Classification and Representation of Semi-Simple Jordan Algebras

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 a→a ^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.