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.