@article{Jiang2025, 
author = {Yan Jiang and Ying Hou and Keli Zheng},
title = {Classical and special Rota-Baxter operators on the real Lie algebra        s    o    (  3  )},
year = {2025},
journal = {Electronic Research Archive},
volume = {33},
number = {10},
pages = {6036-6057},
keywords = {Lie algebra so(3), classical Rota-Baxter operators, multiplicative Rota-Baxter operators, pseudo-Rota-Baxter operators, Yang-Baxter equations, left-symmetric algebra structures},
url = {https://www.sciopen.com/article/10.3934/era.2025268},
doi = {10.3934/era.2025268},
abstract = {This paper presents a classification of real matrix representations of Rota-Baxter operators on the real Lie algebra        s    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        s    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        ∈  [  0  ,  1  ] and        g          22        ∈  [      1    2    ,  1  ]. These findings contribute to the theoretical exploration of Rota-Baxter operators on real orthogonal Lie algebras by presenting explicit    3  ×  3 matrix solutions, which offer valuable tools for their practical implementation in robotic mechanisms.}
}