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| Natura: | Preprint |
| Pubblicazione: |
2026
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| Soggetti: | |
| Accesso online: | https://arxiv.org/abs/2603.20793 |
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| _version_ | 1866908905287712768 |
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| author | Zhu, Haoran |
| author_facet | Zhu, Haoran |
| contents | We show that whenever \[ [\,\cdot,\cdot]_t = [\,\cdot,\cdot]_0 + t[\,\cdot,\cdot]_1,\qquad α_t = \mathrm{id} + tα_1 \] define an infinitesimal Hom--Lie deformation of $\mathfrak{sl}_2(\mathbb K)$ over $\mathbb K[t]/(t^2)$ and $(\mathfrak{sl}_2(\mathbb K),[\,\cdot,\cdot]_0,α_1)$ is a Hom--Lie algebra, then the deformed bracket $[\,\cdot,\cdot]_t$ satisfies the ordinary Jacobi identity over $\mathbb K[t]$. This solves a conjecture of Makhlouf and Silvestrov from 2010. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_20793 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Infinitesimal deformations of $\mathfrak{sl}_2$ with a twisted Jacobi identity Zhu, Haoran Rings and Algebras Mathematical Physics Quantum Algebra We show that whenever \[ [\,\cdot,\cdot]_t = [\,\cdot,\cdot]_0 + t[\,\cdot,\cdot]_1,\qquad α_t = \mathrm{id} + tα_1 \] define an infinitesimal Hom--Lie deformation of $\mathfrak{sl}_2(\mathbb K)$ over $\mathbb K[t]/(t^2)$ and $(\mathfrak{sl}_2(\mathbb K),[\,\cdot,\cdot]_0,α_1)$ is a Hom--Lie algebra, then the deformed bracket $[\,\cdot,\cdot]_t$ satisfies the ordinary Jacobi identity over $\mathbb K[t]$. This solves a conjecture of Makhlouf and Silvestrov from 2010. |
| title | Infinitesimal deformations of $\mathfrak{sl}_2$ with a twisted Jacobi identity |
| topic | Rings and Algebras Mathematical Physics Quantum Algebra |
| url | https://arxiv.org/abs/2603.20793 |