Classical and special Rota-Baxter operators on the real Lie algebra so(3)

i

This article has been updated. A Correction to this article was published on .

i

The Original Article was published on

i

The Original Article was published on

i

This article has been updated

i

This article was retracted on

i

This article has been updated

Abstract

This paper presents a classification of real matrix representations of Rota-Baxter operators on the real Lie algebra 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 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 g₁₁ ∈ [0, 1] and g₂₂ ∈ [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.

Keywords

Lie algebra so(3) / classical Rota-Baxter operators / multiplicative Rota-Baxter operators / pseudo-Rota-Baxter operators / Yang-Baxter equations / left-symmetric algebra structures

CHANGE HISTORY

A Correction to this paper has been published:
https://doi.org

The Retraction Note was updated to reflect reception of statements of disagreement from all authors.

This article has been retracted.Please see the Retraction Notice for more detail:
https://doi.org

References

[1] R. M. Murray, Z. Li, S. S. Sastry, A Mathematical Introduction to Robotic Manipulation, 1st 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

[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

[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

[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

[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

[8] L. Guo, W. Keigher, Baxter algebras and shuffle products, Adv. Math., 150 (2000), 117-149. https://doi.org/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

[10] J. Pei, C. Bai, L. Guo, Rota-Baxter operators on sl(2, C) and solutions of the classical Yang-Baxter equation, J. Math. Phys., 55 (2014), 021701. http://doi.org/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

[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

[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

[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

[15] A. Connes, D. Kreimer, Renormalization in quantum field theory and the Riemann-Hilbert problem II: the β-function, diffeomorphisms and the renormalization group, Commun. Math. Phys., 216 (2001), 215-241. http://doi.org/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

[17] Y. Pan, Q. Liu, C. Bai, L. Guo, Post-Lie algebra structures on the Lie algebra SL(2, C), Electron. J. Linear Algebra, 23 (2012), 180-197. http://doi.org/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

[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

[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

[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

[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.