Special Jordan Algebras
Given an associative algebra A (not of characteristic 2), one can construct a Jordan algebra A+ using the same underlying addition vector space. Notice first that an associative algebra is a Jordan algebra if and only if it is commutative. If it is not commutative we can define a new multiplication on A to make it commutative, and in fact make it a Jordan algebra. The new multiplication x ∘ y is as follows:
This defines a Jordan algebra A+, and we call these Jordan algebras, as well as any subalgebras of these Jordan algebras, special Jordan algebras. All other Jordan algebras are called exceptional Jordan algebras. The Shirshov–Cohn theorem states that any Jordan algebra with two generators is special. Related to this, Macdonald's theorem states that any polynomial in three variables, which has degree one in one of the variables, and which vanishes in every special Jordan algebra, vanishes in every Jordan algebra.
Read more about this topic: Jordan Algebra
Famous quotes containing the words special and/or jordan:
“O my Brothers! love your Country. Our Country is our home, the home which God has given us, placing therein a numerous family which we love and are loved by, and with which we have a more intimate and quicker communion of feeling and thought than with others; a family which by its concentration upon a given spot, and by the homogeneous nature of its elements, is destined for a special kind of activity.”
—Giuseppe Mazzini (1805–1872)
“Like a lot of Black women, I have always had to invent the power my freedom requires ...”
—June Jordan (b. 1936)