Skip to main content
SpringerLink
Log in

On Extensions of Some Classes of Algebras

  • Conference paper
  • First Online:
Algebra, Complex Analysis, and Pluripotential Theory (USUZCAMP 2017)

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 264))

Included in the following conference series:

Abstract

The paper consists of three parts. In the first part we discuss on extensions of Lie algebras and their importance in Physics. Then we deal with the extensions of some classes of algebras with one binary operation. The third part is devoted to the study of extensions of two classes of algebras, possessing two algebraic operations, called dialgebras. In all the cases we propose 2-cocycles and respective extensions. The motivation to study the extensions is to use them further for the classification problem of the classes algebras considered in low dimensional cases.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Aguiar, M.: Pre-poisson algebras. Lett. Math. Phys. 54(4), 263–277 (2000)

    Article  MathSciNet  Google Scholar 

  2. Basri, W., Rakhimov, I.S., Rikhsiboev, I.M.: On nilpotent diassociative algebras. J. Gen. Lie Theor. Appl. 9(1), 1 (2015)

    MathSciNet  MATH  Google Scholar 

  3. Baxter, G.: An analytic problem whose solution follows from a simple algebraic identity. Pacific. J. Math. 10, 731–742 (1960)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  5. De Graaf, W.A.: Classification of nilpotent associative algebras of small dimension. Int. J. Algebr. Comput. 28(01), 133–161 (2017)

    Article  MathSciNet  Google Scholar 

  6. Dzhumadildaev, A.S., Tulenbaev, K.M.: Nilpotency of Zinbiel algebras. J. Dyn. Control Syst. 11(2), 195–213 (2005)

    Article  MathSciNet  Google Scholar 

  7. Gong, M.-P.: Classification of nilpotent Lie algebras of dimension 7 (over algebraic closed fields and \(\mathbb{R}\)). Ph.D. Thesis, Waterloo, Ontario, Canada (1992)

    Google Scholar 

  8. Hegazi, A., Abdelwahab, H.: The classification of \(n\)-dimensional non-associative Jordan algebras with \((n-3)\)-dimensional annihilator. Commun. Algebra 46(2), 629–643 (2018)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  10. Koszul, J.-L.: Domaines bornes homogenes et orbites de groupes de transformation afflnes. Bull. Soc. Math. France 89, 515–533 (1961)

    Article  MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

  12. Loday, J.-L.: Overview on Leibniz algebras, dialgebras and their homology. Fields Inst. Commun. 17, 91–102 (1997)

    MathSciNet  MATH  Google Scholar 

  13. Loday, J.-L., Frabetti, A., Chapoton, F., Goichot, F.: Dialgebras and Related Operads. Lecture Notes in Math., 1763. Springer, Berlin (2001)

    Google Scholar 

  14. Rakhimov, I.S.: On central extensions of associative dialgebras. J. Phys. Conf. Ser. 697(1), 012009 (2016)

    Article  MathSciNet  Google Scholar 

  15. Rakhimov, I.S., Langari, S.J.: A cohomological approach for classifying Nilpotent Leibniz algebras. Int. J. Algebra 4(4), 153–163 (2010)

    MathSciNet  MATH  Google Scholar 

  16. Rakhimov, I.S., Hassan, M.A.: On one-dimensional central extension of a filiform Lie algebra. Bull. Aust. Math. Soc. 84, 205–224 (2011)

    Article  MathSciNet  Google Scholar 

  17. Rakhimov, I.S., Hassan, M.A.: On isomorphism criteria for Leibniz central extensions of a linear deformation of \(\mu _n\). Int. J. Algebra Comput. 21(5), 715–729 (2011)

    Article  Google Scholar 

  18. Rota, G.: Baxter algebras and combinatorial identities I-II. Bull. Amer. Math. Soc. 75(2), 325–334 (1969)

    Article  MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

  20. Vinberg, E.B.: Convex homogeneous cones. Transl. Moscow Math. Soc. 12, 340–403 (1963)

    MATH  Google Scholar 

Download references

Acknowledgements

The author would like to thank the referee for the comments made. The research was supported by FRGS Grant 01-01-16-1869FR, MOHE, Malaysia.

Author information

Authors and Affiliations

  1. Institute for Mathematical Research (INSPEM), Universiti Putra Malaysia, Serdang, Malaysia

    Isamiddin Rakhimov

Authors
  1. Isamiddin Rakhimov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Isamiddin Rakhimov .

Editor information

Editors and Affiliations

  1. Department of Mathematics, California State University, Fullerton, CA, USA

    Zair Ibragimov

  2. Department of Mathematics, Indiana University, Bloomington, IN, USA

    Norman Levenberg

  3. Institute of Mathematics, Uzbekistan Academy of Sciences, Tashkent, Uzbekistan

    Utkir Rozikov

  4. Department of Mathematics, National University of Uzbekistan, Tashkent, Uzbekistan

    Azimbay Sadullaev

Rights and permissions

Reprints and permissions

Copyright information

© 2018 Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Rakhimov, I. (2018). On Extensions of Some Classes of Algebras. In: Ibragimov, Z., Levenberg, N., Rozikov, U., Sadullaev, A. (eds) Algebra, Complex Analysis, and Pluripotential Theory. USUZCAMP 2017. Springer Proceedings in Mathematics & Statistics, vol 264. Springer, Cham. https://doi.org/10.1007/978-3-030-01144-4_12

Download citation

Publish with us

Policies and ethics

Search

Navigation