Skip to main content
Log in

On the Automorphism Groups for Some Leibniz Algebras of Low Dimensions

  • Published:
Ukrainian Mathematical Journal Aims and scope

We study the automorphism groups of Leibniz algebras of low dimensions and obtain complete descriptions of the automorphism groups of Leibniz algebras of dimension 2 and some types of nilpotent Leibniz algebras of dimension 3.

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

Access this article

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

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Sh. Ayupov, K. Kudaybergenov, B. Omirov, and K. Zhao, “Semisimple Leibniz algebras, their derivations and automorphisms,” Lin. Multilin. Algebra, 68, No. 10, 2005–2019 (2020); https://doi.org/10.1080/03081087.2019.1567674.

  2. Sh. Ayupov, B. Omirov, and I. Rakhimov, Leibniz Algebras: Structure and Classification, CRC Press, Taylor & Francis Group (2020).

    MATH  Google Scholar 

  3. A. Blokh, “On a generalization of the concept of Lie algebra,” Dokl. Akad. Nauk SSSR, 165, No. 3, 471–473 (1965).

    MathSciNet  MATH  Google Scholar 

  4. J. M. Casas, M. A. Insua, M. Ladra, and S. Ladra, “An algorithm for the classification of 3-dimensional complex Leibniz algebras,” Lin. Algebra Appl., 436, No. 9, 3747–3756 (2012); https://doi.org/10.1016/j.laa.2011.11.039.

    Article  MathSciNet  MATH  Google Scholar 

  5. C. Cuvier, “Algèbres de Leibnitz: définitions, propriétés,” Ann. Sci. Éc. Norm. Supér. (4), 27, No. 1, 1–45 (1994); https://doi.org/10.24033/asens.1687.

  6. I. Demir, K. C. Misra, and E. Stitzinger, “On some structures of Leibniz algebras,” in: Recent Advances in Representation Theory, Quantum Groups, Algebraic Geometry, and Related Topics, Contemp. Math., 623 (2014), pp 41–54; https://doi.org/10.1090/conm/623/12456.

  7. A. Kh. Khudoyberdiyev, T. K. Kurbanbaev, and B. A. Omirov, “Classification of three-dimensional solvable p-adic Leibniz algebras,” p-Adic Numbers Ultrametric Anal. Appl., 2, No. 3, 207–221 (2010); https://doi.org/10.1134/S2070046610030039.

    Article  MathSciNet  MATH  Google Scholar 

  8. L. A. Kurdachenko, J. Otal, and A. A. Pypka, “Relationships between the factors of the canonical central series of Leibniz algebras,” Europ. J. Math., 2, No. 2, 565–577 (2016); https://doi.org/10.1007/s40879-016-0093-5.

    Article  MathSciNet  MATH  Google Scholar 

  9. L. A. Kurdachenko, A. A. Pypka, and I. Ya. Subbotin, “On the automorphism groups of some Leibniz algebras,” Int. J. Group Theory (to appear); https://doi.org/10.22108/IJGT.2021.130057.1735.

  10. M. Ladra, I. M. Rikhsiboev, and R. M. Turdibaev, “Automorphisms and derivations of Leibniz algebras,” Ukr. Mat. Zh., 68, No. 7, 933–944 (2016); Ukr. Math. J., 68, No. 7, 1062–1076 (2016); https://doi.org/10.1007/s11253-016-1277-3.

  11. J.-L. Loday, “Cyclic homology,” Grundlehren Math. Wiss., 301, Springer (1992); https://doi.org/10.1007/978-3-662-11389-9.

  12. J.-L. Loday, “Une version non commutative des algèbres de Lie: les algèbras de Leibniz,” Enseign. Math., 39, 269–293 (1993).

    MathSciNet  MATH  Google Scholar 

  13. J.-L. Loday and T. Pirashvili, “Universal enveloping algebras of Leibniz algebras and (co)homology,” Math. Ann., 296, No. 1, 139–158 (1993); 10.1007/ BF01445099.

    Article  MathSciNet  MATH  Google Scholar 

  14. I. S. Rakhimov, I. M. Rikhsiboev, and M. A. Mohammed, “An algorithm for classifications of three-dimensional Leibniz algebras over arbitrary fields,” JP J. Algebra, Number Theory Appl., 40, No. 2, 181–198 (2018); https://doi.org/10.17654/NT040020181.

  15. I. M. Rikhsiboev and I. S. Rakhimov, “Classification of three dimensional complex Leibniz algebras,” AIP Conf. Proc., 1450, No. 1, 358–362 (2012); https://doi.org/10.1063/1.4724168.

    Article  MATH  Google Scholar 

  16. V. S. Yashchuk, “On some Leibniz algebras, having small dimension,” Algebra Discrete Math., 27, No. 2, 292–308 (2019).

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to O. O. Pypka.

Additional information

Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 74, No. 10, pp. 1339–1355, October, 2022. Ukrainian DOI: https://doi.org/10.37863/umzh.v74i10.7282.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kurdachenko, L.A., Pypka, O.O. & Velychko, T.V. On the Automorphism Groups for Some Leibniz Algebras of Low Dimensions. Ukr Math J 74, 1526–1546 (2023). https://doi.org/10.1007/s11253-023-02153-2

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11253-023-02153-2

Navigation