Classification and Representation of Semi-Simple Jordan Algebras

  • F. D. Jacobson
  • N. Jacobson
Part of the Contemporary Mathematicians book series (CM)

Abstract

In the present paper we use the term special Jordan algebra to denote a (non-associative) algebra \( \mathfrak{K} \) over a field of characteristic not two for which there exists a 1–1 correspondence aa R of \( \mathfrak{K} \) into an associative algebra \( \mathfrak{A} \) such that
$$ {\left( {a + b} \right)^R} = {a^R} + {b^R},\;{\left( {\alpha a} \right)^R} = \alpha {a^R} $$
(1)
for α in the underlying field and
$$ {\left( {a \cdot b} \right)^R} = \left( {{a^R}{b^R} + {b^R}{a^R}} \right)/2 $$
(2)
In the last equation the • denotes the product defined in the algebra \( \mathfrak{K} \). When there is no risk of confusion we shall also use the · to denote the Jordan product (xy+yx)/2 in an associative algebra. Jordan multiplication is in general non-associative but it is easy to verify that the following special rules hold:
$$ a \cdot b = b \cdot a,\;\left( {a \cdot b} \right) \cdot {a^2} = a \cdot \left( {b \cdot {a^2}} \right) $$
(3)
Hence these rules hold for the product in a special Jordan algebra.

Keywords

Bedding 

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Copyright information

© Birkhäuser Boston 1989

Authors and Affiliations

  • F. D. Jacobson
    • 1
  • N. Jacobson
    • 1
  1. 1.Yale UniversityNew HavenUSA

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