Skip to main content
SpringerLink
Log in

On Some Properties of Lie-Centroids of Leibniz Algebras

  • Published:
Bulletin of the Malaysian Mathematical Sciences Society Aims and scope Submit manuscript

A Correction to this article was published on 24 October 2022

This article has been updated

Abstract

We study some properties of \(\mathsf {Lie}\)-centroids related to central \(\mathsf {Lie}\)-derivations, generalized \(\mathsf {Lie}\)-derivations and almost inner \(\mathsf {Lie}\)-derivations. We also determine the \(\mathsf {Lie}\)-centroid of the tensor product of a commutative associative algebra and a Leibniz algebra.

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

Change history

References

  1. Ayupov, S., Omirov, B.A., Rakhimov, I.S.: Leibniz algebras: structure and classification. Mathematics and its applications. Chapman and Hall/CRC, New York (2020)

    MATH  Google Scholar 

  2. Benkart, G., Neher, E.: The centroid of extended affine and root graded Lie algebras. J. Pure Appl. Algebra 205(1), 117–145 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  3. Biyogmam, G.R., Casas, J.M.: On \({\sf {Lie}}\)-isoclinic Leibniz algebras. J. Algebra 499, 337–357 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  4. Biyogmam, G.R., Casas, J.M.: The \(c\)-nilpotent Schur \({\sf {Lie}}\)-multiplier of Leibniz algebras. J. Geom. Phys. 138, 55–69 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  5. Biyogmam, G.R., Casas, J.M., Pacheco-Rego, N.: \({\sf {Lie}}\)-central derivations, \({\sf {Lie}}\)-centroids and \({\sf {Lie}}\)-stem Leibniz algebras. Publ. Math. Debrecen 97(1–2), 217–239 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  6. Casas, J.M., Insua, M.A.: The Schur \({\sf {Lie}}\)-multiplier of Leibniz algebras. Quaest. Math. 41(7), 917–936 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  7. Casas, J.M., Insua, M.A., Ladra, M., Ladra, S.: An algorithm for the classification of 3-dimensional complex Leibniz algebras. Linear Algebra Appl. 436(9), 3747–3756 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  8. Casas, J.M., Khmaladze, E.: On \({\sf {Lie}}\)-central extensions of Leibniz algebras. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 111(1), 39–56 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  9. Casas, J.M., Van der Linden, T.: Universal central extensions in semi-abelian categories. Appl. Categ. Struct. 22(1), 253–268 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  10. Cuvier, C.: Algèbres de Leibnitz: définitions, propriétés. Ann. Sci. École Norm. Sup. (4) 27(1), 1–45 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  11. Everaert, T.: Relative commutator theory in varieties of \(\Omega \)-groups. J. Pure Appl. Algebra 210(1), 1–10 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  12. Everaert, T., Van der Linden, T.: Baer invariants in semi-abelian categories. I. General theory. Theory Appl. Categ. 12(1), 1–33 (2004)

    MathSciNet  MATH  Google Scholar 

  13. Everaert, T., Van der Linden, T.: Relative commutator theory in semi-abelian categories. J. Pure Appl. Algebra 216(8–9), 1791–1806 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  14. Fröhlich, A.: Baer-invariants of algebras. Trans. Amer. Math. Soc. 109, 221–244 (1963)

    Article  MathSciNet  MATH  Google Scholar 

  15. Furtado-Coelho, J.: Homology and generalized Baer invariants. J. Algebra 40(2), 596–609 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  16. García-Martínez, X., Gray, J.R.A.: Algebraic exponentiation for Lie algebras. Theory Appl. Categ. 36(11), 288–305 (2021)

    MathSciNet  MATH  Google Scholar 

  17. García-Martínez, X., Tsishyn, M., Van der Linden, T., Vienne, C.: Algebras with representable representations. Proc. Edinburgh Math. Soc. 64(3), 555–573 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  18. García-Martínez, X., Van der Linden, T.: A characterisation of Lie algebras amongst anti-commutative algebras. J. Pure Appl. Algebra 223(11), 4857–4870 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  19. García-Martínez, X., Van der Linden, T.: A characterisation of Lie algebras via algebraic exponentiation. Adv. Math. 341, 92–117 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  20. Higgins, P.J.: Groups with multiple operators. Proc. London Math. Soc. 3(6), 366–416 (1956)

    Article  MathSciNet  MATH  Google Scholar 

  21. Huq, S.A.: Commutator, nilpotency, and solvability in categories. Quart. J. Math. Oxford Ser. 2(19), 363–389 (1968)

    Article  MathSciNet  MATH  Google Scholar 

  22. Janelidze, G.: Pure Galois theory in categories. J. Algebra 132(2), 270–286 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  23. Janelidze, G., Kelly, G.M.: Galois theory and a general notion of central extension. J. Pure Appl. Algebra 97(2), 135–161 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  24. Loday, J.-L.: Une version non commutative des algèbres de Lie: les algèbres de Leibniz. Enseign. Math. (2) 39(3–4), 269–293 (1993)

    MathSciNet  MATH  Google Scholar 

  25. Loday, J.-L.: Cyclic homology, 2nd ed. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 301, Springer-Verlag, Berlin, 1998, Appendix E by María O. Ronco, Chapter 13 by the author in collaboration with Teimuraz Pirashvili

  26. Loday, J.-L., Pirashvili, T.: The tensor category of linear maps and Leibniz algebras. Georgian Math. J. 5(3), 263–276 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  27. Lue, A.S.-T.: Baer-invariants and extensions relative to a variety. Proc. Cambridge Philos. Soc. 63, 569–578 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  28. McCrimmon, K.: Jordan centroids. Comm. Algebra 27(2), 933–954 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  29. Neher, E.: Lie tori. C. R. Math. Acad. Sci. Soc. R. Can. 26(3), 84–89 (2004)

    MathSciNet  MATH  Google Scholar 

  30. Ni, J.: Centroids of Zinbiel algebras. Comm. Algebra 42(4), 1844–1853 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  31. Omirov, B.A.: Conjugacy of Cartan subalgebras of complex finite-dimensional Leibniz algebras. J. Algebra 302(2), 887–896 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  32. Rakhimov, I.S., Rikhsiboev, I.M., Basri, W.: Complete lists of low dimensional complex associative algebras. (2009) arXiv:0910.0932v2

  33. Richardson, P.A.: Centroids of quadratic Jordan superalgebras. Comm. Algebra 36(1), 179–207 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  34. Riyahi, Z., Casas Mirás, J.M.: \({\sf {Lie}}\)-isoclinism of pairs of Leibniz algebras. Bull. Malays. Math. Sci. Soc. 43(1), 283–296 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  35. Ruipu, B., Daoji, M.: The centroid of \(n\)-Lie algebras. Algebras Groups Geom. 21(1), 29–38 (2004)

    MathSciNet  MATH  Google Scholar 

  36. Zhang Zhi, X., Liu Li, Q.: Invariant bilinear forms on Lie superalgebras. Chinese Ann. Math. Ser. A 25(2), 139–146 (2004)

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

We would like to thank the anonymous referees for their helpful comments and remarks that greatly improved the quality of the paper. The first and second authors were supported by Ministerio de Ciencia e Innovación (Spain), with grant number PID2020-115155GB-I00. The second author is a Postdoctoral Fellow of the Research Foundation–Flanders (FWO).

Author information

Author notes

  1. Xabier García-Martínez and Natalia Pacheco-Rego have contributed equally to this work.

Authors and Affiliations

  1. Departamento de Matemática Aplicada I and CITMaga, Universidade de Vigo, E. E. Forestal, 36005 , Campus de Pontevedra, Pontevedra, Spain

    José Manuel Casas Mirás

  2. Departamento de Matemáticas and CITMaga, Universidade de Vigo, Esc. Sup. de Enx. Informática, Campus de Ourense, 32004, Ourense, Spain

    Xabier García-Martínez

  3. Faculty of Engineering, Vrije Universiteit Brussel, Pleinlaan 2, 1050, Brussel, Belgium

    Xabier García-Martínez

  4. Polytechnical Institute of Cavado and Ave, Campus do IPCA, 4750-810, Vila Frescainha, S. Martinho, Barcelos, Portugal

    Natalia Pacheco-Rego

Authors
  1. José Manuel Casas Mirás
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Xabier García-Martínez
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Natalia Pacheco-Rego
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to José Manuel Casas Mirás.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Additional information

Communicated by Rosihan M. Ali.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

The original version of this article was revised: In this article the affiliations were incorrectly assigned. The original article has been corrected.

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

Mirás, J.M.C., García-Martínez, X. & Pacheco-Rego, N. On Some Properties of Lie-Centroids of Leibniz Algebras. Bull. Malays. Math. Sci. Soc. 45, 3499–3520 (2022). https://doi.org/10.1007/s40840-022-01389-8

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40840-022-01389-8

Keywords

Mathematics Subject Classification

Search

Navigation