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

Abstract

This paper presents a classification of real matrix representations of Rota-Baxter operators on the real Lie algebra $ \mathfrak{so}(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{so}(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.