# Classification and Representation of Semi-Simple Jordan Algebras

Chapter
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

Associative Algebra Clifford Algebra Jordan Algebra Simple Algebra Universal Algebra
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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