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:
“The books may say that nine-month-olds crawl, say their first words, and are afraid of strangers. Your exuberantly concrete and special nine-month-old hasn’t read them. She may be walking already, not saying a word and smiling gleefully at every stranger she sees. . . . You can support her best by helping her learn what she’s trying to learn, not what the books say a typical child ought to be learning.”
—Amy Laura Dombro (20th century)
“As a child I was taught that to tell the truth was often painful. As an adult I have learned that not to tell the truth is more painful, and that the fear of telling the truth—whatever the truth may be—that fear is the most painful sensation of a moral life.”
—June Jordan (b. 1936)