Skip to main content
SpringerLink
Log in

The Structure of Simple Noncommutative Jordan Superalgebras

  • Published:
Mediterranean Journal of Mathematics Aims and scope Submit manuscript

Abstract

In this paper, we describe all subalgebras and automorphisms of simple noncommutative Jordan superalgebras \(K_3(\alpha ,\beta ,\gamma )\) and \(D_t(\alpha ,\beta ,\gamma )\). We also compute the derivations under some restrictions of the nontrivial simple finite-dimensional noncommutative Jordan superalgebras. All superalgebras are considered over a field of the characteristic 0.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Barreiro, E., Elduque, A., Martínez, C.: Derivations of the Cheng-Kac Jordan superalgebras. J. Algebra 338, 144–156 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  2. Cantarini, N., Kac, V.: Classification of linearly compact simple Jordan and generalized Poisson superalgebras. J. Algebra 313, 100–124 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  3. Elduque, A., Laliena, J., Sacristán, S.: The Kac Jordan superalgebra: automorphisms and maximal subalgebras. Proc. Am. Math. Soc. 135(12), 3805–3813 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  4. Elduque, A., Laliena, J., Sacristán, S.: Maximal subalgebras of Jordan superalgebras. J. Pure Appl. Algebra 212(11), 2461–2478 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  5. Kaygorodov, I.: On \(\delta \)-derivations of simple finite-dimensional Jordan superalgebras. Algebra Logic 46(5), 318–329 (2007)

    Article  MathSciNet  Google Scholar 

  6. Kaygorodov, I.: \(\delta \)-superderivations of simple finite-dimensional Jordan and Lie superalgebras. Algebra Logic 49(2), 130–144 (2010)

    Article  MathSciNet  Google Scholar 

  7. Kaygorodov, I.: On \(\delta \)-superderivations of semisimple finite-dimensional Jordan superalgebras. Math. Notes 91(2), 187–197 (2012)

    Article  MathSciNet  Google Scholar 

  8. Kaygorodov, I., Lopatin, A., Popov, Y.: Jordan algebras admitting derivations with invertible values. Commun. Algebra 46(1), 69–81 (2018)

    Article  MathSciNet  Google Scholar 

  9. Kaygorodov, I., Popov, Y.: Alternative algebras admitting derivations with invertible values and invertible derivations. Izv. Math. 78(5), 75–90 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  10. Kaygorodov, I., Popov, Y.: A characterization of nilpotent nonassociative algebras by invertible Leibniz-derivations. J. Algebra 456, 323–347 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  11. Kaygorodov, I., Shestakov, I., Umirbaev, U.: Free generic poisson fields and algebras. Commun. Algebra (2018). https://doi.org/10.1080/00927872.2017.1358269

  12. Kac, V.: Lie superalgebras. Adv. Math. 26(1), 8–96 (1977)

    Article  MATH  Google Scholar 

  13. Kac, V.: Classification of simple \(\mathbb{Z}\)-graded Lie superalgebras and simple Jordan superalgebras. Commun. Algebra. 13(5), 1375–1400 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  14. Kantor, I.L.: Jordan and Lie superalgebras defined by the Poisson algebra. Am. Math. Soc. Trans. 151, 55–80 (1992)

    MATH  Google Scholar 

  15. Kokoris, L.A.: Nodal noncommutative Jordan algebras. Can. J. Math. 12, 488–492 (1960)

    Article  MathSciNet  MATH  Google Scholar 

  16. Kokoris, L.A.: Simple nodal noncommutative Jordan algebras. Proc. Am. Math. Soc. 9(4), 652–654 (1958)

    MathSciNet  MATH  Google Scholar 

  17. Martinez, C., Zelmanov, E.: Simple finite-dimensional Jordan superalgebras of prime characteristic. J. Algebra 326(2), 575–629 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  18. McCrimmon, K.: Structure and representations of noncommutative Jordan algebras. Trans. Am. Math. Soc. 121, 187–199 (1966)

    Article  MathSciNet  MATH  Google Scholar 

  19. McCrimmon, K.: Noncommutative Jordan rings. Trans. Am. Math. Soc. 158(1), 1–33 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  20. Oehmke, R.H.: On flexible algebras. Ann. Math. (2) 68(2), 221–230 (1958)

    Article  MathSciNet  MATH  Google Scholar 

  21. Popov, A.: Differentiably simple Jordan algebras. Sib. Math. J. 54(4), 713–721 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  22. Popov, Y.: Representations of noncommutative Jordan superalgebras. Preprint

  23. Pozhidaev, A., Shestakov, I.: Noncommutative Jordan superalgebras of degree \(n > 2\). Algebra Logic 49(1), 26–59 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  24. Pozhidaev, A., Shestakov, I.: Simple finite-dimensional noncommutative Jordan superalgebras of characteristic 0. Sib. Math. J. 54(2), 389–406 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  25. Racine, M., Zelmanov, E.: Simple Jordan superalgebras with semisimple even part. J. Algebra 270(2), 374–444 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  26. Retakh, A.: Derivations of KKM doubles. Commun. Algebra 38, 3660–3760 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  27. Shestakov, A.: Ternary derivations of simple Jordan superalgebras. Algebra Logic 53(4), 323–348 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  28. Smith, K.C.: Noncommutative Jordan algebras of capacity two. Trans. Am. Math. Soc. 158(1), 151–159 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  29. Schafer, R.D.: Noncommutative Jordan algebras of characteristic 0. Proc. Am. Math. Soc. 6, 472–475 (1955)

    MathSciNet  MATH  Google Scholar 

  30. Zhelyabin, V., Kaygorodov, I.: On \(\delta \)-superderivations of simple superalgebras of Jordan brackets. St. Peterburg Math. J. 23(4), 40–58 (2012)

    MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Universidade Federal do ABC, CMCC, Santo Andre, SP, Brazil

    Ivan Kaygorodov

  2. Universidade Estadual de Campinas, IMECC, Campinas, SP, Brazil

    Artem Lopatin & Yury Popov

Authors
  1. Ivan Kaygorodov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Artem Lopatin
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Yury Popov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ivan Kaygorodov.

Additional information

The results from the third section of the paper (Theorems 5–8) were obtained by the A. Lopatin with financial support by RFBR No. 16-31-60111 (mol_a_dk). The results from the 4 and 5 sections (Theorems 9–12) were obtained by the first and the third authors with financial support by CNPq No 300603/2016-9; RFBR 17-01-00258 and the Presidents Programme Support of Young Russian Scientists (Grant MK-1378.2017.1).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kaygorodov, I., Lopatin, A. & Popov, Y. The Structure of Simple Noncommutative Jordan Superalgebras. Mediterr. J. Math. 15, 33 (2018). https://doi.org/10.1007/s00009-018-1084-1

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00009-018-1084-1

Mathematics Subject Classification

Search

Navigation