The description of Rota-Baxter operators of nonzero weight on complex general linear Lie algebra of order~$2$
Sibirskie èlektronnye matematičeskie izvestiâ, Tome 19 (2022) no. 2, pp. 870-879.

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In the paper, a classification of Rota-Baxter operators of weight $1$ on general linear complex Lie algebra of order $2$ is given. The description was made up to the action of $Aut(gl_2(\mathbb C))$.
Keywords: Lie algebra, Rota—Baxter operator, general linear Lie algebra.
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M. Goncharov. The description of Rota-Baxter operators of nonzero weight on complex general linear Lie algebra of order~$2$. Sibirskie èlektronnye matematičeskie izvestiâ, Tome 19 (2022) no. 2, pp. 870-879. http://geodesic.mathdoc.fr/item/SEMR_2022_19_2_a10/

[1] F.G. Tricomi, “On the finite Hilbert transformation”, Q. J. Math., Oxf. II. Ser., 2 (1951), 199–211 | DOI | MR | Zbl

[2] G. Baxter, “An analytic problem whose solution follows from a simple algebraic identity”, Pac. J. Math., 10 (1960), 731–742 | DOI | MR | Zbl

[3] F.V. Atkinson, “Some aspects of Baxter's functional equation”, J. Math. Anal. Appl., 7 (1963), 1–30 | DOI | MR | Zbl

[4] P. Cartier, “On the structure of free Baxter algebras”, Adv. Math., 9 (1972), 253–265 | DOI | MR | Zbl

[5] G.-C. Rota, “Baxter algebras and combinatorial identities. I”, Bull. Am. Math. Soc., 75 (1969), 325–329 | DOI | MR | Zbl

[6] A.A. Belavin, V.G. Drinfel'd, “Solutions of the classical Yang-Baxter equation for simple Lie algebras”, Funct. Anal. Appl., 16:3 (1983), 159–180 | DOI | MR | Zbl

[7] M.A. Semenov-Tyan-Shanskii, What is a classical r-matrix?, Funct. Anal. Appl., 17:4 (1983), 259–272 | DOI | MR | Zbl

[8] P.S. Kolesnikov, “Homogeneous averaging operators on simple finite conformal Lie algebras”, J. Math. Phys., 56:7 (2015), 071702 | DOI | MR | Zbl

[9] E.I. Konovalova, Double Lie algebras, PhD thesis, Samara, 2009

[10] Yu Pan, Q. Liu, C. Bai, L. Guo, “PostLie algebra structures on the Lie algebra $\mathrm{sl}(2,\mathbb{C})$”, Electron. J. Linear Algebra, 23 (2012), 180–197 | MR | Zbl

[11] J. Pei, C. Bai, L. Guo, “Rota-Baxter operators on $\mathrm{sl}(2,\mathbb{C})$ and solutions of the classical Yang-Baxter equation”, J. Math. Phys., 55:2 (2014), 021701 | DOI | MR | Zbl

[12] P. Benito, V. Gubarev, A. Pozhidaev, “Rota-Baxter operators on quadratic algebras”, Mediterr. J. Math., 15:5 (2018), 189 | DOI | MR | Zbl

[13] X. Tang, Y. Zhang, Q. Sun, “Rota-Baxter operators on 4-dimensional complex simple associative algebras”, Appl. Math. Comp., 229 (2014), 173–186 | DOI | MR | Zbl

[14] V.V. Sokolov, “Classification of constant solutions of the associative Yang-Baxter equation on $\mathrm{Mat}_3$”, Theor. Math. Phys., 176:3 (2013), 1156–1162 | DOI | MR | Zbl

[15] M. Goncharov, V. Gubarev, “Rota-Baxter operators of nonzero weight on the matrix algebra of order three”, Linear Multilinear Algebra, 70:6 (2022), 1055–1080 | DOI | MR | Zbl

[16] V. Gubarev, R. Kozlov, Rota-Baxter operators on $\mathrm{Cur}(sl_2(\mathbb C))$, 2022, arXiv: 2209.13141 [math.RA]