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Generalized Derivations of Lie Color Algebras

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Abstract

In this paper, we give some basic properties of the generalized derivation algebra GDer(L) of a Lie color algebra L. In particular, we prove that GDer(L) = QDer(L) + QC(L), the sum of the quasiderivation algebra and the quasicentroid. We also prove that QDer(L) can be embedded as derivations in a larger Lie color algebra.

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References

  1. Bahturin Y.A., Mikhalev A.A., Petrogradshy V.M., Zaicev M.N.: Infinite-dimensional Lie superalgebras, de Gruyter Expositions in Mathematics, vol. 7. Walter de Gruyter & Co., Berlin (1992)

    Book  Google Scholar 

  2. Benoist Y.: La partie semi-simple de l’algèbre des dérivations d’une algèbre de Lie nilpotente. C. R. Acad. Sci. Paris. 307, 901–904 (1988)

    MathSciNet  MATH  Google Scholar 

  3. Bergen J., Passman D.S.: Delta ideal of Lie color algebras. J. Algebra 177, 740–754 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bergen J., Grzeszczuk P.: Simple Jordan color algebras arising from associative graded algebras. J. Algebra 246, 915–950 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  5. Feldvoss J.: Representations of Lie color algebras. Adv. Math. 157, 95–137 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  6. Fialkow, L.A.: Generalized derivations. Topics in modern operator theory (Timişoara/Herculane, 1980), pp. 95–103. Operator Theory: Adv. Appl. vol. 2. Birkhä user, Basel (1981)

  7. Filippov, V.T.: On δ-derivations of Lie algebras, (Russian) Sibirsk. Mat. Zh. 39(6), 1409–1422 (1998) iv; translation in Siberian Math. J. 39(6), 1218–1230 (1998)

    Google Scholar 

  8. Filippov, V.T.: On δ-derivations of prime Lie algebras. (Russian) Sibirsk. Mat. Zh. 40(1), 201–213 (1999) v; translation in Siberian Math. J. 40(1), 174–184 (1999)

  9. Filippov V.T.: On δ-derivations of prime Lie algebras. (Russian) Dokl. Akad. Nauk 364(4), 453–454 (1999)

    Google Scholar 

  10. Filippov, V.T.: On δ-derivations of prime alternative and Maltsev algebras. (Russian) Algebra Log. 39(5), 618–625 (2000), 632; translation in Algebra and Logic 39(5), 354–358 (2000)

  11. Hopkins N.C.: Generalized derivations of nonassociative algebras. Nova J. Math. Game Theory Algebra 5, 215–224 (1996)

    MathSciNet  MATH  Google Scholar 

  12. Jimenez-Gestal C., Perez-Izquierdo J.M.: Ternary derivations of generalized Cayley–Dickson algebras. Commun. Algebra 31, 5071–5094 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  13. Jimenez-Gestal C., Perez-Izquierdo J.M.: Ternary derivations of finite-dimensional real division algebras. Linear Algebra Appl. 428, 2192–2219 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  14. Kac V.G.: Lie superalgebras. Adv. Math. 26, 8–96 (1977)

    Article  MATH  Google Scholar 

  15. Kaigorodov, I.B.: On δ-derivations of simple finite-dimensional Jordan superalgebras. (Russian) Algebra Logika 46(5), 585–605 (2007) 664; translation in Algebra Logic 46(5), 318–329 (2007)

  16. Kaigorodov, I.B.: On δ-derivations of classical Lie superalgebras. (Russian) Sibirsk. Mat. Zh. 50(3), 547–565 (2009) translation in Sib. Math. J. 50(3), 434–449 (2009)

  17. Leger G.F., Luks E.M.: Generalized derivations of Lie algebras. J. Algebra 228, 165–203 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  18. Mcanally D.S., Bracken A.J.: Uncolouring of Lie colour algebras. Bull. Austral. Math. Soc. 55, 425–428 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  19. Melville D.J.: Centroids of nilpotent Lie algebras. Commun. Algebra 20, 3649–3682 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  20. Montgomery S.: Constructing simple Lie superalgebras from associative graded algebras. J. Algebra 195, 558–579 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  21. Passman D.: Simple Lie color algebras of Witt type. J. Algebra 208, 698–721 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  22. Ree R.: Generalized Lie elements. Can. J. Math. 12, 493–502 (1960)

    Article  MathSciNet  MATH  Google Scholar 

  23. Scheunert M.: Generalized Lie algebras. J. Math. Phys. 20, 712–720 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  24. Su Y., Zhao K., Zhu L.: Simple color algebras of Weyl type. Israel. J. Math. 137, 109–123 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  25. Su Y., Zhao K., Zhu L.: Classification of derivation-simple color algebras related to locally finite derivations. J. Math. Phys. 45, 525–536 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  26. Vinberg, É.B.: Generalized derivations of algebras, Algebra and analysis (Irkutsk, 1989), 185–188, Amer. Math. Soc. Transl. Ser. 2, vol. 163. American Mathematical Society, Providence (1995)

  27. Wilson M.C.: Delta methods in enveloping algebras of Lie color algebras. J. Algebra 75, 661–696 (1995)

    Article  Google Scholar 

  28. Zhang R.X., Zhang Y.Z.: Generalized derivations of Lie superalgebras. Commun. Algebra 38, 3737–3751 (2010)

    Article  MATH  Google Scholar 

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Authors and Affiliations

  1. School of Mathematics and Statistics, Northeast Normal University, Changchun, 130024, China

    Liangyun Chen, Yao Ma & Lin Ni

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  1. Liangyun Chen
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  2. Yao Ma
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  3. Lin Ni
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Correspondence to Liangyun Chen.

Additional information

Supported by NNSF of China (No. 11171055), NSF of Jilin province (No.201115006), Scientific Research Foundation for Returned Scholars Ministry of Education of China and the Fundamental Research Funds for the Central Universities (No. IISSXT146).

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Chen, L., Ma, Y. & Ni, L. Generalized Derivations of Lie Color Algebras. Results. Math. 63, 923–936 (2013). https://doi.org/10.1007/s00025-012-0241-2

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