Advertisement

A Note on Lie Algebras of Characteristic p

Chapter
Part of the Contemporary Mathematicians book series (CM)

Abstract

Let \( \mathfrak{L} \) be a finite dimensional Lie algebra over a field of characteristic p, let \( \mathfrak{A} \) be the universal associative algebra of \( \mathfrak{L} \) ([1] and [4]) and let \( \mathfrak{C} \) be the center of \( \mathfrak{A} \). In this note we prove that if a is a linear element of \( \mathfrak{A} \) then there exists a non-zero polynomial φ such that <j> (a) ε \( \mathfrak{C} \). We use this result to obtain the following: (1) a simple direct proof of Iwasawa’s theorem ([2], p. 420) that every finite dimensional Lie algebra of characteristic p has a faithful finite dimensional representation, (2) a proof of a conjecture of Chevalley that every finite dimensional Lie algebra of characteristic p has a representation which is not completely reducible, (3) a proof that \( \mathfrak{A} \) can be imbedded in a division algebra.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    G. Birkhoff, “Representability of Lie algebras…,” Annals of Mathematics, vol. 38 (1937), pp. 326–332.Google Scholar
  2. [2]
    K. Iwasawa, “On the representation of Lie algebras,” Japanese Journal of Mathematics, vol. 19 (1948), pp. 405–426.Google Scholar
  3. [3]
    N. Jacobson, “Restricted Lie algebras of characteristic p,” Transactions of the American Mathematical Society, vol. 50 (1941), pp. 15–25.Google Scholar
  4. [4]
    E. Witt, „Treue Darstellung Liescher Ringe,“ Journal für die reine und angewandedte Mathematik, vol. 177 (1937), pp. 152–160.CrossRefGoogle Scholar

Copyright information

© Birkhäuser Boston 1989

Authors and Affiliations

  1. 1.Yale UniversityUSA

Personalised recommendations