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Super Kupershmidt Operators on Pre-malcev Superalgebras

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Abstract

The main purpose of this paper is to study a generalization of pre-Malcev algebras and pre-Lie superalgebras, which is called pre-Malcev superalgebras. We first introduce the definition of pre-Malcev superalgebras with representations, and obtain some important properties and constructions of the super Kupershmidt operators on pre-Malcev superalgebras. Then we give the relationship among pre-Malcev superalgebras, alternative superalgebras and M-dendriform superalgebras. Furthermore, we give infinitesimal deformations of super Kupershmidt operators. Finally, we define bimodules of pre-Malcev superalgebras and study their relationships with super Kupershmidt operators.

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References

  1. Abdaoui E., Mabrouk S., Makhlouf A., Rota–Baxter operators on pre-Lie superalgebras. Bull. Malays. Math. Sci. Soc., 2019, 42(4): 1567–1606

    Article  MathSciNet  Google Scholar 

  2. Albuquerque H., Malcev superalgebras. Math. Appl., 1994, 303: 1–7

    MathSciNet  Google Scholar 

  3. Albuquerque H., Benayadi S., Quadratic Malcev superalgebras. J. Pure Appl. Algebra, 2004, 187(1–3): 19–45

    Article  MathSciNet  Google Scholar 

  4. Albuquerque H., Elduque A., Classification of Mal’tsev superalgebras of small dimensions. Algebra and Logic, 1996, 35(6): 351–365

    Article  MathSciNet  Google Scholar 

  5. Bai C.M., Bellier O., Guo L., Ni X., Splitting of operations, Manin products, and Rota–Baxter operators. Int. Math. Res. Not. IMRN, 2013, 2013(3): 485–524

    Article  MathSciNet  Google Scholar 

  6. Balavoine D., Deformations of algebras over a quadratic operads. Contemp. Math., 1997, 202: 207–234

    Article  MathSciNet  Google Scholar 

  7. Burde D., Simple left-symmetric algebras with solvable Lie algebra. Manuscripta Math., 1998, 95(3): 397–411

    Article  MathSciNet  Google Scholar 

  8. Dorfman I., Dirac Structures and Integrability of Nonlinear Evolution Equations. Nonlinear Sci. Theory Appl., Chichester: John Wiley & Sons, Ltd., 1993, xii+176 pp.

    Google Scholar 

  9. Elduque A., A note on semiprime Malcev superalgebras. Proc. Roy. Soc. Edinburgh Sect. A, 1993, 123(5): 887–891

    Article  MathSciNet  Google Scholar 

  10. Elhamdadi M., Makhlouf A., Deformations of Hom-alternative and Hom-Malcev algebras. Algebras Groups Geom., 2011, 28(2): 117–145

    MathSciNet  Google Scholar 

  11. Gerstenhaber M., The cohomology structure of an associative ring. Ann. of Math. (2), 1963, 78: 267–288

    Article  MathSciNet  Google Scholar 

  12. Hegazi A.S., Abdelwahab H., Calderon Martin A.J., The classification of n-dimensional non-Lie Malcev algebras with (n–4)-dimensional annihilator. Linear Algebra Appl., 2016, 505: 32–56

    Article  MathSciNet  Google Scholar 

  13. Hu Y.W., Liu J.F., Sheng Y.H., Kupershmidt-(dual-)Nijenhuis structures on a Lie algebra with a representation. J. Math. Phys., 2018, 59(8): 081702, 14 pp.

    Article  MathSciNet  Google Scholar 

  14. Kupershmidt B., What a classical r-matrix really is. J. Nonlinear Math. Phys., 1999, 6(4): 448–488

    Article  MathSciNet  Google Scholar 

  15. Kuzmin E.N., Malcev algebras and their representations. Algebra i Logika, 1968, 7(4): 48–69

    MathSciNet  Google Scholar 

  16. Mabrouk S., Deformation of Kupershmidt operators and Kupershmidt–Nijenhuis structures of a Malcev algebra. Hacet. J. Math. Stat., 2022, 51(1): 199–217

    Article  MathSciNet  Google Scholar 

  17. Madariaga S., Splitting of operations for alternative and Malcev structures. Comm. Algebra, 2017, 45(1): 183–197

    Article  MathSciNet  Google Scholar 

  18. Magri F., Morosi C., On the reduction theory of the Nijenhuis operators and its applications to Gel’fand-Dikiĭ equations. Atti Accad. Sci. Torino Cl. Sci. Fis. Mat. Natur., 1983, 117: 599–626

    MathSciNet  Google Scholar 

  19. Sagle A., Malcev algebras. Trans. Amer. Math. Soc., 1961, 101: 426–458

    Article  MathSciNet  Google Scholar 

  20. Trushina M.N., Shestakov I.P., Representations of alternative algebras and superalgebras. J. Math. Sci. (N.Y.), 2012, 185(3): 504–512

    Article  MathSciNet  Google Scholar 

  21. Wang Q., Sheng Y.H., Bai C.M., Liu J.F., Nijenhuis operators on pre-Lie algebras. Commun. Contemp. Math., 2019, 21(7): 1850050, 37 pp.

    Article  MathSciNet  Google Scholar 

  22. Zhao Y.N., Chen L.Y., Super Kupershmidt operators and Kupershmidt–Nijenhuis structures on Malcev superalgebras. Comm. Algebra, 2023, 51(11): 4686–4701

    Article  MathSciNet  Google Scholar 

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Acknowledgements

This work was supported by NSF of Jilin Province (No. YDZJ202201-ZYTS589), NNSF of China (Nos. 12271085, 12071405) and the Fundamental Research Funds for the Central Universities.

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  1. School of Mathematics and Statistics, Northeast Normal University, Changchun, 130024, China

    Yinuo Zhao & Liangyun Chen

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  1. Yinuo Zhao
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  2. Liangyun Chen
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Correspondence to Liangyun Chen.

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Conflict of Interest The authors declare no conflict of interest.

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Zhao, Y., Chen, L. Super Kupershmidt Operators on Pre-malcev Superalgebras. Front. Math (2024). https://doi.org/10.1007/s11464-023-0045-6

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