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

TL;DR

This paper classifies all Rota-Baxter operators of any weight on the simple Jordan algebra of 2x2 matrices, revealing their relation to operators on the associative algebra and introducing symmetrized operators.

Contribution

It provides a complete description of Rota-Baxter operators on the simple Jordan algebra of 2x2 matrices, including new symmetrized operators and their relation to classical Rota-Baxter operators.

Findings

All Rota-Baxter operators of weight 0 are either on the associative algebra or symmetrized.

Nonzero weight Rota-Baxter operators are either on the associative algebra or symmetrized, up to an involution.

Classification extends to symmetrized operators, enriching the structure theory of Jordan algebras.

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 $M_2(F)$.