Editing Jordan algebra (section)

==Special Jordan algebras==

Notice first that an [[associative algebra]] is a Jordan algebra if and only if it is commutative. 

Given any associative algebra ''A'' (not of [[Characteristic (algebra)|characteristic]] 2), one can construct a Jordan algebra ''A''<sup>+</sup> using the same underlying addition and a new multiplication, the '''Jordan product''' defined by:
:<math>x\circ y = \frac{xy+yx}{2}.</math>

These Jordan algebras and their subalgebras are called '''special Jordan algebras''', while all others are '''exceptional Jordan algebras'''. This construction is analogous to the [[Lie algebra]] associated to ''A'', whose product (Lie bracket) is defined by the commutator <math>[x,y] = xy - yx</math>.

The [[Anatoly Shirshov|Shirshov]]–Cohn theorem states that any Jordan algebra with two [[Generating set|generators]] is special.<ref name="mcc100">{{harvnb|McCrimmon|2004|p=100}}</ref> Related to this, Macdonald's theorem states that any polynomial in three variables, having degree one in one of the variables, and which vanishes in every special Jordan algebra, vanishes in every Jordan algebra.<ref name="mcc99">{{harvnb|McCrimmon|2004|p=99}}</ref>

===Hermitian Jordan algebras===

If (''A'', ''&sigma;'') is an associative algebra with an [[involution (mathematics)|involution]] ''&sigma;'', then if ''&sigma;''(''x'') = ''x'' and ''&sigma;''(''y'') = ''y'' it follows that <math display="inline">\sigma(xy + yx) = xy + yx.</math> Thus the set of all elements fixed by the involution (sometimes called the ''hermitian'' elements) form a subalgebra of ''A''<sup>+</sup>, which is sometimes denoted H(''A'',''&sigma;'').