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DERIVATIONS OF TERNARY LIE ALGEBRAS AND GENERALIZATIONS

Year 2017, , 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, , 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.
There are 31 citations in total.

Details

Subjects Mathematical Sciences
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

Cite

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. https://doi.org/10.24330/ieja.296030
AMA Abdeljelil AB, Elhamdadi M, Makhlouf A. DERIVATIONS OF TERNARY LIE ALGEBRAS AND GENERALIZATIONS. IEJA. January 2017;21(21):55-75. doi:10.24330/ieja.296030
Chicago Abdeljelil, Amine Ben, Mohamed Elhamdadi, and Abdenacer Makhlouf. “DERIVATIONS OF TERNARY LIE ALGEBRAS AND GENERALIZATIONS”. International Electronic Journal of Algebra 21, no. 21 (January 2017): 55-75. https://doi.org/10.24330/ieja.296030.
EndNote Abdeljelil AB, Elhamdadi M, Makhlouf A (January 1, 2017) DERIVATIONS OF TERNARY LIE ALGEBRAS AND GENERALIZATIONS. International Electronic Journal of Algebra 21 21 55–75.
IEEE A. B. Abdeljelil, M. Elhamdadi, and A. Makhlouf, “DERIVATIONS OF TERNARY LIE ALGEBRAS AND GENERALIZATIONS”, IEJA, vol. 21, no. 21, pp. 55–75, 2017, doi: 10.24330/ieja.296030.
ISNAD Abdeljelil, Amine Ben et al. “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.
JAMA Abdeljelil AB, Elhamdadi M, Makhlouf A. DERIVATIONS OF TERNARY LIE ALGEBRAS AND GENERALIZATIONS. IEJA. 2017;21:55–75.
MLA Abdeljelil, Amine Ben et al. “DERIVATIONS OF TERNARY LIE ALGEBRAS AND GENERALIZATIONS”. International Electronic Journal of Algebra, vol. 21, no. 21, 2017, pp. 55-75, doi:10.24330/ieja.296030.
Vancouver Abdeljelil AB, Elhamdadi M, Makhlouf A. DERIVATIONS OF TERNARY LIE ALGEBRAS AND GENERALIZATIONS. IEJA. 2017;21(21):55-7.