Classical and special Rota-Baxter operators on the real Lie algebra $\mathfrak{s}\mathfrak{o}(3)$

This paper presents a classification of real matrix representations of Rota-Baxter operators on the real Lie algebra $\mathfrak{s}\mathfrak{o}(3)$. We obtained all non-isomorphic classical Rota-Baxter operators of weight $0$ up to orthogonal similarity, as well as special Rota-Baxter operators (multiplicative and pseudo-Rota-Baxter operators) of weights $0$ and $1$. For weight $0$, we found three canonical symmetric matrix forms that solve the classical Yang-Baxter equation and used them to construct explicit left-symmetric algebra structures. Interestingly, no non-trivial multiplicative Rota–Baxter operators exist on $\mathfrak{s}\mathfrak{o}(3)$ for weights $0$ or $1$. Pseudo-Rota-Baxter operators exhibit a rich structure: for weight 1, we found two families of symmetric matrices, while for weight 0, there were five families parameterized by continuous parameters $g_{11}\in [0,1]$ and $g_{22}\in [\frac{1}{2},1]$. These findings contribute to the theoretical exploration of Rota-Baxter operators on real orthogonal Lie algebras by presenting explicit $3\times 3$ matrix solutions, which offer valuable tools for their practical implementation in robotic mechanisms.