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Engel subalgebras of n-Lie algebras

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Abstract

Engel subalgebras of finite-dimensional n-Lie algebras are shown to have similar properties to those of Lie algebras. Using these, it is shown that an n-Lie algebra, all of whose maximal subalgebras are ideals, is nilpotent. A primitive 2-soluble n-Lie algebra is shown to split over its minimal ideal, and all the complements to its minimal ideal are conjugate. A subalgebra is shown to be a Cartan subalgebra if and only if it is minimal Engel, provided that the field has sufficiently many elements. Cartan subalgebras are shown to have a property analogous to intravariance.

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References

  1. Kasymov, Sh. M.: On a theory of n-Lie algebras. Algebra and Logic, 26, 155–166 (1987)

    Article  MATH  Google Scholar 

  2. Barnes, D. W.: Sortability of representations of Lie algebras. J. Algebra, 27, 486–490 (1973)

    Article  MATH  MathSciNet  Google Scholar 

  3. Williams, M.: Nilpotent n-Lie Algebras. Thesis, North Carolina State University, 2004

  4. Bai, R., Chen, L. Y., Meng, D.: The Frattini subalgebras of n-Lie algebras. Acta Mathematica Sinica, English Series, 23(5), 847–856 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  5. Barnes, D. W.: On the cohomology of soluble Lie algebras. Math. Zeitschr., 101, 343–349 (1967)

    Article  MathSciNet  Google Scholar 

  6. Barnes, D. W., Newell, M. L.: Some theorems on saturated homomorphs of soluble Lie algebras. Math. Zeitschr., 115, 179–187 (1970)

    Article  MATH  MathSciNet  Google Scholar 

  7. Gaschütz, W.: Zur Theorie der endlichen auflösbaren Gruppen. Math. Zeitschr., 80, 300–305 (1963)

    Article  MATH  Google Scholar 

  8. Doerk, K., Hawkes, T.: Finite soluble groups, De Gruyter, Berlin-New York, 1992

    MATH  Google Scholar 

  9. Barnes, D. W., Gastineau-Hills, H. M.: On the theory of soluble Lie algebras. Math. Zeitschr., 106, 343–354 (1968)

    Article  MATH  MathSciNet  Google Scholar 

  10. Barnes, D. W.: The Frattini argument for Lie algebras. Math. Zeitschr., 133, 277–283 (1973)

    Article  MATH  MathSciNet  Google Scholar 

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  1. 1 Little Wonga Rd, Cremorne, NSW, 2090, Australia

    Donald W. Barnes

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  1. Donald W. Barnes
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Correspondence to Donald W. Barnes.

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This work was done while the author was an Honorary Associate of the School of Mathematics and Statistics, University of Sydney

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Barnes, D.W. Engel subalgebras of n-Lie algebras. Acta. Math. Sin.-English Ser. 24, 159–166 (2008). https://doi.org/10.1007/s10114-007-1008-7

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  • DOI: https://doi.org/10.1007/s10114-007-1008-7

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