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

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 $\varphi :R\to -R-\lambda \mathrm{id}$, a symmetrized Rota-Baxter operator of weight $\lambda$ on $M_2(F)$.

12 p

References in corpus (3)

• Spectrum of Rota-Baxter operators
• Nonunital decompositions of the matrix algebra of order three
• Rota-Baxter operators of nonzero weight on the split Cayley-Dickson algebra

Cited by in corpus (1)

• Kupershmidt-Nijenhuis structures on pre-Malcev algebras