Skip to main content
SpringerLink
Log in

On Maximal Extensions of Nilpotent Lie Algebras

  • Research Articles
  • Published:
Functional Analysis and Its Applications Aims and scope

Abstract

Extensions of finite-dimensional nilpotent Lie algebras, in particular, solvable extensions, are considered. Some properties of maximal extensions are proved. A counterexample to L. Šnobl’s conjecture concerning the uniqueness of maximal solvable extensions is constructed.

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

References

  1. È. B. Vinberg, V. V. Gorbatsevich, and A. L. Onishchik, “Structure of Lie groups and Lie algebras”, Lie groups and Lie algebras-3, Itogi Nauki i Tekhniki. Ser. Sovrem. Probl. Mat. Fund. Napr., VINITI, Moscow, 1990, 5–253.

    MATH  Google Scholar 

  2. J. Humphrey, Linear Algebraic Groups, Springer-Verlag, Berlin, 1975.

    Book  Google Scholar 

  3. A. Malcev, “On solvable Lie algebras”, Izv. Akad. Nauk SSSR Ser. Mat., 9:5 (1945), 329–356.

    Google Scholar 

  4. A. L. Onishchik and Yu. B. Khakimdzhanov, “Semidirect sums of Lie algebras”, Mat. Zametki, 18:1 (1975), 31–40; English transl.: Math. Notes, 18:1 (1975), 600–604.

    MathSciNet  MATH  Google Scholar 

  5. V. V. Gorbatsevich, “Splittings of Lie groups and their application to the study of homogeneous spaces”, Izv. Akad. Nauk SSSR Ser. Mat., 43:6 (1979), 1227–1258; English transl.: Math. USSR-Izv., 15:3 (1980), 441–467.

    MathSciNet  MATH  Google Scholar 

  6. G. Hochschild and G. Mostow, “Extensions of representations of Lie groups and Lie algebras. I”, Amer. J. Math., 79 (1957), 924–942.

    Article  MathSciNet  MATH  Google Scholar 

  7. L. Šnobl, “On the structure of maximal solvable extensions and of Levi extensions of nilpotent Lie algebras”, J. Phys. A., 43:50 (2010).

    Article  MathSciNet  MATH  Google Scholar 

  8. L. Šnobl and P. Winternitz, “Classification and Identification of Lie Algebras”, CRM Monograph Series, Amer. Math. Soc., Providence, RI, 2014.

    Google Scholar 

  9. G. Favre, “Un algèbre de Lie caractéristiquement nilpotente de dimention \(7\)”, C. R. Acad. Sci. Paris, Ser. A, 274 (1972), 1338–1339.

    MathSciNet  MATH  Google Scholar 

  10. L. Magnin, “Determination of \(7\)-dimensional indecomposable nilpotent complex Lie algebras by adjoining a derivation to \(6\)-dimensional Lie algebras”, Algebr. Represent. Theory, 13:6 (2010), 723–753.

    Article  MathSciNet  MATH  Google Scholar 

  11. S. Tôgô, “On the derivations of Lie algebras”, Canadian J. Math., 13 (1961), 201–216.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

The author is grateful to B. Omirov for useful discussions of the results and to the referee for pointing out mistakes in the first version of the paper.

Author information

Authors and Affiliations

  1. Moscow, Russia

    V. V. Gorbatsevich

Authors
  1. V. V. Gorbatsevich
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to V. V. Gorbatsevich.

Additional information

Translated from Funktsional'nyi Analiz i ego Prilozheniya, 2022, Vol. 56, pp. 25–34 https://doi.org/10.4213/faa4005.

Translated by O. V. Sipacheva

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Gorbatsevich, V.V. On Maximal Extensions of Nilpotent Lie Algebras. Funct Anal Its Appl 56, 257–263 (2022). https://doi.org/10.1134/S0016266322040037

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0016266322040037

Keywords

Search

Navigation