Derivation Lie algebras of semidirect sums


Suppose that \(L=L_1 \ltimes L_2\) is a semidirect sum of two Lie algebras. In this article, we first obtain the structure of \({\text {Der}}(L:L_2)\) the subalgebra of \({\text {Der}}(L)\) that consists of those derivations mapping \(L_2\) to itself. Then we investigate some conditions under which \({\text {Der}}(L:L_2)\) is also a semidirect sum.

This is a preview of subscription content, log in to check access.


  1. 1.

    Alemi, M.R., Saeedi, F.: Derivation algebra of direct sum of Lie algebra (submitted)

  2. 2.

    Bardakov, V., Singh, M.: Extensions and automorphisms of Lie algebras. J. Algebra Appl. 16, 1750162 (2017)

    MathSciNet  Article  Google Scholar 

  3. 3.

    Bass, H., Oesterle, J., Weinstein, A.: Poisson Structures and Their Normal Forms. Birkhauser, Basel (2005).

    Google Scholar 

  4. 4.

    Bidwell, J.N.S., Curran, M.J., McCoughan, D.J.: Automorphisms of direct products of finite groups. Arche. Math. (Basel) 86(6), 481–489 (2006)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Curran, J.: Automorphisms of semidirect products. Math. Proc. R. Iy. Acad. 108, 205–210 (2008)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Degrijs, D., Petrosyan, N.: On the cohomology of split Lie algebra extensions. J. Lie Theory 22, 1–15 (2012)

    MathSciNet  Google Scholar 

  7. 7.

    Dietz, J.: Automorphism groups of semidirect product. Commun. Algebra 40, 3308–3316 (2012)

    Article  Google Scholar 

  8. 8.

    Dietz, J.: On automorphisms of products of group. In: Campbell, C.M., et al. (eds.) Groups St, Andrews, vol. 1. London Mathematical Society, London (2005)

    Google Scholar 

  9. 9.

    Dixmier, J., Lister, W.G.: Derivations of nilpotent Lie algebras. Proc. Am. Math. Soc. 8, 155–158 (1957)

    MathSciNet  Article  Google Scholar 

  10. 10.

    Fouladi, S., Jamali, A.R., Orfi, R.: Automorphisma of abelian Lie ring extensions. J. Algebra Appl. 16(8), 1750176 (2017)

    MathSciNet  Article  Google Scholar 

  11. 11.

    Hilton, P.J., Stammback, V.: A Course in Homological Algebra. Graduate texts in Mathematics, vol. 4, 2nd edn. Springer, New York (1997)

    Google Scholar 

  12. 12.

    Li, W., Wilson, R.L.: Central extensions of some Lie algebras. Proc. Am. Math. Soc. 126, 2569–2577 (1998)

    MathSciNet  Article  Google Scholar 

  13. 13.

    Mundt, E.: Constant Young–Mills potentials. J. Lie Theory 107(3), 107–115 (1993)

    MathSciNet  MATH  Google Scholar 

  14. 14.

    Onishchik, A.L., Vinberg, E.B. (eds.): Lie Groups and Lie Algebras. Springer, New York (1994).

    Google Scholar 

  15. 15.

    Passi, I., Singh, M., Yadav, M.K.: Automorphisms of abelian group extensions. J. Algebra 324(4), 820–830 (2010)

    MathSciNet  Article  Google Scholar 

  16. 16.

    Schenkman, E.: On the derivation algebra and the holomorph of a nilpotent Lie algebra. Mem. Amer. Math. Soc. 14, 15–22 (1955)

    MathSciNet  MATH  Google Scholar 

  17. 17.

    Saeedi, F., Sheikh-Mohseni, S.: A characterization of stem algebras in terms of central derivations. Algebras Represent. Theory 20, 1143–1150 (2017)

    MathSciNet  Article  Google Scholar 

  18. 18.

    Saeedi, F., Sheikh-Mohseni, S.: On ID\(^*\)-derivations of Filippov algebras. Southeast Asian Bull. Math. 11(2), 1850050 (2018)

    MathSciNet  MATH  Google Scholar 

  19. 19.

    Varadarajan, V.S.: Lie Groups, Lie Algebras, and Their Representations. Springer, New York (1984).

    Google Scholar 

Download references

Author information



Corresponding author

Correspondence to F. Saeedi.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Barati, M., Saeedi, F. & Alemi, M.R. Derivation Lie algebras of semidirect sums. Rend. Circ. Mat. Palermo, II. Ser 69, 653–663 (2020).

Download citation


  • Semidirect sum
  • Derivation
  • Cohomology of Lie algebra

Mathematics Subject Classification

  • Primary 17B40
  • 17B56
  • Secondary 18G60