SpringerLink
Log in

On Some Properties of Generalized \({\textsf{Lie}}\)-Derivations of Leibniz Algebras

  • Conference paper
  • First Online:
Non-Associative Algebras and Related Topics (NAART 2020)

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 427))

  • 181 Accesses

Abstract

We classify and deduce some properties on \(\textsf{Lie}\)-derivations and \(\textsf{Lie}\)-centroids of low-dimensional Leibniz algebras. We also study the relationship between the generalized \(\textsf{Lie}\)-derivations of the direct sum of two finite-dimensional Leibniz algebras such that they have no non-trivial common direct factor and the direct sum of the generalized \(\textsf{Lie}\)-derivations of each summand.

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

Access this chapter

Log in via an institution

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
GBP 19.95
Price includes VAT (United Kingdom)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
GBP 127.50
Price includes VAT (United Kingdom)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
GBP 159.99
Price includes VAT (United Kingdom)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free ship** worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Bai, R., Meng, D.: The centroid of n-Lie algebras. Algebras Groups Geom. 21(1), 29–38 (2004)

    MathSciNet  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 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 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)

    Google Scholar 

  6. Casas, J.M., García-Martínez, X., Pacheco Rego, N.: On some properties of \(\sf {Lie}\)-centroids of Leibniz algebras. Bull. Malays. Math. Sci. Soc. 45(6), 3499–3520 (2022)

    Google Scholar 

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

    Google Scholar 

  8. 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 

  9. Casas, J.M., Khmaladze, E.: On \(\sf {Lie}\)-central extensions of Leibniz algebras. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Math. RACSAM 111(1), 36–56 (2017)

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

  17. Zhang, Z.X., Li, L.Q.: Invariant bilinear forms on Lie superalgebras. Chinese Ann. Math. Ser. A 25(2), 139–146 (2004); translation in Chinese J. Contemp. Math. 25(2), 109–116 (2004)

    Google Scholar 

Download references

Acknowledgements

The first author was supported by Agencia Estatal de Investigación (Spain), grant PID2020-115155GB-I00 (European FEDER support included, UE).

Author information

Authors and Affiliations

  1. Departamento de Matemática Aplicada I & CITMAga, Universidade de Vigo, Campus Universitario A Xunqueira, 36005, Pontevedra, Spain

    José Manuel Casas Mirás

  2. Polytechnic Institute of Cavado and Ave, 4750-810, Vila Frescainha, Portugal

    Natália Pacheco Rego

Authors
  1. José Manuel Casas Mirás
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Natália Pacheco Rego
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Natália Pacheco Rego .

Editor information

Editors and Affiliations

  1. Mathematics, University of Coimbra, Coimbra, Portugal

    Helena Albuquerque

  2. Department of Mathematics, Centre for Mathematics of the University of Coimbra, Coimbra, Portugal

    Jose Brox

  3. Mathematics, University of Oviedo, Oviedo, Spain

    Consuelo Martínez

  4. CMUC-Centre for Mathematics of the University of Coimbra, CeBER-Centre for Business and Economics Research, Faculty of Economics, University of Coimbra, Coimbra, Portugal

    Paulo Saraiva

Rights and permissions

Reprints and permissions

Copyright information

© 2023 The Author(s), under exclusive license to Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Casas Mirás, J.M., Pacheco Rego, N. (2023). On Some Properties of Generalized \({\textsf{Lie}}\)-Derivations of Leibniz Algebras. In: Albuquerque, H., Brox, J., Martínez, C., Saraiva, P. (eds) Non-Associative Algebras and Related Topics. NAART 2020. Springer Proceedings in Mathematics & Statistics, vol 427. Springer, Cham. https://doi.org/10.1007/978-3-031-32707-0_7

Download citation

Publish with us

Policies and ethics

Search

Navigation