Derivations of Some Filiform Leibniz Algebras

AL-hossain Ahmad; Khiyar AL-hossain

doi:doi:10.11648/j.pamj.20140306.12

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Derivations of Some Filiform Leibniz Algebras

AL-hossain Ahmad, Khiyar AL-hossain

Published in Pure and Applied Mathematics Journal (Volume 3, Issue 6)

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.

Published in	Pure and Applied Mathematics Journal (Volume 3, Issue 6)
DOI	10.11648/j.pamj.20140306.12
Page(s)	121-125
Creative Commons	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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[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.
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[15]	J.- L. Loday and T. Pirashvili , "Universal enveloping algebra of Leibniz algebras and (co)homology,", Math. Ann., 296 (1993), no. 1,139-158.
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[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).
[19]	E. Schenkman , "On the derivation algebra and the holomorph of a nilpotent Lie algebras," , Mem. Amer. Math. Soc. No. 14 (1955), 15-22.
[20]	David J.Winter, Abstract Lie algebras, printed in the United States of America, (1972), 150 p.

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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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ACS 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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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,
  author = {AL-hossain Ahmad and Khiyar AL-hossain},
  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://article.sciencepublishinggroup.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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TY  - JOUR
T1  - Derivations of Some Filiform Leibniz Algebras
AU  - AL-hossain Ahmad
AU  - Khiyar AL-hossain
Y1  - 
PY  - 
N1  - https://doi.org/10.11648/j.pamj.20140306.12
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AB  - 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.
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Author Information

AL-hossain Ahmad

Department of Mathematics, AL Qunfudha University College, Umm AL Qura University, City of AL Qunfudha, Kingdom of Saudi Arabia
Khiyar AL-hossain

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

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APA Style

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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ACS 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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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

Copy | Download

@article{10.11648/j.pamj.20140306.12,
  author = {AL-hossain Ahmad and Khiyar AL-hossain},
  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://article.sciencepublishinggroup.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 = {}
}

Copy | Download

TY  - JOUR
T1  - Derivations of Some Filiform Leibniz Algebras
AU  - AL-hossain Ahmad
AU  - Khiyar AL-hossain
Y1  - 
PY  - 
N1  - https://doi.org/10.11648/j.pamj.20140306.12
DO  - 10.11648/j.pamj.20140306.12
T2  - Pure and Applied Mathematics Journal
JF  - Pure and Applied Mathematics Journal
JO  - Pure and Applied Mathematics Journal
SP  - 121
EP  - 125
PB  - Science Publishing Group
SN  - 2326-9812
UR  - https://doi.org/10.11648/j.pamj.20140306.12
AB  - 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.
VL  - 3
IS  - 6
ER  -

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