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Quasi-trace functions on Lie algebras and their applications to 3-Lie algebras

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Abstract

We introduce the notion of quasi-trace functions on Lie algebras. As applications we study realizations of 3-dimensional and 4-dimensional 3-Lie algebras. Some comparison results on cohomologies of 3-Lie algebras and Leibniz algebras arising from quasi-trace functions are obtained.

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References

  1. R. K. Amayo: Quasi-ideals of Lie algebras., I. Proc. Lond. Math. Soc., III. Ser. 33 (1976), 28–36.

    Article  MathSciNet  Google Scholar 

  2. R. K. Amayo: Quasi-ideals of Lie algebras., II. Proc. Lond. Math. Soc., III. Ser. 33 (1976), 37–64.

    Article  MathSciNet  Google Scholar 

  3. J. Arnlind, A. Kitouni, A. Makhlouf, S. Silvestrov: Structure and cohomology of 3-Lie algebras induced by Lie algebras. Algebra, Geometry and Mathematical Physics. Springer Proceedings in Mathematics and Statistics 85. Springer, Berlin, 2014, pp. 123–144.

    MATH  Google Scholar 

  4. J. Arnlind, A. Makhlouf, S. Silvestrov: Ternary Hom-Nambu-Lie algebras induced by Hom-Lie algebras. J. Math. Phys. 51 (2010), Article ID 043515, 11 pages.

    Article  MathSciNet  Google Scholar 

  5. H. Awata, M. Li, D. Minic, T. Yoneya: On the quantization of Nambu brackets. J. High Energy Phys. 2001 (2001), Article ID 13, 17 pages.

    Article  MathSciNet  Google Scholar 

  6. R. Bai, C. Bai, J. Wang: Realizations of 3-Lie algebras. J. Math. Phys. 51 (2010), Article ID 063505, 12 pages.

    Article  MathSciNet  Google Scholar 

  7. D. Burde, C. Steinhoff: Classification of orbit closures of 4-dimensional complex Lie algebras. J. Algebra 214 (1999), 729–739.

    Article  MathSciNet  Google Scholar 

  8. R. Carter: Lie Algebras of Finite and Affine Type. Cambridge Studies in Advanced Mathematics 96. Cambridge Univesity Press, Cambridge, 2005.

    Google Scholar 

  9. Y. L. Daletskii, L. A. Takhtajan: Leibniz and Lie algebra structures for Nambu algebra. Lett. Math. Phys. 39 (1997), 127–141.

    Article  MathSciNet  Google Scholar 

  10. J. A. de Azcárraga, J. M. Izquierdo: n-ary algebras: A review with applications. J. Phys. A, Math. Theor. 43 (2010), Article ID 293001, 117 pages.

    Article  Google Scholar 

  11. J. Dixmier: Enveloping Algebras. Graduate Studies in Mathematics 11. American Mathematical Society, Providence, 1996.

    Google Scholar 

  12. W. A. Dudek: On some old and new problems in n-ary groups. Quasigroups Relat. Syst. 8 (2001), 15–36.

    MathSciNet  MATH  Google Scholar 

  13. K. Erdmann, M. J. Wildon: Introduction to Lie Algebras. Springer UndergraduateMathematics Series. Springer, London, 2006.

    Google Scholar 

  14. J. M. Figueroa-O’Farrill: Deformations of 3-algebras. J. Math. Phys. 50 (2009), Article ID 113514, 27 pages.

    Article  Google Scholar 

  15. V. T. Filippov: n-Lie algebras. Sib. Math. J. 26 (1985), 879–891.

    Article  Google Scholar 

  16. X. García-Martínez, R. Turdibaev, T. Van der Linden: Do n-Lie algebras have universal enveloping algebras?. J. Lie Theory 28 (2018), 43–55.

    MathSciNet  MATH  Google Scholar 

  17. N. Jacobson: Lie Algebras. Interscience Tracts in Pure and Applied Mathematics 10. Interscience Publishers, New York, 1962.

    Google Scholar 

  18. S. M. Kasymov: Theory of n-Lie algebras. Algebra Logic 26 (1987), 155–166.

    Article  Google Scholar 

  19. J. Liu, A. Makhlouf, Y. Sheng: A new approach to representations of 3-Lie algebras and abelian extensions. Algebr. Represent. Theory 20 (2017), 1415–1431.

    Article  MathSciNet  Google Scholar 

  20. J.-L. Loday: Une version non commutative des algèbres de Lie: Les algèbres de Leibniz. Enseign. Math., II. Sér. 39 (1993), 269–293. (In French.)

    MATH  Google Scholar 

  21. J.-L. Loday, T. Pirashvili: Universal enveloping algebras of Leibniz algebras and (co)homology. Math. Ann. 296 (1993), 139–158.

    Article  MathSciNet  Google Scholar 

  22. L. Song, J. Jiang: Generalized derivations extensions of 3-Lie algebras and corresponding Nambu-Poisson structures. J. Geom. Phys. 124 (2018), 74–85.

    Article  MathSciNet  Google Scholar 

  23. L. Takhtajan: On foundation of the generalized Nambu mechanics. Commun. Math. Phys. 160 (1994), 295–315.

    Article  MathSciNet  Google Scholar 

  24. Y. Tan, S. Xu: The Wells map for abelian extensions of 3-Lie algebras. Czech. Math. J. 69 (2019), 1133–1164.

    Article  MathSciNet  Google Scholar 

  25. T. Zhang: Cohomology and deformations of 3-Lie colour algebras. Linear Multilinear Algebra 63 (2015), 651–671.

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Mathematical College, Sichuan University, No. 24 South Section 1, Yihuan Road, Chengdu, 610064, P.R. China

    Youjun Tan

  2. School of Mathematical Sciences, Jiangsu University, 301 Xuefu Road, Zhenjiang, 212013, P.R. China

    Senrong Xu

Authors
  1. Youjun Tan
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  2. Senrong Xu
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Corresponding author

Correspondence to Senrong Xu.

Additional information

The research has been supported by Grant number 11171233 of the NSF of China.

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Tan, Y., Xu, S. Quasi-trace functions on Lie algebras and their applications to 3-Lie algebras. Czech Math J 72, 559–591 (2022). https://doi.org/10.21136/CMJ.2022.0059-21

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  • DOI: https://doi.org/10.21136/CMJ.2022.0059-21

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