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:
“Mr. Christian, it is about time for many people to begin to come to the White House to discuss different phases of the coal strike. When anybody comes, if his special problem concerns the state, refer him to the governor of Pennsylvania. If his problem has a national phase, refer him to the United States Coal Commission. In no event bring him to me.”
—Calvin Coolidge (1872–1933)
“We do not deride the fears of prospering white America. A nation of violence and private property has every reason to dread the violated and the deprived.”
—June Jordan (b. 1939)