Rota-Baxter operators on the simple Jordan algebra of matrices of order two
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arXiv
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| Format: | Preprint |
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2025
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| author | Gubarev, Vsevolod Panasenko, Alexander |
| author_facet | Gubarev, Vsevolod Panasenko, Alexander |
| contents | We describe all Rota-Baxter operators of any weight on the space of matrices from $M_2(F)$ considered under the product $a\circ b = (ab + ba)/2$ and usually denoted as $M_2(F)^{(+)}$. This algebra is known to be a simple Jordan one.
We introduce symmetrized Rota-Baxter operators of weight $λ$ and show that every Rota-Baxter operator of weight 0 on $M_2(F)^{(+)}$ either is a Rota-Baxter operator of weight 0 on $M_2(F)$ or is a symmetrized Rota-Baxter operator of weight 0 on the same $M_2(F)$.
We also prove that every Rota-Baxter operator of nonzero weight $λ$ on $M_2(F)^{(+)}$ is either a Rota-Baxter operator of weight $λ$ on $M_2(F)$ or is, up to the action of $ϕ\colon R\to -R-λ\mathrm{id}$, a symmetrized Rota-Baxter operator of weight $λ$ on $M_2(F)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_18866 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Rota-Baxter operators on the simple Jordan algebra of matrices of order two Gubarev, Vsevolod Panasenko, Alexander Rings and Algebras 16W99, 17C20 We describe all Rota-Baxter operators of any weight on the space of matrices from $M_2(F)$ considered under the product $a\circ b = (ab + ba)/2$ and usually denoted as $M_2(F)^{(+)}$. This algebra is known to be a simple Jordan one. We introduce symmetrized Rota-Baxter operators of weight $λ$ and show that every Rota-Baxter operator of weight 0 on $M_2(F)^{(+)}$ either is a Rota-Baxter operator of weight 0 on $M_2(F)$ or is a symmetrized Rota-Baxter operator of weight 0 on the same $M_2(F)$. We also prove that every Rota-Baxter operator of nonzero weight $λ$ on $M_2(F)^{(+)}$ is either a Rota-Baxter operator of weight $λ$ on $M_2(F)$ or is, up to the action of $ϕ\colon R\to -R-λ\mathrm{id}$, a symmetrized Rota-Baxter operator of weight $λ$ on $M_2(F)$. |
| title | Rota-Baxter operators on the simple Jordan algebra of matrices of order two |
| topic | Rings and Algebras 16W99, 17C20 |
| url | https://arxiv.org/abs/2502.18866 |