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:
“There is special providence in the fall of a sparrow. If it be now, ‘tis not to come; if it be not to come, it will be
now; if it be not now, yet it will come—the readiness is
all.”
—William Shakespeare (1564–1616)
“All of Western tradition, from the late bloom of the British Empire right through the early doom of Vietnam, dictates that you do something spectacular and irreversible whenever you find yourself in or whenever you impose yourself upon a wholly unfamiliar situation belonging to somebody else. Frequently it’s your soul or your honor or your manhood, or democracy itself, at stake.”
—June Jordan (b. 1939)