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Local Automorphisms on Finite-Dimensional Lie and Leibniz Algebras

  • Shavkat Ayupov
  • Karimbergen Kudaybergenov
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 264)

Abstract

We prove that a linear mapping on the algebra \(\mathfrak {sl}_n\) of all trace zero complex matrices is a local automorphism if and only if it is an automorphism or an anti-automorphism. We also show that a linear mapping on a simple Leibniz algebra of the form \(\mathfrak {sl}_n\dot{+}\mathscr {I}\) is a local automorphism if and only if it is an automorphism. We give examples of finite-dimensional nilpotent Lie algebras \(\mathscr {L}\) with \(\dim \mathscr {L} \ge 3\) which admit local automorphisms which are not automorphisms.

Keywords

Simple Lie algebra Simple Leibniz algebra Nilpotent Lie algebra Automorphism Local automorphism 

Notes

Acknowledgements

The motivation to study the problems considered in this paper came out from discussions made with Professor E.Zelmanov during the Second USA–Uzbekistan Conference held at the Urgench State University on August, 2017, which the authors gratefully acknowledge. The authors are indebted to Professor Mauro Costantini from the University of Padova for valuable comments to the initial version of the present paper.

References

  1. 1.
    Ayupov, Sh.A, Kudaybergenov, K.K.: 2-local derivations and automorphisms on B(H). J. Math. Anal. Appl. 395, 15–18 (2012)Google Scholar
  2. 2.
    Ayupov, Sh.A., Kudaybergenov, K.K., Nurjanov, B.O., Alauadinov, A.K.: Local and 2-local derivations on noncommutative Arens algebras. Math. Slovaca 64, 423–432 (2014)Google Scholar
  3. 3.
    Ayupov, Sh.A., Arzikulov, F.N.: 2-local derivations on semi-finite von Neumann algebras. Glasgow Math. J. 56, 9–12 (2014)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Ayupov, Sh.A., Kudaybergenov, K.K.: 2-local derivations on von Neumann algebras. Positivity 19, 445–455 (2015)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Ayupov, Sh.A., Kudaybergenov, K.K., Rakhimov, I.S.: 2-Local derivations on finite-dimensional Lie algebras. Linear Algebr. Appl. 474, 1–11 (2015)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Ayupov, Sh.A., Kudaybergenov, K.K.: Local derivations on finite dimensional Lie algebras. Linear Algebr. Appl. 493, 381–398 (2016)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Ayupov, Sh.A., Kudaybergenov, K.K.: 2-Local automorphisms on finite-dimensional Lie algebras. Linear Algebr. Appl. 507, 121–131 (2016)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Ayupov, Sh.A., Kudaybergenov, K.K., Omirov, A.B.: Local and 2-local derivations and automorphisms on simple Leibniz algebras (2017). arXiv:1703.10506
  9. 9.
    Ayupov, Sh.A., Kudaybergenov, K.K., Omirov, A.B., Zhao, K.: Semisimple Leibniz algebras and their derivations and automorphisms (2017). arXiv:1708.08082
  10. 10.
    Chen, Z., Wang, D.: Nonlinear maps satisfying derivability on standard parabolic subalgebras of finite-dimensional simple Lie algebras. Linear Multilinear Algebr. 59, 261–270 (2011)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Chen, Z., Wang, D.: 2-Local automorphisms of finite-dimensional simple Lie algebras. Linear Algebr. Appl. 486, 335–344 (2015)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Humphreys, J.E.: Introduction to Lie Algebras and Representation Theory. Springer, New York (1972)CrossRefGoogle Scholar
  13. 13.
    Jacobson, N.: Lie Algebras. Dover Publications. Inc., New York (1979)Google Scholar
  14. 14.
    Kadison, R.V.: Local derivations. J. Algebr. 130, 494–509 (1990)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Kim, S.O., Kim, J.S.: Local automorphisms and derivations on \(M_n\). Proc. Am. Math. Soc. 132, 1389–1392 (2004)Google Scholar
  16. 16.
    Larson, D.R., Sourour, A.R.: Local derivations and local automorphisms of B(X). Proc. Sympos. Pure Math. 51, 187–194 (1990)Google Scholar
  17. 17.
    Molnar, L.: Selected Preserver Problems on Algebraic Structures of Linear Operators and on Function Spaces. Springer, Berlin (2007)Google Scholar
  18. 18.
    Šemrl, P.: Linear mappings preserving square-zero matrices. Bull. Aust. Math. Soc. 48, 365–370 (1993)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Šemrl, P.: Local automorphisms and derivations on B(H). Proc. Am. Math. Soc. 125, 2677–2680 (1997)Google Scholar
  20. 20.
    Vergne, M.: Réductibilité de la varireté des algébres de Lie nilpotentes. C. R. Acad. Sci. Paris. 263, 4–6 (1966)Google Scholar
  21. 21.
    Wan, Z.: Lie Algebras. Pergamon Press Ltd, Oxford (1975)CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.V.I.Romanovskiy Institute of Mathematics, Uzbekistan Academy of SciencesTashkentUzbekistan
  2. 2.Ch. Abdirov 1, Department of MathematicsKarakalpak State UniversityNukusUzbekistan

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