{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,27]],"date-time":"2026-01-27T18:11:16Z","timestamp":1769537476126,"version":"3.49.0"},"reference-count":36,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2026,1,22]],"date-time":"2026-01-22T00:00:00Z","timestamp":1769040000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"crossref","award":["12571042"],"award-info":[{"award-number":["12571042"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"crossref"}]},{"DOI":"10.13039\/501100004731","name":"Natural Science Foundation of Zhejiang Province","doi-asserted-by":"crossref","award":["LZ25A010002"],"award-info":[{"award-number":["LZ25A010002"]}],"id":[{"id":"10.13039\/501100004731","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>In this paper, we describe all Nijenhuis operators on 2-dimensional complex pre-Lie algebras and 3-dimensional complex associative algebras. As an application, using these operators, we obtain solutions to the classical Yang-Baxter equation on the corresponding sub-adjacent Lie algebras.<\/jats:p>","DOI":"10.3390\/axioms15010080","type":"journal-article","created":{"date-parts":[[2026,1,26]],"date-time":"2026-01-26T16:13:48Z","timestamp":1769444028000},"page":"80","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Nijenhuis Operators on 2D Pre-Lie Algebras and 3D Associative Algebras"],"prefix":"10.3390","volume":"15","author":[{"given":"Xiaoguang","family":"Zou","sequence":"first","affiliation":[{"name":"Zhejiang College, Shanghai University of Finance and Economics, Jinhua 321013, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Xiang","family":"Gao","sequence":"additional","affiliation":[{"name":"School of Mathematical Sciences, Zhejiang Normal University, Jinhua 321004, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Chuangchuang","family":"Kang","sequence":"additional","affiliation":[{"name":"School of Mathematical Sciences, Zhejiang Normal University, Jinhua 321004, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2637-142X","authenticated-orcid":false,"given":"Jiafeng","family":"L\u00fc","sequence":"additional","affiliation":[{"name":"School of Mathematical Sciences, Zhejiang Normal University, Jinhua 321004, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2026,1,22]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"200","DOI":"10.1016\/S1385-7258(51)50028-8","article-title":"Xn\u22121-forming sets of eigenvectors","volume":"54","author":"Nijenhuis","year":"1951","journal-title":"Indag. Math."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"108001","DOI":"10.1016\/j.aim.2021.108001","article-title":"Nijenhuis Geometry","volume":"394","author":"Bolsinov","year":"2022","journal-title":"Adv. Math."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Wells, R.O. (1980). Differential Analysis on Complex Manifolds, Springer. Graduate Texts in Mathematics.","DOI":"10.1007\/978-1-4757-3946-6"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"59","DOI":"10.2307\/1970484","article-title":"On the deformation of rings and algebras","volume":"79","author":"Gerstenhaber","year":"1964","journal-title":"Ann. Math."},{"key":"ref_5","first-page":"89","article-title":"Deformations of Lie algebra structures","volume":"17","author":"Nijenhuis","year":"1967","journal-title":"J. Math. Mech."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"7267","DOI":"10.1088\/0305-4470\/37\/29\/007","article-title":"Dirac-Nijenhuis Structures","volume":"37","year":"2004","journal-title":"J. Phys. A"},{"key":"ref_7","unstructured":"Dorfman, I. (1993). Dirac Structures and Integrability of Nonlinear Evolution Equations, John Wiley & Sons."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"1850050","DOI":"10.1142\/S0219199718500505","article-title":"Nijenhuis operators on pre-Lie algebras","volume":"21","author":"Wang","year":"2019","journal-title":"Commun. Contemp. Math."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"659","DOI":"10.1088\/0253-6102\/65\/6\/659","article-title":"Nijenhuis operators on n-Lie algebras","volume":"65","author":"Liu","year":"2016","journal-title":"Commun. Theor. Phys."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"1038","DOI":"10.1080\/00927872.2021.1977942","article-title":"Nijenhuis operators on Hom-Lie algebras","volume":"50","author":"Das","year":"2022","journal-title":"Commun. Algebra"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"625","DOI":"10.1007\/s00574-011-0032-5","article-title":"Nijenhuis structures on Courant algebroids","volume":"42","year":"2011","journal-title":"Bull. Braz. Math. Soc."},{"key":"ref_12","first-page":"713","article-title":"Nijenhuis operators on Banach homogeneous spaces","volume":"35","author":"Larotonda","year":"2024","journal-title":"Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"105057","DOI":"10.1016\/j.geomphys.2023.105057","article-title":"Nijenhuis operators on Leibniz algebras","volume":"196","author":"Mondal","year":"2024","journal-title":"J. Geom. Phys."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"150","DOI":"10.1016\/j.jalgebra.2023.09.047","article-title":"Nijenhuis operators and associative D-bialgebras","volume":"639","author":"Ma","year":"2024","journal-title":"J. Algebra"},{"key":"ref_15","first-page":"127","article-title":"The constructions and deformations of Nijenhuis operators on 3-Hom-Lie algebras","volume":"62","author":"Li","year":"2025","journal-title":"J. Korean Math. Soc."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"172","DOI":"10.1080\/14786445708642275","article-title":"On the theory of analytic forms called trees","volume":"13","author":"Cayley","year":"1857","journal-title":"Philos. Mag."},{"key":"ref_17","first-page":"340","article-title":"Convex homogeneous cones","volume":"12","author":"Vinberg","year":"1963","journal-title":"Trans. Mosc. Math. Soc."},{"key":"ref_18","first-page":"445","article-title":"Domaines born\u00e9s homog\u00e8nes et orbites de groupes de transformations affines","volume":"91","author":"Koszul","year":"1963","journal-title":"Bull. Soc. Math. Fr."},{"key":"ref_19","first-page":"445","article-title":"Flat left-invariant connections adapted to the automorphism structure of a Lie group","volume":"16","author":"Medina","year":"1981","journal-title":"J. Differ. Geom."},{"key":"ref_20","unstructured":"Winterhalder, A. (1997). Linear Nijenhuis-tensors and the construction of integrable systems. arXiv."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"20170430","DOI":"10.1098\/rsta.2017.0430","article-title":"Open problems, questions and challenges in finite-dimensional integrable systems","volume":"376","author":"Bolsinov","year":"2018","journal-title":"Philos. Trans. R. Soc. A"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"1016","DOI":"10.1080\/00927870802279030","article-title":"Bijective 1-cocycles and classification of 3-dimensional left-symmetric algebras","volume":"37","author":"Bai","year":"2009","journal-title":"Comm. Algebra"},{"key":"ref_23","first-page":"397","article-title":"Simple left-symmetric algebras with solvable Lie algebra","volume":"95","author":"Burde","year":"1998","journal-title":"Manuscr. Math."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"373","DOI":"10.4310\/jdg\/1214440553","article-title":"Complete left-invariant affine structures on nilpotent Lie groups","volume":"24","author":"Kim","year":"1986","journal-title":"J. Differ. Geom."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"1641","DOI":"10.1080\/00927872.2011.554472","article-title":"On real left-symmetric algebras","volume":"40","author":"Kong","year":"2012","journal-title":"Comm. Algebra"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"189","DOI":"10.1016\/0550-3213(81)90414-4","article-title":"Dynamical symmetry of integrable quantum systems","volume":"180","author":"Belavin","year":"1981","journal-title":"Nucl. Phys. B"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"11","DOI":"10.1070\/RM1979v034n05ABEH003909","article-title":"The quantum inverse scattering method of the inverse problem and the Heisenberg XYZ model","volume":"34","author":"Faddeev","year":"1979","journal-title":"Russ. Math. Surv."},{"key":"ref_28","unstructured":"Faddeev, L.D., and Takhtajan, L.A. (2007). Hamiltonian Methods in the Theory of Solitons, Springer. Classics in Mathematics."},{"key":"ref_29","first-page":"259","article-title":"What is a classical R-matrix?","volume":"17","year":"1983","journal-title":"Funct. Anal. Appl."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"11073","DOI":"10.1088\/1751-8113\/40\/36\/007","article-title":"A unified algebraic approach to the classical Yang-Baxter equation","volume":"40","author":"Bai","year":"2007","journal-title":"J. Phys. A Math. Theor."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"021701","DOI":"10.1063\/1.4863898","article-title":"Rota-Baxter operators on sl(2,C) and solutions of the classical Yang-Baxter equation","volume":"55","author":"Pei","year":"2014","journal-title":"J. Math. Phys."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"104287","DOI":"10.1016\/j.geomphys.2021.104287","article-title":"The algebraic and geometric classification of nilpotent left-symmetric algebras","volume":"167","author":"Adashev","year":"2021","journal-title":"J. Geom. Phys."},{"key":"ref_33","first-page":"2207","article-title":"The classification of left-symmetric algebras in dimension 2","volume":"41","author":"Bai","year":"1996","journal-title":"Chin. Sci. Bull."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"6036","DOI":"10.3934\/era.2025268","article-title":"Classical and special Rota-Baxter operators on the real Lie algebra so(3)","volume":"33","author":"Jiang","year":"2025","journal-title":"Electron. Res. Arch."},{"key":"ref_35","first-page":"81","article-title":"Rota-Baxter operators on 3-dimensional Lie algebras and solutions of the classical Yang-Baxter equation","volume":"35","author":"Cheng","year":"2019","journal-title":"Commun. Math. Res."},{"key":"ref_36","doi-asserted-by":"crossref","unstructured":"Bai, C. (2020). 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