Forms of the Classical Lie Algebras

  • G. B. Seligman
Part of the Ergebnisse der Mathematik und ihrer Grenzgebiete book series (MATHE2, volume 40)

Abstract

The discussion in this chapter may be regarded as motivated by the problem of determining all Lie algebras L with non-singular Killing form over an arbitrary field F (characteristic ≠ 2, 3). By Chapter I, § 7, the problem reduces to the case where 1D50F is simple. If ℨ is the centroid of L, then L is normal simple when regarded as a Lie algebra over ℨ moreover, if S is a ℨ-linear transformation of L, we have TrF (S) = Tℨ/F (Tr (S)), where Tℨ/F is the trace in the field extension ℨ/F (cf. [223, p. 66]). It follows that L has non-singular Killing form over ℨ, and that T ℨ/F is not zero, hence that ℨ/F is a separable extension. Thus the problem is reduced to the study of finite separable extensions of F, and to the case where L is normal simple. Assuming L normal simple over F, let Ω be an algebraically closed extension of F. Then L Ω has non-singular Killing form, and so is a classical simple Lie algebra over Ω therefore L Ω belongs to one of an already determined set of isomorphism classes. The “problem of forms” is that of describing L when L Ω is known, i.e., of determining the F-isomorphism classes of algebras L such that L Ω belongs to a given Ω-isomorphism class.

Keywords

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Copyright information

© Springer-Verlag Berlin Heidelberg 1967

Authors and Affiliations

  • G. B. Seligman
    • 1
  1. 1.Department of MathematicsYale UniversityNew HavenUSA

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