiono, Galindo and Vendramin, originally stated for cocommutative Hopf braces. These Yetter-Drinfeld braces produce Hopf algebras in the category of Yetter-Drinfeld modules over $H$, through an operation that generalises Majid's transmutation. A characterisation of Yetter-Drinfeld braces via 1-cocycles, in analogy to the one for Hopf braces, is given. Every coquasitriangular Hopf algebra $H$ will be seen to yield a Yetter-Drinfeld brace, where the additional structure on $H$ is given by the transmutation. We compute explicit examples of Yetter-Drinfeld braces on the Sweedler's Hopf algebra, on the algebras $E(n)$, on $\mathrm{SL}_{q}(2)$, and an example in the class of Suzuki algebras. |
| title | Matched pairs and Yetter-Drinfeld braces |
| topic | Quantum Algebra Category Theory Rings and Algebras Representation Theory Primary 16T05, Secondary 16T25, 18M15 |
| url | https://arxiv.org/abs/2406.10009 |