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.
Citation: Yan Jiang, Ying Hou, Keli Zheng. Classical and special Rota-Baxter operators on the real Lie algebra $ \mathfrak{so}(3) $[J]. Electronic Research Archive, 2025, 33(10): 6036-6057. doi: 10.3934/era.2025268
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.
| [1] | R. M. Murray, Z. Li, S. S. Sastry, A Mathematical Introduction to Robotic Manipulation, $1^{st}$ edition, CRC Press, 1994. https://doi.org/10.1201/9781315136370 |
| [2] | T. D. Barfoot, State Estimation for Robotics, Cambridge University Press, 2017. https://doi.org/10.1017/9781316671528 |
| [3] |
P. Sun, Y. Li, K. Chen, W. Zhu, Q. Zhong, B. Chen, Generalized kinematics analysis of hybrid mechanisms based on screw theory and Lie groups Lie algebras, Chin. J. Mech. Eng., 34 (2021), 98. https://doi.org/10.1186/s10033-021-00610-2 doi: 10.1186/s10033-021-00610-2
|
| [4] |
G. E. Baxter, An analytic problem whose solution follows from a simple algebraic identity, Pac. J. Math., 10 (1960), 731–742. https://doi.org/10.2140/pjm.1960.10.731 doi: 10.2140/pjm.1960.10.731
|
| [5] |
G. Rota, Baxter algebras and combinatorial identities. Ⅰ, Bull. Am. Math. Soc., 75 (1969), 325–329. http://doi.org/10.1090/S0002-9904-1969-12156-7 doi: 10.1090/S0002-9904-1969-12156-7
|
| [6] |
G. Rota, Baxter algebras and combinatorial identities. Ⅱ, Bull. Am. Math. Soc., 75 (1969), 330–334. https://doi.org/10.1090/s0002-9904-1969-12158-0 doi: 10.1090/s0002-9904-1969-12158-0
|
| [7] |
P. Cartier, On the structure of free Baxter algebras, Adv. Math., 9 (1972), 253–265. https://doi.org/10.1016/0001-8708(72)90018-7 doi: 10.1016/0001-8708(72)90018-7
|
| [8] |
L. Guo, W. Keigher, Baxter algebras and shuffle products, Adv. Math., 150 (2000), 117–149. https://doi.org/10.1006/aima.1999.1858 doi: 10.1006/aima.1999.1858
|
| [9] |
X. Gao, L. Guo, M. Rosenkranz, On rings of differential Rota-Baxter operators, Int. J. Algebra Comput., 28 (2018), 1–36. https://doi.org/10.1142/S0218196718500017 doi: 10.1142/S0218196718500017
|
| [10] |
J. Pei, C. Bai, L. Guo, Rota-Baxter operators on $\mathfrak{sl}(2, \mathbb{C})$ and solutions of the classical Yang-Baxter equation, J. Math. Phys., 55 (2014), 021701. http://doi.org/10.1063/1.4863898 doi: 10.1063/1.4863898
|
| [11] |
L. Wu, M. Wang, Y. Cheng, Rota-Baxter operators on 3-dimensional Lie algebras and the classical $R$-matrices, Adv. Math. Phys., 2017 (2017), 6128102. https://doi.org/10.1155/2017/6128102 doi: 10.1155/2017/6128102
|
| [12] |
T. Ma, H. Yang, L. Zhang, H. Zheng, Quasitriangular covariant monoidal bihom-bialgebras, associative monoidal bihom-Yang-Baxter equations and Rota-Baxter paired monoidal bihom-modules, Colloq. Math., 161 (2020), 189–221. http://doi.org/10.4064/cm7993-9-2019 doi: 10.4064/cm7993-9-2019
|
| [13] |
R. Winkel, Sequences of symmetric polynomials and combinatorial properties of tableaux, Adv. Math., 134 (1998), 46–89. https://doi.org/10.1006/aima.1997.1715 doi: 10.1006/aima.1997.1715
|
| [14] |
A. Connes, D. Kreimer, Renormalization in quantum field theory and the Riemann-Hilbert problem Ⅰ: the hopf algebra structure of graphs and the main theorem, Commun. Math. Phys., 210 (2000), 249–273. http://doi.org/10.1007/s002200050779 doi: 10.1007/s002200050779
|
| [15] |
A. Connes, D. Kreimer, Renormalization in quantum field theory and the Riemann-Hilbert problem Ⅱ: the $\beta$-function, diffeomorphisms and the renormalization group, Commun. Math. Phys., 216 (2001), 215–241. http://doi.org/10.1007/PL00005547 doi: 10.1007/PL00005547
|
| [16] |
X. Li, D. Hou, C. Bai, Rota-Baxter operators on pre-Lie algebras, J. Nonlinear Math. Phys., 14 (2007), 269–289. http://doi.org/10.2991/jnmp.2007.14.2.9 doi: 10.2991/jnmp.2007.14.2.9
|
| [17] |
Y. Pan, Q. Liu, C. Bai, L. Guo, Post-Lie algebra structures on the Lie algebra $SL(2, \mathbb{C})$, Electron. J. Linear Algebra, 23 (2012), 180–197. http://doi.org/10.13001/1081-3810.1514 doi: 10.13001/1081-3810.1514
|
| [18] |
P. Xu, X. Tang, Graded post-Lie algebra structures and homogeneous Rota-Baxter operators on the Schrodinger-Virasoro algebra, Electron. Res. Arch., 29 (2021), 2771–2789. http://doi.org/10.3934/era.2021013 doi: 10.3934/era.2021013
|
| [19] |
M. Kontsevich, Deformation quantization of Poisson manifolds, Lett. Math. Phys., 66 (2003), 157–216. http://doi.org/10.1023/B:MATH.0000027508.00421.bf doi: 10.1023/B:MATH.0000027508.00421.bf
|
| [20] |
X. Tang, Y. Zhang, Q. Sun, Rota-baxter operators on 4-dimensional complex simple associative algebras, Appl. Math. Comput., 229 (2014), 173–186. https://doi.org/10.1016/j.amc.2013.12.032 doi: 10.1016/j.amc.2013.12.032
|
| [21] |
V. Gubarev, Rota-Baxter operators of weight zero on the matrix algebra of order three without unit in kernel, J. Algebra, 683 (2025), 253–277. https://doi.org/10.1016/j.jalgebra.2025.06.019 doi: 10.1016/j.jalgebra.2025.06.019
|
| [22] | J. E. Humphreys, Introduction to Lie Algebras and Representation Theory, Springer-Verlag, 1972. https://doi.org/10.1007/978-1-4612-6398-2 |
| [23] | S. V. Skresanov. Rota-Baxter operators on compact simple Lie groups and algebras, preprint, arXiv: 2506.14324. |
Yan Jiang, Ying Hou, Keli Zheng. Classical and special Rota-Baxter operators on the real Lie algebra $ \mathfrak{so}(3) $[J]. Electronic Research Archive, 2025, 33(10): 6036-6057. doi: 10.3934/era.2025268
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