Comptes Rendus
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Cohomology and deformations of modified Rota–Baxter Lie–Yamaguti algebras
[Cohomologie et déformations des algèbres de Lie–Yamaguti de Rota–Baxter modifiées]
Comptes Rendus. Mathématique, Volume 363 (2025), pp. 641-662.

In this paper, first we introduce a notion of modified Rota–Baxter Lie–Yamaguti algebra of any weight. Also, we introduce the concept of a representation modified Rota–Baxter Lie–Yamaguti algebra of any weight. Then, we define a cohomology theory for modified Rota–Baxter Lie–Yamaguti algebras of any weight. As applications of the cohomology, we study formal deformations of modified Rota–Baxter Lie–Yamaguti algebras of arbitrary weights.

Dans cet article, nous introduisons d’abord la notion d’algèbre de Rota–Baxter Lie–Yamaguti modifiée de poids quelconque. Nous introduisons également le concept de représentation d’algèbre de Rota–Baxter Lie–Yamaguti modifiée de poids quelconque. Nous définissons ensuite une théorie de cohomologie pour les algèbres de Rota–Baxter Lie–Yamaguti modifiées de poids quelconque. Comme applications de la cohomologie, nous étudions les déformations formelles d’algèbres de Rota–Baxter Lie–Yamaguti modifiées de poids quelconque.

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DOI : 10.5802/crmath.743
Keywords: Lie–Yamaguti algebra, modified Rota–Baxter operator, representation, cohomology, deformation
Mots-clés : Algèbre de Lie–Yamaguti, opérateur de Rota–Baxter modifié, représentation, cohomologie, déformation

Khaled Basdouri 1 ; Sami Benabdelhafidh 1

1 University of Sfax, Faculty of Sciences of Sfax, BP 1171, 3038 Sfax, Tunisia
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits
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     title = {Cohomology and deformations of modified {Rota{\textendash}Baxter} {Lie{\textendash}Yamaguti} algebras},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {641--662},
     publisher = {Acad\'emie des sciences, Paris},
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Khaled Basdouri; Sami Benabdelhafidh. Cohomology and deformations of modified Rota–Baxter Lie–Yamaguti algebras. Comptes Rendus. Mathématique, Volume 363 (2025), pp. 641-662. doi : 10.5802/crmath.743. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.743/

[1] Chengming Bai; Li Guo; Xiang Ni Nonabelian generalized Lax pairs, the classical Yang–Baxter equation and post-Lie algebras, Commun. Math. Phys., Volume 297 (2010) no. 2, pp. 553-596 | DOI | MR | Zbl

[2] Imed Basdouri; Sami Benabdelhafidh; Wen Teng Cohomology of modified λ-differential Jacobi–Jordan algebras and its applications, Acta Comment. Univ. Tartu. Math., Volume 28 (2024) no. 2, pp. 215-232 | MR | Zbl

[3] Glen Baxter An analytic problem whose solution follows from a simple algebraic identity, Pac. J. Math., Volume 10 (1960), pp. 731-742 | DOI | MR | Zbl

[4] Pilar Benito; Murray Bremner; Sara Madariaga Symmetric matrices, orthogonal Lie algebras and Lie–Yamaguti algebras, Linear Multilinear Algebra, Volume 63 (2015) no. 6, pp. 1257-1281 | DOI | MR | Zbl

[5] Pilar Benito; Cristina Draper; Alberto Elduque Lie–Yamaguti algebras related to 𝔤2, J. Pure Appl. Algebra, Volume 202 (2005) no. 1-3, pp. 22-54 | DOI | MR | Zbl

[6] Pilar Benito; Alberto Elduque; Fabián Martín-Herce Irreducible Lie–Yamaguti algebras, J. Pure Appl. Algebra, Volume 213 (2009) no. 5, pp. 795-808 | DOI | MR | Zbl

[7] Pierre Cartier On the structure of free Baxter algebras, Adv. Math., Volume 9 (1972), pp. 253-265 | DOI | MR | Zbl

[8] Shan Chen; Qiong Lou; Qinxiu Sun Cohomologies of Rota–Baxter Lie triple systems and applications, Commun. Algebra, Volume 51 (2023) no. 10, pp. 4299-4315 | DOI | MR | Zbl

[9] Apurba Das Cohomology of weighted Rota–Baxter Lie algebras and Rota–Baxter paired operators (2021) | arXiv | Zbl

[10] Apurba Das A cohomological study of modified Rota–Baxter algebras (2022) | arXiv

[11] Apurba Das Cohomology and deformations of weighted Rota–Baxter operators, J. Math. Phys., Volume 63 (2022) no. 9, 091703, 16 pages | DOI | MR | Zbl

[12] Apurba Das; Samir Kumar Hazra; Satyendra Kumar Mishra Non-abelian extensions of Rota–Baxter Lie algebras and inducibility of automorphisms (2023) | arXiv | Zbl

[13] Kurusch Ebrahimi-Fard Loday-type algebras and the Rota–Baxter relation, Lett. Math. Phys., Volume 61 (2002) no. 2, pp. 139-147 | DOI | MR | Zbl

[14] Shuangjian Guo; Yufei Qin; Kai Wang; Guodong Zhou Deformations and cohomology theory of Rota–Baxter 3-Lie algebras of arbitrary weights, J. Geom. Phys., Volume 183 (2023), 104704, 24 pages | DOI | MR | Zbl

[15] Jun Jiang; Yunhe Sheng Deformations of modified r-matrices and cohomologies of related algebraic structures, J. Noncommut. Geom., Volume 19 (2025) no. 2, pp. 429-450 | DOI | MR | Zbl

[16] Michael K. Kinyon; Alan Weinstein Leibniz algebras, Courant algebroids, and multiplications on reductive homogeneous spaces, Am. J. Math., Volume 123 (2001) no. 3, pp. 525-550 | DOI | MR | Zbl

[17] Luen-Chau Li Classical r-matrices and compatible Poisson structures for Lax equations on Poisson algebras, Commun. Math. Phys., Volume 203 (1999) no. 3, pp. 573-592 | DOI | MR | Zbl

[18] Yizheng Li; Dingguo Wang Cohomology and deformation theory of modified Rota–Baxter Leibniz algebras (2022) | arXiv

[19] Jie Lin; Liangyun Chen; Yao Ma On the deformation of Lie–Yamaguti algebras, Acta Math. Sin., Engl. Ser., Volume 31 (2015) no. 6, pp. 938-946 | DOI | MR | Zbl

[20] Yao Ma; Liangyun Chen; Jie Lin One-parameter formal deformations of Hom–Lie–Yamaguti algebras, J. Math. Phys., Volume 56 (2015) no. 1, 011701, 12 pages | DOI | MR | Zbl

[21] Bibhash Mondal; Ripan Saha Cohomology of modified Rota–Baxter Leibniz algebra of weight λ (2023) | arXiv

[22] Katsumi Nomizu Invariant affine connections on homogeneous spaces, Am. J. Math., Volume 76 (1954), pp. 33-65 | DOI | MR | Zbl

[23] Nikolai Yu. Reshetikhin; Michael A. Semenov-Tian-Shansky Quantum R-matrices and factorization problems, J. Geom. Phys., Volume 5 (1988) no. 4, pp. 533-550 | DOI | MR | Zbl

[24] Alexey G. Reyman; Michael A. Semenov-Tian-Shansky Reduction of Hamiltonian systems, affine Lie algebras and Lax equations, Invent. Math., Volume 54 (1979), pp. 81-100 | DOI | Zbl

[25] Gian-Carlo Rota Baxter algebras and combinatorial identities. I, II, Bull. Am. Math. Soc., Volume 75 (1969), p. 325-329 and 330–334 | DOI | MR | Zbl

[26] Michael A. Semenov-Tian-Shansky What a classical r-matrix is, Funkts. Anal. Prilozh., Volume 17 (1983) no. 4, pp. 17-33 | MR

[27] Michael A. Semenov-Tian-Shansky Integrable systems and factorization problems, Factorization and integrable systems (Faro, 2000) (Operator Theory: Advances and Applications), Birkhäuser, 2003 no. 141, pp. 155-218 | DOI | MR | Zbl

[28] Yunhe Sheng; Jia Zhao Relative Rota–Baxter operators and symplectic structures on Lie–Yamaguti algebras, Commun. Algebra, Volume 50 (2022) no. 9, pp. 4056-4073 | DOI | MR | Zbl

[29] Yunhe Sheng; Jia Zhao; Yanqiu Zhou Nijenhuis operators, product structures and complex structures on Lie–Yamaguti algebras, J. Algebra Appl., Volume 20 (2021) no. 8, 2150146, 22 pages | DOI | MR | Zbl

[30] Nobuyoshi Takahashi Modules over geometric quandles and representations of Lie–Yamaguti algebras, J. Lie Theory, Volume 31 (2021) no. 4, pp. 897-932 | MR | Zbl

[31] Kai Wang; Guodong Zhou Deformations and homotopy theory of Rota–Baxter algebras of any weight (2021) | arXiv | Zbl

[32] Kiyosi Yamaguti On the cohomology space of Lie triple system, Kumamoto J. Sci., Math., Volume 5 (1960), pp. 44-52 | MR | Zbl

[33] Kiyosi Yamaguti On cohomology groups of general Lie triple systems, Kumamoto J. Sci., Math., Volume 8 (1967/69), pp. 135-146 | MR | Zbl

[34] Kiyosi Yamaguti On the Lie triple system and its generalization, J. Sci. Hiroshima Univ., Ser. A, Volume 21 (1957/58), pp. 155-160 | MR | Zbl

[35] Tao Zhang; Juan Li Deformations and extensions of Lie–Yamaguti algebras, Linear Multilinear Algebra, Volume 63 (2015) no. 11, pp. 2212-2231 | DOI | MR | Zbl

[36] Tao Zhang; Juan Li Representations and cohomologies of Hom–Lie–Yamaguti algebras with applications, Colloq. Math., Volume 148 (2017) no. 1, pp. 131-155 | DOI | MR | Zbl

[37] Xigou Zhang; Xing Gao; Li Guo Free modified Rota–Baxter algebras and Hopf algebras, Int. Electron. J. Algebra, Volume 25 (2019), pp. 12-34 | DOI | MR | Zbl

[38] Xigou Zhang; Xing Gao; Li Guo Modified Rota–Baxter algebras, shuffle products and Hopf algebras, Bull. Malays. Math. Sci. Soc., Volume 42 (2019) no. 6, pp. 3047-3072 | DOI | MR | Zbl

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