On Reduced Exceptional Simple Jordan Algebras

  • A. A. Albert
  • N. Jacobson
Part of the Contemporary Mathematicians book series (CM)


In this paper we shall be concerned only with finite dimensional algebras over an arbitrary field \( \mathfrak{F} \) of characteristic not two. Let \( \mathfrak{A} \) be an associative algebra over \( \mathfrak{F} \) and ab the associative product composition of \( \mathfrak{A} \). Then the vector space \( \mathfrak{A} \) is a Jordan algebra \( {\mathfrak{A}^{\left( + \right)}} \) relative to the composition a·b = 1/2(ab + ba), that is, this composition satisfies the defining identities
$$ a \cdot b = b \cdot a,\,\left[ {\left( {a \cdot a} \right) \cdot } \right] \cdot a = \left( {a \cdot a} \right) \cdot \left( {b \cdot a} \right) $$
The algebra \( {\mathfrak{A}^{\left( + \right)}} \) and its subalgebras are called special Jordan algebras.


Lution Bedding 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Albert, A. A., A structure theory for Jordan algebras, these Annals, vol. 48 (1947), pp. 446–467.Google Scholar
  2. [2]
    Albert, A. A., A theory of power associative commutative algebras, Trans. Amer. Math. Soc, vol. 69 (1950), pp. 503–527.Google Scholar
  3. [3]
    Albert, A. A., Quadratic forms permitting composition, these Annals, vol. 43 (1942), pp. 161–177.CrossRefGoogle Scholar
  4. [4]
    Albert, A. A., A construction of exceptional Jordan division algebras, to appear in these Annals.Google Scholar
  5. [5]
    Hasse, H., Darstellbarkeit von Zahlen durch quadratische Formen in einem beliebigen algebraischen Zahlkorper, J. Reine Angew Math., vol. 152 (1923), pp. 113–130.Google Scholar
  6. [6]
    Jacobson, N., Cayley numbers and simple Lie algebras of type G, Duke Math. J., vol. 5 (1939), pp. 775–783.CrossRefGoogle Scholar
  7. [7]
    Jacobson, N., Structure of alternative and Jordan bimodules, Osaka Math. J., vol. 6 (1954), pp. 1–71.Google Scholar
  8. [8]
    Schafer, R. D., The exceptional simple Jordan algebras, Amer. J. Math., vol. 70 (1948), pp. 82–94.CrossRefGoogle Scholar
  9. [9]
    Tomber, M. L., Lie algebras of type F, Proc. Amer. Math. Soc, vol. 4 (1953), pp. 759–768.Google Scholar
  10. [10]
    Zorn, M., Alternativkorper und quadratische system, Hamb. Abh., vol. 9 (1933), pp. 395–402.CrossRefGoogle Scholar

Copyright information

© Birkhäuser Boston 1989

Authors and Affiliations

  • A. A. Albert
    • 1
  • N. Jacobson
    • 1
  1. 1.Yale UniversityUSA

Personalised recommendations