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The Algebraic and Geometric Classification of Nilpotent Assosymmetric Algebras

Algebras and Representation Theory volume 24, pages135–148(2021)Cite this article

Abstract

We present algebraic and geometric classifications of the 4-dimensional complex nilpotent assosymmetric algebras.

References

  1. 1.

    Abdelwahab, H., Calderón, A.J., Kaygorodov, I.: The algebraic and geometric classification of nilpotent binary Lie algebras. International Journal of Algebra and Computation 29(6), 1113–1129 (2019)

    MathSciNet  Article  Google Scholar 

  2. 2.

    Adashev, J., Camacho, L., Omirov, B.: Central extensions of null-filiform and naturally graded filiform non-Lie Leibniz algebras. Journal of Algebra 479, 461–486 (2017)

    MathSciNet  Article  Google Scholar 

  3. 3.

    Alvarez, M.A.: The variety of 7-dimensional 2-step nilpotent Lie algebras. Symmetry 10(1), 26 (2018)

    Article  Google Scholar 

  4. 4.

    Alvarez, M.A., Hernández, I., Kaygorodov, I.: Degenerations of Jordan superalgebras. Bulletin of the Malaysian Mathematical Sciences Society 42(6), 3289–3301 (2019)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Beneš, T., Burde, D.: Classification of orbit closures in the variety of three-dimensional Novikov algebras. Journal of Algebra and its Applications 13, 2,1350081,33 (2014)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Boers, A.: On assosymmetric rings. Indag. Math. 5(1), 9–27 (1994)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Burde, D., Steinhoff, C.: Classification of orbit closures of 4–dimensional complex Lie algebras. Journal of Algebra 214(2), 729–739 (1999)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Calderón Martín, A., Fernández Ouaridi, A., Kaygorodov, I.: The classification of n-dimensional anticommutative algebras with (n − 3)-dimensional annihilator. Communications in Algebra 47(1), 173–181 (2019)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Calderón Martín, A., Fernández Ouaridi, A., Kaygorodov, I.: The classification of 2-dimensional rigid algebras, Linear and Multilinear Algebra. https://doi.org/10.1080/03081087.2018.1519009 (2018)

  10. 10.

    Calderón Martín, A., Fernández Ouaridi, A., Kaygorodov, I.: The classification of bilinear maps with radical of codimension 2, (2018). arXiv:1806.07009

  11. 11.

    Camacho, L., Kaygorodov, I., Lopatkin, V., Salim, M.: The variety of dual mock-Lie algebras, Communications in Mathematics, to appear. arXiv:1910.01484

  12. 12.

    Cicalò, S., De Graaf, W., Schneider, C.: Six-dimensional nilpotent Lie algebras. Linear Algebra Appl. 436(1), 163–189 (2012)

    MathSciNet  Article  Google Scholar 

  13. 13.

    Darijani, I., Usefi, H: The classification of 5-dimensional p-nilpotent restricted Lie algebras over perfect fields, I. Journal of Algebra 464, 97–140 (2016)

    MathSciNet  Article  Google Scholar 

  14. 14.

    De Graaf, W.: Classification of 6-dimensional nilpotent Lie algebras over fields of characteristic not 2. Journal of Algebra 309(2), 640–653 (2007)

    MathSciNet  Article  Google Scholar 

  15. 15.

    De Graaf, W.: Classification of nilpotent associative algebras of small dimension. International Journal of Algebra and Computation 28(1), 133–161 (2018)

    MathSciNet  Article  Google Scholar 

  16. 16.

    Demir, I., Misra, K., Stitzinger, E.: On classification of four-dimensional nilpotent Leibniz algebras. Communications in Algebra 45(3), 1012–1018 (2017)

    MathSciNet  Article  Google Scholar 

  17. 17.

    Dzhumadildaev, A.: Assosymmetric algebras under Jordan product. Communications in Algebra 46(1), 1532–4125 (2018)

    MathSciNet  Google Scholar 

  18. 18.

    Dzhumadildaev, A., Zhakhayev, B.: Free assosymmetric algebras as modules of groups. arXiv:1810.05254

  19. 19.

    Fernández Ouaridi, A., Kaygorodov, I., Khrypchenko, M., Volkov, Yu.: Degenerations of nilpotent algebras. arXiv:1905.05361

  20. 20.

    Gorshkov, I., Kaygorodov, I., Khrypchenko, M.: The algebraic classification of nilpotent Tortkara algebras. arXiv:1904.00845

  21. 21.

    Gorshkov, I., Kaygorodov, I., Khrypchenko, M.: The geometric classification of Tortkara algebras. Communications in Algebra 48(1), 204–209 (2020)

    MathSciNet  Article  Google Scholar 

  22. 22.

    Gorshkov, I., Kaygorodov, I., Kytmanov, A., Salim, M: The variety of nilpotent Tortkara algebras. Journal of Siberian Federal University Mathematics & Physics 12(2), 173–184 (2019)

    MathSciNet  Article  Google Scholar 

  23. 23.

    Gorshkov, I., Kaygorodov, I., Popov, Yu.: Degenerations of Jordan algebras and Marginal algebras, Algebra Colloquium, 2019, to appear

  24. 24.

    Grunewald, F., O’Halloran, J.: Varieties of nilpotent Lie algebras of dimension less than six. Journal of Algebra 112, 315–325 (1988)

    MathSciNet  Article  Google Scholar 

  25. 25.

    Grunewald, F., O’Halloran, J.: A Characterization of orbit closure and applications. Journal of Algebra 116, 163–175 (1988)

    MathSciNet  Article  Google Scholar 

  26. 26.

    Hegazi, A., Abdelwahab, H.: Classification of five-dimensional nilpotent Jordan algebras. Linear Algebra Appl. 494, 165–218 (2016)

    MathSciNet  Article  Google Scholar 

  27. 27.

    Hegazi, A., Abdelwahab, H.: Is it possible to find for any \(n,m \in \mathbb {N}\) a Jordan algebra of nilpotency type (n, 1,m)?. Beitrage zur Algebra und Geometrie 57(4), 859–880 (2016)

    MathSciNet  Article  Google Scholar 

  28. 28.

    Hegazi, A., Abdelwahab, H., Calderón Martín, A.: The classification of n-dimensional non-Lie Malcev algebras with (n − 4)-dimensional annihilator. Linear Algebra Appl. 505, 32–56 (2016)

    MathSciNet  Article  Google Scholar 

  29. 29.

    Hegazi, A., Abdelwahab, H., Calderón Martín, A.: Classification of nilpotent Malcev algebras of small dimensions over arbitrary fields of characteristic not 2. Algebras and Representation Theory 21(1), 19–45 (2018)

    MathSciNet  Article  Google Scholar 

  30. 30.

    Hentzel, I., Jacobs, D., Peresi, L.: A basis for free assosymmetric algebras. Journal of Algebra 183, 306–318 (1996)

    MathSciNet  Article  Google Scholar 

  31. 31.

    Ismailov, N., Kaygorodov, I., Volkov, Yu.: The geometric classification of Leibniz algebras. International Journal of Mathematics 29, 5,1850035 (2018)

    MathSciNet  Article  Google Scholar 

  32. 32.

    Ismailov, N., Kaygorodov, I., Volkov, Yu.: Degenerations of Leibniz and anticommutative algebras. Can. Math. Bull. 62(3), 539–549 (2019)

    MathSciNet  Article  Google Scholar 

  33. 33.

    Karimjanov, I., Kaygorodov, I., Khudoyberdiyev, K: The algebraic and geometric classification of nilpotent Novikov algebras. J. Geom. Phys. 143, 11–21 (2019)

    MathSciNet  Article  Google Scholar 

  34. 34.

    Karimjanov, I., Kaygorodov, I., Ladra, M.: Central extensions of filiform associative algebras, Linear and Multilinear Algebra. https://doi.org/10.1080/03081087.2019.1620674 (2019)

  35. 35.

    Kaygorodov, I., Khrypchenko, M., Popov, Yu.: The algebraic and geometric classification of nilpotent terminal algebras. arXiv:1909.00358

  36. 36.

    Kaygorodov, I., Lopes, S., Popov, Yu.: Degenerations of nilpotent associative commutative algebras, Communications in Algebra, to appear. https://doi.org/10.1080/00927872.2019.1691581

  37. 37.

    Kaygorodov, I., Paez-Guillán, P., Voronin, V.: The algebraic and geometric classification of nilpotent bicommutative algebras, Algebras and Representation Theory, to appear. https://doi.org/10.1007/s10468-019-09944-x

  38. 38.

    Kaygorodov, I., Popov, Yu., Pozhidaev, A., Volkov, Yu.: Degenerations of Zinbiel and nilpotent Leibniz algebras. Linear and Multilinear Algebra 66(4), 704–716 (2018)

    MathSciNet  Article  Google Scholar 

  39. 39.

    Kaygorodov, I., Popov, Yu., Volkov, Yu.: Degenerations of binary-Lie and nilpotent Malcev algebras. Communications in Algebra 46(11), 4929–4941 (2018)

    MathSciNet  Article  Google Scholar 

  40. 40.

    Kaygorodov, I., Volkov, Yu.: The variety of 2-dimensional algebras over an algebraically closed field. Can. J. Math. 71(4), 819–842 (2019)

    MathSciNet  Article  Google Scholar 

  41. 41.

    Kaygorodov, I., Volkov, Yu.: Complete classification of algebras of level two. Moscow Mathematical Journal 19(3), 485–521 (2019)

    MathSciNet  Article  Google Scholar 

  42. 42.

    Kaygorodov, I., Volkov, Yu.: Degenerations of Filippov algebras. arXiv:1911.00358

  43. 43.

    Kim, H., Kim, K.: The structure of assosymmetric algebras. Journal of Algebra 319(6), 2243–2258 (2008)

    MathSciNet  Article  Google Scholar 

  44. 44.

    Kleinfeld, E.: Assosymmetric rings. Proceedings of the American Mathematical Society 8, 983–986 (1957)

    MathSciNet  Article  Google Scholar 

  45. 45.

    Mazzola, G.: The algebraic and geometric classification of associative algebras of dimension five. Manuscripta Mathematica 27(1), 81–101 (1979)

    MathSciNet  Article  Google Scholar 

  46. 46.

    Mazzola, G.: Generic finite schemes and Hochschild cocycles. Commentarii Mathematici Helvetici 55(2), 267–293 (1980)

    MathSciNet  Article  Google Scholar 

  47. 47.

    Pokrass, D., Rodabaugh, D.: Solvable assosymmetric rings are nilpotent. Proceedings of the American Mathematical Society 64(1), 30–34 (1977)

    MathSciNet  Article  Google Scholar 

  48. 48.

    Seeley, C.: Degenerations of 6-dimensional nilpotent Lie algebras over ℂ. Communications in Algebra 18, 3493–3505 (1990)

    MathSciNet  Article  Google Scholar 

  49. 49.

    Skjelbred, T., Sund, T.: Sur la classification des algebres de Lie nilpotentes. C. R. Acad. Sci. Paris Ser. A-B 286(5), A241–A242 (1978)

    MathSciNet  MATH  Google Scholar 

  50. 50.

    Zusmanovich, P.: Central extensions of current algebras. Trans. Am. Math. Soc. 334(1), 143–152 (1992)

    MathSciNet  Article  Google Scholar 

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Acknowledgments

This work was supported by FAPESP 18/15712-0; RFBR 18-31-20004; AP05131123 ”Cohomological and structural problems of non-associative algebras”. The authors thank Prof. Dr. Mykola Khrypchenko for constructive discussions about degenerations of algebras.

Author information

Affiliations

  1. Suleyman Demirel University, Kaskelen, Kazakhstan

    Nurlan Ismailov & Farukh Mashurov

  2. Institute of Mathematics and Mathematical Modeling, Almaty, Kazakhstan

    Nurlan Ismailov

  3. CMCC, Universidade Federal do ABC, Santo André, Brasil

    Ivan Kaygorodov

  4. Kazakh-British Technical University, Almaty, Kazakhstan

    Farukh Mashurov

Authors
  1. Nurlan Ismailov
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  2. Ivan Kaygorodov
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  3. Farukh Mashurov
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Corresponding author

Correspondence to Ivan Kaygorodov.

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Presented by: Michel Brion

Appendix

Appendix

Table 1 The list of 4-dimensional nilpotent “pure” assosymmetric algebras
Full size table
Table 2 The list of 4-dimensional nilpotent “pure” assosymmetric algebras
Full size table

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Ismailov, N., Kaygorodov, I. & Mashurov, F. The Algebraic and Geometric Classification of Nilpotent Assosymmetric Algebras. Algebr Represent Theor 24, 135–148 (2021). https://doi.org/10.1007/s10468-019-09935-y

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Keywords

  • Assosymmetric algebras
  • Nilpotent algebras
  • Algebraic classification
  • Central extension
  • Geometric classification
  • Degeneration

Mathematics Subject Classification 2010

  • 17A30
  • 17D25
  • 14D06
  • 14L30