Pure and Applied Mathematics Journal

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Derivations of Some Filiform Leibniz Algebras

Received: 26 September 2014    Accepted: 23 October 2014    Published:
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Abstract

In this paper the two classes of filiform Leibniz algebras μ_(0 )^n and μ_(1 )^n in (n+1) dimensions of filiform Leibniz algebras such that n≥2 will be considered. The study includes derivations of naturally graded Leibniz algebras of first class L_n and second class W_n, be algebras whose multiplications rules are defined by the μ_(0 )^n and μ_(1 )^n, respectively. The algebras of derivations of naturally graded Leibniz algebras are described by linear transformations and dimensions derivations. Finally, we determine number of derivations of naturally graded Leibniz algebras.

DOI 10.11648/j.pamj.20140306.12
Published in Pure and Applied Mathematics Journal (Volume 3, Issue 6, December 2014)
Page(s) 121-125
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This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.

Copyright

Copyright © The Author(s), 2024. Published by Science Publishing Group

Keywords

Leibniz Algebra, Filiform Leibniz Algebra, Characteristically Nilpotent Algebra, Graded Leibniz Algebra, Derivation

References
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[2] Sh. A. Ayupov and B. A. Omirov , "On some classes of nilpotent Leibniz algebras," , Sibirsk. Mat. Zh. [Siberian Math. J.], 42 (2001), no. 1, 18-29.
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[4] C. Chevalley , "Theorie des Groupes de Lie, Tome II, Groupes algebriques," , Paris, 1951.
[5] J. Dixmier , "Sous-algebras de Cartan et decompositions de levi dans les algebrasde Lie," ,Trans. Roy. Soc. Canada Ser. III, 20 (1956), 17-21.
[6] J. Dixmier and W. G. Lister , "Derivations of nilpotent Lie algebras," , Proc. Amer. Math. Soc., 8 (1957), 155-158.
[7] M. Goze and Yu. Hakimdjanov , "Nilpotent Lie Algebras," , Mathematics and its Applications, vol. 361, Kluwer, Dordrecht, 1996.
[8] You. B. Hakimjanov , "variete des lois d'algebres de lie nilpotentes," , Geometrie Dedicata, 40 (1991), no. 3, 269-295.
[9] Harish-Chandra , "On the radical of a Lie algebra," , Prov. Amer. Math. Soc., 1 (1950), 14-17.
[10] J. E. Humphreys , "Introduction to Lie Algebras and Representation Theory," , Springer-Verlag New York. Heidelberg. Berlin. (1972), 25-27.
[11] N. Jacobson, A note on automorphisms and derivations of Lie algebras, Proc. Amer. Math. Soc. 6(1955), 281-283.
[12] Yu. B. Khakimdzhanov , "Characteristically nilpotent Lie algebras," , Mat. Sb. [Math. USSR-Sb.], 181 (1990), no. 5, 642-655.
[13] G. Leger , "A note on the derivations of Lie algebras," , Proc. Amer. Math. Soc., 4 (1953), 511-514.
[14] G. Leger , "Derivations of Lie algebras III," , Duke Math.J., 30 (1963), 637-645.
[15] J.- L. Loday and T. Pirashvili , "Universal enveloping algebra of Leibniz algebras and (co)homology,", Math. Ann., 296 (1993), no. 1,139-158.
[16] A. I. Malcev , "Solvable Lie algebras," , Izv. Akad. Nauk SSSR Ser. Mat., 9 (1945), 329-352.
[17] B. A. Omirov , "On the Derivations of Filiform Leibniz Algebras," , Mathematical Notes, Vol. 77, No5, 2005, 677-685.
[18] I.S. Rakhimov, and S.K. Husain, , "On isomorphism classes and invariants of low- dimensional Complex _liform Leibniz algebras (Part 2)." , arXiv math RA. (2008).
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Author Information
  • Department of Mathematics, AL Qunfudha University College, Umm AL Qura University, City of AL Qunfudha, Kingdom of Saudi Arabia

  • Department of Mathematics, AL Qunfudha University College, Umm AL Qura University, City of AL Qunfudha, Kingdom of Saudi Arabia

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    AL-hossain Ahmad, Khiyar AL-hossain. (). Derivations of Some Filiform Leibniz Algebras. Pure and Applied Mathematics Journal, 3(6), 121-125. https://doi.org/10.11648/j.pamj.20140306.12

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    AL-hossain Ahmad; Khiyar AL-hossain. Derivations of Some Filiform Leibniz Algebras. Pure Appl. Math. J. , 3(6), 121-125. doi: 10.11648/j.pamj.20140306.12

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    AMA Style

    AL-hossain Ahmad, Khiyar AL-hossain. Derivations of Some Filiform Leibniz Algebras. Pure Appl Math J. ;3(6):121-125. doi: 10.11648/j.pamj.20140306.12

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  • @article{10.11648/j.pamj.20140306.12,
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      title = {Derivations of Some Filiform Leibniz Algebras},
      journal = {Pure and Applied Mathematics Journal},
      volume = {3},
      number = {6},
      pages = {121-125},
      doi = {10.11648/j.pamj.20140306.12},
      url = {https://doi.org/10.11648/j.pamj.20140306.12},
      eprint = {https://download.sciencepg.com/pdf/10.11648.j.pamj.20140306.12},
      abstract = {In this paper the two classes of filiform Leibniz algebras μ_(0  )^n and μ_(1  )^n in (n+1) dimensions of filiform Leibniz algebras such that n≥2  will be considered. The study includes derivations of naturally graded Leibniz algebras of first class L_n and second class W_n, be algebras whose multiplications rules are defined by the  μ_(0  )^n and μ_(1  )^n, respectively. The algebras of derivations of naturally graded Leibniz algebras are described by linear transformations and dimensions derivations. Finally, we determine number of derivations of naturally graded Leibniz algebras.},
     year = {}
    }
    

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