Skip to main content
SpringerLink
Log in

Projections of Semisimple Lie Algebras

  • Published:
Algebra and Logic Aims and scope

It is proved that the property of being a semisimple algebra is preserved under projections (lattice isomorphisms) for locally finite-dimensional Lie algebras over a perfect field of characteristic not equal to 2 and 3, except for the projection of a three-dimensional simple nonsplit algebra. Over fields with the same restrictions, we give a lattice characterization of a three-dimensional simple split Lie algebra and a direct product of a one-dimensional algebra and a three-dimensional simple nonsplit one.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. D. W. Barnes, “Lattice isomorphisms of Lie algebras,” J. Aust. Math. Soc., 4, No. 4, 470-475 (1964).

    Article  MathSciNet  MATH  Google Scholar 

  2. M. Goto, “Lattices of subalgebras of real Lie algebras,” J. Alg., 11, No. 1, 6-24 (1969).

    Article  MathSciNet  MATH  Google Scholar 

  3. A. G. Gein, “Projections of Lie algebras of characteristic 0,” Izv. Vyssh. Uch. Zav., Mat., No. 4, 26-31 (1978).

  4. G. Gr¨atzer, General Lattice Theory, Academie-Verlag, Berlin (1978).

    Book  Google Scholar 

  5. R. K. Amayo and J. Schwarz, “Modularity in Lie algebras,” Hiroshima Math. J., 10, 311-322 (1980).

    Article  MathSciNet  MATH  Google Scholar 

  6. A. G. Gein, “On modular subalgebras of Lie algebras,” in A Study of Algebraic Systems via Properties of Their Subsystems [in Russian], Ural State Univ., Sverdlovsk (1987), pp. 27-33.

  7. A. G. Gein, “The modular law and relative complements in the lattice of subalgebras of a Lie algebra,” Izv. Vyssh. Uch. Zav., Mat., No. 3, 18-25 (1987).

  8. N. Jacobson, Lie Algebras, Interscience, New York (1962).

    MATH  Google Scholar 

  9. A. A. Premet and K. N. Semenov, “Varieties of residually finite Lie algebras,” Mat. Sb., 137(179), No. 1(9), 103-113 (1988).

  10. V. R. Varea, “Lie algebras whose proper subalgebras are either semisimple, abelian or almostabelian,” Hiroshima Math. J., 24, No. 2, 221-241 (1994).

    Article  MathSciNet  MATH  Google Scholar 

  11. A. G. Gein, “Finite-dimensional simple Lie algebras with a subalgebra lattice of length 3,” Izv. Vyssh. Uch. Zav., Mat., No. 10, 74-78 (2012).

    MathSciNet  MATH  Google Scholar 

  12. A. G. Gein, “Modular subalgebras and projections of locally finite dimensional Lie algebras of characteristic zero,” Mat. Zap. Ural Univ., 13, 39-51 (1983).

    MathSciNet  Google Scholar 

  13. A. G. Gein, “Projections of semisimple Lie algebras,” Mal’tsev Readings (2017), p. 110; http://www.math.nsc.ru/conference/malmeet/17/malmeet17.pdf.

Download references

Author information

Authors and Affiliations

  1. El’tsin Ural Federal University, ul. Mira 19, Yekaterinburg, 620002, Russia

    A. G. Gein

Authors
  1. A. G. Gein
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. G. Gein.

Additional information

Dedicated to the memory of Yu. N. Mukhin, my scientific supervisor

Supported through the Competitiveness Project (Agreement No. 02.A03.21.0006 of 27.08.2013 between the Ministry of Education and Science of the Russian Federation and the Ural Federal University).

Translated from Algebra i Logika, Vol. 58, No. 2, pp. 149-166,March-April, 2019.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Gein, A.G. Projections of Semisimple Lie Algebras. Algebra Logic 58, 103–114 (2019). https://doi.org/10.1007/s10469-019-09529-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10469-019-09529-z

Keywords

Search

Navigation