On Reduced Exceptional Simple Jordan Algebras

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

Abstract

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
$$ a \cdot b = b \cdot a,\,\left[ {\left( {a \cdot a} \right) \cdot } \right] \cdot a = \left( {a \cdot a} \right) \cdot \left( {b \cdot a} \right) $$
(1)
The algebra \( {\mathfrak{A}^{\left( + \right)}} \) and its subalgebras are called special Jordan algebras.

Keywords

Lution Bedding 

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Copyright information

© Birkhäuser Boston 1989

Authors and Affiliations

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

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