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, Volume 1, Issue 3, pp 211–229 | Cite as

Involutionen von Jordan-Algebren

  • Karl-Heinz Helwig


In this paper we will give some algebraic results on certain Lie algebras defined by involutions of Jordan algebras. Most of the results will be used elsewhere for applications in analysis. Let A be the (-1)-eigenspace of an involution J≠Id of a central simple Jordan algebra A of degree at least 3, let D be the Lie algebra of all inner derivations of A leaving A_ invariant, and let h be the Lie algebra D+L(A_), where L denotes the regular representation of A. In the case where the 1-eigenspace A+ of J is central simple too, we will show A+=A_A_ and prove that h is semi-simple and irreducible on A. If A+ is not central simple, then A_A_ has codimension 1 in A+ and h is semi-simple, but irreducible on A_A_+A_ only if the characteristic of the groundfield does not divide the degree of A. At characteristic O we will view D as an extension of the derivationalgebra of A+ and determine the structure of the kernel of this extension.


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  1. 1.
    BRAUN, H. u. M. KOECHER: Jordan-Algebren. Bd. 128, Berlin-Heidelberg-New York: Springer 1966Google Scholar
  2. 2.
    HELWIG, K.-H.: Über Automorphismen und Derivationen von Jordan-Algebren. Indag. Math., 29, No.4, 381–394 (1967)Google Scholar
  3. 3.
    HELWIG, K.-H.: Halbeinfache reelle Jordan-Algebren. Math.Z. 109, 1–28 (1969)Google Scholar
  4. 4.
    HIRZEBRUCH, U.: Über Jordan-Algebren und kompakte Riemannsche symmetrische Räume vom Rang 1. Math.Z. 90, 339–354 (1965)Google Scholar
  5. 5.
    JACOBSON, N.: Completely reducible Lie algebras of linear transformations. Proc.Amer.Math.Soc. 2, 105–113 (1951)Google Scholar
  6. 6.
    KOECHER, M.: Imbedding of Jordan algebras into Lie algebras II. Amer.J. of Math. Vol. XC, No.2, 476–510 (1968)Google Scholar
  7. 7.
    MEYBERG, K.: Zur Dimensionsbestimmung einiger Primärkomponenten. Indag.Math. 29, No.1, 115–122 (1967)Google Scholar
  8. 8.
    MEYBERG, K.: Bemerkungen zu einem Isomorphiesatz der linearen Algebra. Indag.Math. 30, No.4, 449–451 (1968)Google Scholar

Copyright information

© Springer-Verlag 1969

Authors and Affiliations

  • Karl-Heinz Helwig
    • 1
  1. 1.Mathematisches Institut der Universität8 München

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