Cohomologies of modified $ \lambda $-differential Lie triple systems and applications

Wen Teng; Fengshan Long; Yu Zhang; Wen Teng; Fengshan Long; Yu Zhang

doi:10.3934/math.20231280

AIMS Mathematics

2023, Volume 8, Issue 10: 25079-25096. doi: 10.3934/math.20231280

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Research article

Cohomologies of modified $ \lambda $-differential Lie triple systems and applications

School of Mathematics and Statistics, Guizhou University of Finance and Economics, Guiyang 550025, China

Received: 09 July 2023 Revised: 13 August 2023 Accepted: 21 August 2023 Published: 28 August 2023
MSC : 17A30, 17A42, 17B10, 17B56

In this paper, we introduce the concept and representation of modified $ \lambda $-differential Lie triple systems. Next, we define the cohomology of modified $ \lambda $-differential Lie triple systems with coefficients in a suitable representation. As applications of the proposed cohomology theory, we study 1-parameter formal deformations and abelian extensions of modified $ \lambda $-differential Lie triple systems.
- Lie triple system,
- modified $ \lambda $-differential operator,
- cohomology,
- deformation,
- extension
Citation: Wen Teng, Fengshan Long, Yu Zhang. Cohomologies of modified $ \lambda $-differential Lie triple systems and applications[J]. AIMS Mathematics, 2023, 8(10): 25079-25096. doi: 10.3934/math.20231280

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Abstract

In this paper, we introduce the concept and representation of modified $ \lambda $-differential Lie triple systems. Next, we define the cohomology of modified $ \lambda $-differential Lie triple systems with coefficients in a suitable representation. As applications of the proposed cohomology theory, we study 1-parameter formal deformations and abelian extensions of modified $ \lambda $-differential Lie triple systems.

References

[1]	N. Jacobson, Lie and Jordan triple systems, J. Am. Math. Soc., 71 (1949), 49–170. https://doi.org/10.2307/2372102 doi: 10.2307/2372102
[2]	N. Jacobson, General representation theory of Jordan algebras, T. Am. Math. Soc., 70 (1951), 509–530. https://doi.org/10.2307/1990612 doi: 10.2307/1990612
[3]	W. Lister, A structure theory of Lie triple systems, T. Am. Math. Soc., 72 (1952), 217–242. https://doi.org/10.2307/1990753 doi: 10.2307/1990753
[4]	K. Yamaguti, On the cohomology space of Lie triple system, Kumamoto J. Sci. Ser. A, 5 (1960), 44–52. https://doi.org/10.1007/s40840-016-0334-2 doi: 10.1007/s40840-016-0334-2
[5]	T. Hodge, B. Parshall, On the representation theory of Lie triple systems, T. Am. Math. Soc., 354 (2002), 4359–4391. https://doi.org/10.2307/3072903 doi: 10.2307/3072903
[6]	B. Harris, Cohomology of Lie triple systems and Lie algebras with involution, T. Am. Math. Soc., 98 (1961), 148–162. https://doi.org/10.2307/1993515 doi: 10.2307/1993515
[7]	F. Kubo, Y. Taniguchi, A controlling cohomology of the deformation theory of Lie triple systems, J. Algebra, 278 (2004), 242–250. https://doi.org/10.1016/j.jalgebra.2004.01.005 doi: 10.1016/j.jalgebra.2004.01.005
[8]	J. Lin, Y. Wang, S. Deng, $T^$-extension of Lie triple systems, Linear Algebra Appl., 431* (2009), 2071–2083. https://doi.org/10.1016/j.laa.2009.07.001 doi: 10.1016/j.laa.2009.07.001
[9]	T. Zhang, Notes on cohomologies of Lie triple systems, J. Lie Theory, 24 (2014), 909–929.
[10]	J. F. Ritt, Differential algebra, AMS Colloquium Publications, 1950. https://doi.org/10.1007/978-94-010-0854-9-5
[11]	T. Voronov, Higher derived brackets and homotopy algebras, J. Pure Appl. Algebra, 202 (2005), 133–153. https://doi.org/10.1016/j.jpaa.2005.01.010 doi: 10.1016/j.jpaa.2005.01.010
[12]	V. Coll, M. Gerstenhaber, A. Giaquinto, An explicit deformation formula with noncommuting derivations, Ring Theory, 1989,396–403.
[13]	A. R. Magid, Lectures on differential Galois theory, University Lecture Series, American Mathematical Society, 7 (1994).
[14]	V. Ayala, E. Kizil, I. A. Tribuzy, On an algorithm for finding derivations of Lie algebras, Proyecciones, 31 (2012), 81–90. https://doi.org/10.4067/S0716-09172012000100008 doi: 10.4067/S0716-09172012000100008
[15]	V. Ayala, J. Tirao, Linear control systems on Lie groups and controllability, Proc. Sympos. Pure Math., 64, (1999). https://doi.org/10.1090/pspum/064/1654529
[16]	I. Batalin, G. Vilkovisky, Gauge algebra and quantization, Phys. Lett. B, 102 (1981), 27–31. https://doi.org/10.1016/0370-2693(81)90205-7 doi: 10.1016/0370-2693(81)90205-7
[17]	S. Y. Chou, C. Attanayake, C. Thapa, A homotopy perturbation method for a class of truly nonlinear oscillators, Ann. Math. Sci. Appl., 6 (2021), 3–23. https://doi.org/10.4310/AMSA.2021.v6.n1.a1 doi: 10.4310/AMSA.2021.v6.n1.a1
[18]	Y. Lin, Y. Wei, Q. Ye, A homotopy method for multikernel-based approximation, J. Nonlinear Var. Anal., 6 (2022), 139–154.
[19]	E. Kolchin, Differential algebra and algebraic groups, Academic Press, 1973.
[20]	M. Singer, M. V. Put, Galois theory of linear differential equations, Springer, 2003. https://doi.org/10.1017/CBO9780511721564.002
[21]	L. Loday, On the operad of associative algebras with derivation, Georgian Math. J., 17 (2010), 347–372. https://doi.org/10.1515/GMJ.2010.010 doi: 10.1515/GMJ.2010.010
[22]	R. Tang, Y. Frégier, Y. Sheng, Cohomologies of a Lie algebra with a derivation and applications, J. Algebra, 534 (2019), 65–99. https://doi.org/10.1016/j.jalgebra.2019.06.007 doi: 10.1016/j.jalgebra.2019.06.007
[23]	A. Das, Leibniz algebras with derivations, J. Homotopy Relat. Str., 16 (2021), 245–274. https://doi.org/10.1007/s40062-021-00280-w doi: 10.1007/s40062-021-00280-w
[24]	X. Wu, Y. Ma, B. Sun, L. Chen, Cohomology of Leibniz triple systems with derivations, J. Geom. Phys., 179 (2022), 104594. https://doi.org/10.1016/j.geomphys.2022.104594 doi: 10.1016/j.geomphys.2022.104594
[25]	X. Wu, Y. Ma, B. Sun, L. Chen, Abelian extensions of Lie triple systems with derivations, Electron. Res. Arch., 30 (2022), 1087–1103. https://doi.org/10.3934/era.2022058 doi: 10.3934/era.2022058
[26]	Q. Sun, S. Chen, Cohomologies and deformations of Lie triple systems with derivations, J. Algebra Appl., 2024 (2024), 2450053. https://doi.org/10.1142/S0219498824500531 doi: 10.1142/S0219498824500531
[27]	S. Guo, Central extensions and deformations of Lie triple systems with a derivation, J. Math. Res. Appl., 42 (2022), 189–198. https://doi.org/:10.3770/j.issn:2095-2651.2022.02.009
[28]	R. Bai, L. Guo, J. Li, Y. Wu, Rota-Baxter $3$-Lie algebras, J. Math. Phys., 54 (2013), 063504. https://doi.org/10.1063/1.4808053 doi: 10.1063/1.4808053
[29]	L. Guo, W. Keigher, On differential Rota-Baxter algebras, J. Pure Appl. Algebra, 212 (2008), 522–540. https://doi.org/10.1016/j.jpaa.2007.06.008 doi: 10.1016/j.jpaa.2007.06.008
[30]	L. Guo, G. Regensburger, M. Rosenkranz, On integro-differential algebras, J. Pure Appl. Algebra, 218 (2014), 456–471. https://doi.org/10.1016/j.jpaa.2013.06.015 doi: 10.1016/j.jpaa.2013.06.015
[31]	L. Guo, Y. Li, Y. Sheng, G. Zhou, Cohomology, extensions and deformations of differential algebras with any weights, Theor. Appl. Categ., 38 (2020), 1409–1433. https://doi.org/10.48550/arXiv.2003.03899 doi: 10.48550/arXiv.2003.03899
[32]	A. Das, Cohomology and deformations of weighted Rota-Baxter operators, J. Math. Phys., 63 (2022), 091703. https://doi.org/10.1063/5.0093066 doi: 10.1063/5.0093066
[33]	K. Wang, G. Zhou, Deformations and homotopy theory of Rota-Baxter algebras of any weight, arXiv Preprint, 2021. https://doi.org/10.48550/arXiv.2108.06744
[34]	A. Das, Cohomology of weighted Rota-Baxter Lie algebras and Rota-Baxter paired operators, arXiv Preprint, 2021. https://doi.org/10.48550/arXiv.2109.01972
[35]	S. Hou, Y. Sheng, Y. Zhou, 3-post-Lie algebras and relative Rota-Baxter operators of nonzero weight on 3-Lie algebras, J. Algebra, 615 (2023), 103–129. https://doi.org/10.1016/j.jalgebra.2022.10.016 doi: 10.1016/j.jalgebra.2022.10.016
[36]	S. Guo, Y. Qin, K. Wang, G. Zhou, Deformations and cohomology theory of Rota-Baxter $3$-Lie algebras of arbitrary weights, J. Geom. Phys., 183 (2023), 104704. https://doi.org/10.1016/j.geomphys.2022.104704 doi: 10.1016/j.geomphys.2022.104704
[37]	S. Chen, Q. Lou, Q. Sun, Cohomologies of Rota-Baxter Lie triple systems and applications, Commun. Algebra, 51 (2023), 1–17. https://doi.org/10.1080/00927872.2023.2205938 doi: 10.1080/00927872.2023.2205938
[38]	Y. Li, D. Wang, Lie algebras with differential operators of any weights, Electron. Res. Arch., 31 (2022), 1195–1211. https://doi.org/10.3934/era.2023061 doi: 10.3934/era.2023061
[39]	A. Das, A cohomological study of modified Rota-Baxter algebras, arXiv Preprint, 2022. https://doi.org/10.48550/arXiv.2207.02273
[40]	Y. Li, D. Wang, Cohomology and Deformation theory of Modified Rota-Baxter Leibniz algebras, arXiv Preprint, 2022. https://doi.org/10.48550/arXiv.2211.09991
[41]	B. Mondal, R. Saha, Cohomology of modified Rota-Baxter Leibniz algebra of weight $\kappa$, arXiv Preprint, 2022. https://doi.org/10.48550/arXiv.2211.07944
[42]	J. Jiang, Y. Sheng, Cohomologies and deformations of modified $r$-matrices, arXiv Preprint, 2022. https://doi.org/10.48550/arXiv.2206.00411
[43]	X. Peng, Y. Zhang, X. Gao, Y. Luo, Universal enveloping of (modified) $\lambda$-differential Lie algebras, Linear Multilinear A., 70 (2022), 1102–1127. https://doi.org/10.1080/03081087.2020.1753641 doi: 10.1080/03081087.2020.1753641
[44]	J. Zhou, L. Chen, Y. Ma, Generalized derivations of Lie triple systems, B. Malays. Math. Sci. So., 41 (2018), 637–656. https://doi.org/10.1007/s40840-016-0334-2 doi: 10.1007/s40840-016-0334-2
[45]	Y. Sheng, J. Zhao, Relative Rota-Baxter operators and symplectic structures on Lie-Yamaguti algebras, Commun. Algebra, 50 (2022), 4056–4073. https://doi.org/10.1080/00927872.2022.2057517 doi: 10.1080/00927872.2022.2057517

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