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arXiv · 2502.18866

Rota-Baxter operators on the simple Jordan algebra of matrices of order two

Vsevolod Gubarev·Alexander Panasenko
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Abstract

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 $\lambda$ 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 $\lambda$ on $M_2(F)^{(+)}$ is either a Rota-Baxter operator of weight $\lambda$ on $M_2(F)$ or is, up to the action of $\phi\colon R\to -R-\lambda\mathrm{id}$, a symmetrized Rota-Baxter operator of weight $\lambda$ on $M_2(F)$.

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Vsevolod Gubarev, Alexander Panasenko. 2025-02-26. Rota-Baxter operators on the simple Jordan algebra of matrices of order two. https://doi.org/10.1007/s40840-025-01932-3

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