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Ternary Lie Superalgebras and Nambu-Hamilton Equation in Superspace

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Algebraic Structures and Applications (SPAS 2017)

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 317))

Abstract

In  the present paper we give a survey of methods for constructing ternary Lie algebras and ternary Lie superalgebras. We also propose a generalization of Nambu-Hamilton equation to a superspace and show that this generalization induces a family of ternary Nambu-Poisson brackets of even degree functions on a superspace. Then we show that the construction of ternary quantum Nambu-Poisson bracket, based on the trace of a matrix, can be extended to matrix Lie superalgebra \(\mathfrak {gl}(m,n)\) by means of the supertrace of a matrix. We propose a generalization of Nambu-Hamilton equation in superspace. We show that this generalization induces a family of ternary Nambu-Poisson brackets, which is defined with the help of Berezinian.

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References

  1. Abramov, V.: Super 3-Lie algebras induced by super Lie algebras. Adv. Appl. Clifford Algebr. 27, 9–16 (2017)

    Google Scholar 

  2. Abramov, V.: Matrix 3-Lie superalgebras and BRST supersymmetry. Int. J. Geom. Methods Mod. Phys. 14, 1750160 (2017)

    Article  MathSciNet  Google Scholar 

  3. Abramov, V.: Quantum super Nambu bracket of cubic supermatrices and 3-Lie superalgebra. Adv. Appl. Clifford Algebr. 28, 33 (2018)

    Google Scholar 

  4. Abramov, V.: Generalization of Nambu-Hamilton equation and extension of Nambu-Poisson bracket to superspace. Universe 4(10), 106 (2018)

    Google Scholar 

  5. Abramov, V.: Nambu-Poisson bracket on superspace. Int. J. Geom. Methods Mod. Phys. 15(11), 1850190 (2018)

    Google Scholar 

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

    Google Scholar 

  7. Arnlind, J., Kitouni, A., Makhlouf, A., Silvestrov, S.: Structure and Cohomology of \(3\)-Lie Algebras Induced by Lie Algebras. In: Makhlouf, A., Paal, E., Silvestrov, S., Stolin, A. (eds.), Algebra, Geometry and Mathematical Physics. Springer Proceedings in Mathematics & Statistics, vol. 85, 123–144. Springer, Berlin (2014)

    Google Scholar 

  8. Ataguema, H., Makhlouf, A., Silvestrov, S.: Generalization of \(n\)-ary Nambu algebras and beyond. J. Math. Phys. 50, 083501 (2009)

    Google Scholar 

  9. Awata, H., Li, M., Minic, D., Yaneya, T.: On the quantization of Nambu brackets. JHEP02 2001, 013

    Google Scholar 

  10. Bai, C., Guo, L., Sheng, Y.: Bialgebras, the classical Yang-Baxter equation and Manin triples for 3-Lie bialgebras. arXiv:1604.05996v1 [math-ph]

  11. Bai, R., Bai, C., Wang, J.: Realizations of 3-Lie algebras. J. Math. Phys. 51, 063505 (2010)

    Article  MathSciNet  Google Scholar 

  12. Bai, R., Wu, Y.: Constructing 3-Lie algebras. arXiv:1306.1994v1 [math-ph]

  13. Basu, A., Harvey, J.A.: The M2–M5 brane system and a generalized Nahm’s equation. Nucl. Phys. B 713, 136–155 (2005)

    Article  MathSciNet  Google Scholar 

  14. Berezin, F.A.: An introduction to algebra and analysis with anticommuting variables, Izdat. Moskov. Gos. Univ., Moscow (1983) (English translation, Introduction to superanalysis, Reidel, Dordrecht (1987))

    Google Scholar 

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

    Article  Google Scholar 

  16. Kitouni, A., Makhlouf, A.: On structure and central extensions of \((n+1)\)- Lie algebras induced by \(n\)-Lie algebras (2014). arxiv:1405.5930

  17. Ling, W.X.: On the structure of \(n\)-Lie algebras. Ph.D. thesis (1993). Siegen

    Google Scholar 

  18. Nambu, Y.: Generalized Hamiltonian dynamics. Phys. Rev. D 7(8), 2405–2412 (1973)

    Article  MathSciNet  Google Scholar 

  19. Rosenfeld, B.A.: Multidimensional Spaces. Nauka, Moscow (1966)

    Google Scholar 

  20. Takhtajan, L.: On foundation of generalized Nambu mechanics. Commun. Math. Phys. 160(2), 295–315 (1994)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

The authors gratefully acknowledges that this work was financially supported by the institutional funding IUT20-57 of the Estonian Ministry of Education and Research and by the Doctoral School in Mathematics and Statistics of Estonia.

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Authors and Affiliations

  1. Institute of Mathematics and Statistics, University of Tartu, J. Liivi 2, 50409, Tartu, Estonia

    Viktor Abramov & Priit Lätt

Authors
  1. Viktor Abramov
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  2. Priit Lätt
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Corresponding author

Correspondence to Viktor Abramov .

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Editors and Affiliations

  1. Division of Applied Mathematics, School of Education, Culture and Communication, Mälardalen University, Västerås, Sweden

    Sergei Silvestrov

  2. Division of Applied Mathematics, School of Education, Culture and Communication, Mälardalen University, Västerås, Sweden

    Anatoliy Malyarenko

  3. Division of Applied Mathematics, School of Education, Culture and Communication, Mälardalen University, Västerås, Sweden

    Milica Rančić

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Abramov, V., Lätt, P. (2020). Ternary Lie Superalgebras and Nambu-Hamilton Equation in Superspace. In: Silvestrov, S., Malyarenko, A., Rančić, M. (eds) Algebraic Structures and Applications. SPAS 2017. Springer Proceedings in Mathematics & Statistics, vol 317. Springer, Cham. https://doi.org/10.1007/978-3-030-41850-2_3

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