Advertisement

Locally Nilpotent Sets of Derivations

Conference paper
  • 287 Downloads
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 319)

Abstract

Let B be an algebra over a field \(\mathbf {k}\). We define what it means for a subset of \({{\,\mathrm{Der}\,}}_\mathbf {k}(B)\) to be a locally nilpotent set. We prove some basic results about that notion and explore the following questions. Let L be a Lie subalgebra of \({{\,\mathrm{Der}\,}}_\mathbf {k}(B)\); if \(L \subseteq {{\,\mathrm{LND}\,}}(B)\) then does it follow that L is a locally nilpotent set? Does it follow that L is a nilpotent Lie algebra?

Keywords

Locally nilpotent derivation Nilpotent Lie algebra 

2010 Mathematics Subject Classification

Primary: 14R20 13N15. Secondary: 17B30 17B65 17B66 

References

  1. 1.
    Ado, I.D.: The representation of Lie algebras by matrices. Am. Math. Soc. Transl. 1949(2), 21 (1949)MathSciNetGoogle Scholar
  2. 2.
    Golod, E.S.: On nil-algebras and finitely approximable \(p\)-groups. Izv. Akad. Nauk SSSR Ser. Math. 28, 273–276 (1964). (Russian)MathSciNetGoogle Scholar
  3. 3.
    Golod, E.S.: Some problems of Burnside type. In: Proceedings of the International Congress of Mathematicians (Moscow, 1966), pp. 284–289. Izdat. “Mir”, Moscow (1968) (Russian)Google Scholar
  4. 4.
    Iwasawa, K.: On the representation of Lie algebras. Jpn. J. Math. 19, 405–426 (1948)MathSciNetGoogle Scholar
  5. 5.
    Jacobson, N.: Lie Algebras. Dover Publications, Inc., New York (1979). Republication of the 1962 originalzbMATHGoogle Scholar
  6. 6.
    Kostrikin, A.I.: Around Burnside. Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 20. Springer, Berlin (1990). Translated from the Russian and with a preface by James WiegoldGoogle Scholar
  7. 7.
    Rowen, L.H.: Ring Theory, vol. II. Pure and Applied Mathematics, vol. 128. Academic Press, Inc., Boston (1988)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsUniversity of OttawaOttawaCanada

Personalised recommendations