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DERIVATIONS OF TERNARY LIE ALGEBRAS AND GENERALIZATIONS

Year 2017, Volume 21, Issue 21, 55 - 75, 17.01.2017
https://doi.org/10.24330/ieja.296030

Abstract

We study derivations of ternary Lie algebras. Precisely, we investigate the relation between derivations of Lie algebras and the induced ternary Lie algebras. We also explore the spaces of quasi-derivations, the centroid and the quasi-centroid and give some properties. Finally, we compute these spaces for low dimensional ternary Lie algebras g. 

References

  • [1] J. Arnlind, A. Kitouni, A. Makhlouf and S. Silvestrov, Structure and cohomology
  • of 3-Lie algebras induced by Lie algebras, Algebra, geometry and mathematical
  • physics, 123-144, Springer Proc. Math. Stat., 85, Springer, Heidelberg, 2014.
  • [2] J. Arnlind, A.Makhlouf and S. Silvestrov, Ternary Hom-Nambu-Lie algebras
  • induced by Hom-Lie algebras, J. Math. Phys., 51(4) (2010), 043515, 11 pp.
  • [3] J. Arnlind, A. Makhlouf and S. Silvestrov, Construction of n-Lie algebras and
  • n-ary Hom-Nambu-Lie algebras, J. Math. Phys., 52(12) (2011), 123502, 13 pp.
  • [4] R. Bai, J. Wang and Z. Li, Derivations of the 3-Lie algebra realized by gl(n, C),
  • J. Nonlinear Math. Phys., 18(1) (2011), 151-160.
  • [5] V. T. Filippov, n-Lie algebras, Sibirsk. Mat. Zh., 26(6) (1985), 126-140.
  • [6] V. T. Filippov, On δ-derivations of Lie algebras, (Russian) Sibirsk. Mat. Zh.,
  • 39(6) (1998), 1409-1422, iv; translation in Siberian Math. J. 39(6) (1998), 1218-1230.
  • [7] M. E. Gordji, R. Farrokhzad and S. A. Hosseinioun, Ternary (σ, τ, ξ)-
  • derivations on Banach ternary algebras, Int. J. Nonlinear Anal. Appl., 5(1)(2014), 23-35.
  • [8] M. R. Hestenes, A ternary algebra with applications to matrices and linear
  • transformations, Arch. Rational Mech. Anal., 11 (1962), 138-194.
  • [9] I. Kaygorodov, On δ-derivations of n-ary algebras, (Russian) Izv. Ross. Akad.
  • Nauk Ser. Mat., 76(6) (2012), 81-94; translation in Izv. Math., 76(6) (2012),1150-1162.
  • [10] I. Kaygorodov and Y. Popov, Generalized derivations of (color) n-ary algebras,
  • Linear Multilinear Algebra, 64(6) (2016), 1086-1106.
  • [11] G. F. Leger and E. M. Luks, Generalized derivations of Lie algebras, J. Algebra,
  • 228(1) (2000), 165-203.
  • [12] W. G. Lister, A structure theory of Lie triple systems, Trans. Amer. Math.
  • Soc., 72 (1952), 217-242.
  • [13] P. Novotný and J. Hrivnák, On (α, β, γ)-derivations of Lie algebras and corresponding
  • invariant functions, J. Geom. Phys., 58(2) (2008), 208-217.
  • [14] J. M. Perez-Izquierdo, Unital algebras, ternary derivations, and local triality,
  • Algebras, representations and applications, Contemp. Math., 483, Amer.
  • Math. Soc., Providence, RI, (2009), 205-220.
  • [15] Y. Sheng, Representations of hom-Lie algebras, Algebr. Represent. Theory,
  • 15(6) (2012), 1081-1098.

Year 2017, Volume 21, Issue 21, 55 - 75, 17.01.2017
https://doi.org/10.24330/ieja.296030

Abstract

References

  • [1] J. Arnlind, A. Kitouni, A. Makhlouf and S. Silvestrov, Structure and cohomology
  • of 3-Lie algebras induced by Lie algebras, Algebra, geometry and mathematical
  • physics, 123-144, Springer Proc. Math. Stat., 85, Springer, Heidelberg, 2014.
  • [2] J. Arnlind, A.Makhlouf and S. Silvestrov, Ternary Hom-Nambu-Lie algebras
  • induced by Hom-Lie algebras, J. Math. Phys., 51(4) (2010), 043515, 11 pp.
  • [3] J. Arnlind, A. Makhlouf and S. Silvestrov, Construction of n-Lie algebras and
  • n-ary Hom-Nambu-Lie algebras, J. Math. Phys., 52(12) (2011), 123502, 13 pp.
  • [4] R. Bai, J. Wang and Z. Li, Derivations of the 3-Lie algebra realized by gl(n, C),
  • J. Nonlinear Math. Phys., 18(1) (2011), 151-160.
  • [5] V. T. Filippov, n-Lie algebras, Sibirsk. Mat. Zh., 26(6) (1985), 126-140.
  • [6] V. T. Filippov, On δ-derivations of Lie algebras, (Russian) Sibirsk. Mat. Zh.,
  • 39(6) (1998), 1409-1422, iv; translation in Siberian Math. J. 39(6) (1998), 1218-1230.
  • [7] M. E. Gordji, R. Farrokhzad and S. A. Hosseinioun, Ternary (σ, τ, ξ)-
  • derivations on Banach ternary algebras, Int. J. Nonlinear Anal. Appl., 5(1)(2014), 23-35.
  • [8] M. R. Hestenes, A ternary algebra with applications to matrices and linear
  • transformations, Arch. Rational Mech. Anal., 11 (1962), 138-194.
  • [9] I. Kaygorodov, On δ-derivations of n-ary algebras, (Russian) Izv. Ross. Akad.
  • Nauk Ser. Mat., 76(6) (2012), 81-94; translation in Izv. Math., 76(6) (2012),1150-1162.
  • [10] I. Kaygorodov and Y. Popov, Generalized derivations of (color) n-ary algebras,
  • Linear Multilinear Algebra, 64(6) (2016), 1086-1106.
  • [11] G. F. Leger and E. M. Luks, Generalized derivations of Lie algebras, J. Algebra,
  • 228(1) (2000), 165-203.
  • [12] W. G. Lister, A structure theory of Lie triple systems, Trans. Amer. Math.
  • Soc., 72 (1952), 217-242.
  • [13] P. Novotný and J. Hrivnák, On (α, β, γ)-derivations of Lie algebras and corresponding
  • invariant functions, J. Geom. Phys., 58(2) (2008), 208-217.
  • [14] J. M. Perez-Izquierdo, Unital algebras, ternary derivations, and local triality,
  • Algebras, representations and applications, Contemp. Math., 483, Amer.
  • Math. Soc., Providence, RI, (2009), 205-220.
  • [15] Y. Sheng, Representations of hom-Lie algebras, Algebr. Represent. Theory,
  • 15(6) (2012), 1081-1098.

Details

Subjects Mathematics
Journal Section Articles
Authors

Amine Ben Abdeljelil This is me


Mohamed Elhamdadi This is me


Abdenacer Makhlouf This is me

Publication Date January 17, 2017
Published in Issue Year 2017, Volume 21, Issue 21

Cite

Bibtex @research article { ieja296030, journal = {International Electronic Journal of Algebra}, issn = {1306-6048}, eissn = {1306-6048}, address = {1710 Sokak, No:41, Batikent/Ankara}, publisher = {Abdullah HARMANCI}, year = {2017}, volume = {21}, number = {21}, pages = {55 - 75}, doi = {10.24330/ieja.296030}, title = {DERIVATIONS OF TERNARY LIE ALGEBRAS AND GENERALIZATIONS}, key = {cite}, author = {Abdeljelil, Amine Ben and Elhamdadi, Mohamed and Makhlouf, Abdenacer} }
APA Abdeljelil, A. B. , Elhamdadi, M. & Makhlouf, A. (2017). DERIVATIONS OF TERNARY LIE ALGEBRAS AND GENERALIZATIONS . International Electronic Journal of Algebra , 21 (21) , 55-75 . DOI: 10.24330/ieja.296030
MLA Abdeljelil, A. B. , Elhamdadi, M. , Makhlouf, A. "DERIVATIONS OF TERNARY LIE ALGEBRAS AND GENERALIZATIONS" . International Electronic Journal of Algebra 21 (2017 ): 55-75 <https://dergipark.org.tr/en/pub/ieja/issue/27921/296030>
Chicago Abdeljelil, A. B. , Elhamdadi, M. , Makhlouf, A. "DERIVATIONS OF TERNARY LIE ALGEBRAS AND GENERALIZATIONS". International Electronic Journal of Algebra 21 (2017 ): 55-75
RIS TY - JOUR T1 - DERIVATIONS OF TERNARY LIE ALGEBRAS AND GENERALIZATIONS AU - Amine BenAbdeljelil, MohamedElhamdadi, AbdenacerMakhlouf Y1 - 2017 PY - 2017 N1 - doi: 10.24330/ieja.296030 DO - 10.24330/ieja.296030 T2 - International Electronic Journal of Algebra JF - Journal JO - JOR SP - 55 EP - 75 VL - 21 IS - 21 SN - 1306-6048-1306-6048 M3 - doi: 10.24330/ieja.296030 UR - https://doi.org/10.24330/ieja.296030 Y2 - 2016 ER -
EndNote %0 International Electronic Journal of Algebra DERIVATIONS OF TERNARY LIE ALGEBRAS AND GENERALIZATIONS %A Amine Ben Abdeljelil , Mohamed Elhamdadi , Abdenacer Makhlouf %T DERIVATIONS OF TERNARY LIE ALGEBRAS AND GENERALIZATIONS %D 2017 %J International Electronic Journal of Algebra %P 1306-6048-1306-6048 %V 21 %N 21 %R doi: 10.24330/ieja.296030 %U 10.24330/ieja.296030
ISNAD Abdeljelil, Amine Ben , Elhamdadi, Mohamed , Makhlouf, Abdenacer . "DERIVATIONS OF TERNARY LIE ALGEBRAS AND GENERALIZATIONS". International Electronic Journal of Algebra 21 / 21 (January 2017): 55-75 . https://doi.org/10.24330/ieja.296030
AMA Abdeljelil A. B. , Elhamdadi M. , Makhlouf A. DERIVATIONS OF TERNARY LIE ALGEBRAS AND GENERALIZATIONS. IEJA. 2017; 21(21): 55-75.
Vancouver Abdeljelil A. B. , Elhamdadi M. , Makhlouf A. DERIVATIONS OF TERNARY LIE ALGEBRAS AND GENERALIZATIONS. International Electronic Journal of Algebra. 2017; 21(21): 55-75.
IEEE A. B. Abdeljelil , M. Elhamdadi and A. Makhlouf , "DERIVATIONS OF TERNARY LIE ALGEBRAS AND GENERALIZATIONS", International Electronic Journal of Algebra, vol. 21, no. 21, pp. 55-75, Jan. 2017, doi:10.24330/ieja.296030