Given an associative algebra A (not of characteristic 2), one can construct a Jordan algebra A+ with the same underlying addition, and a new multiplication (x.y) as follows.
.
If A has an involution, then the involution fixes elements of the form
(xy + yx) / 2.
Thus the set of all elements fixed by the involution form a subalgebra of A+ .
A Jordan algebra that is isomorphic to an algebra of the form A+ is known as a special Jordan algebra. Otherwise it is an exceptional Jordan algebra.
Examples
The set of self-adjoint real, complex, or quaternionic matrices with multiplication
(xy + yx) / 2
form a special Jordan algebra.
The set of 3×3 self-adjoint matrices over the octonions again with multiplication
(xy + yx) / 2.
Despite the similarity to the previous example, this is an exceptional Jordan algebra. (The octonions are not an associative algebra.)
Jordan had four universities with a combined enrollment of nearly 29,000; more than one-third of the students were women (11,000).
Jordan's location as a buffer zone between the settled region of the Mediterranean littoral west of the Jordan River and the major part of the desert to the east contributed significantly to the country's experience in ancient and more recent times.
Jordan, Hashemite Kingdom of (in Arabic, al-Mamlakah al-Urdunniyah al-Hashemiyah), kingdom in the Middle East, bordered on the north by Syria, on the east by Iraq and Saudi Arabia, on the south by Saudi Arabia and the Gulf of Aqaba, and on the west by Israel and the West Bank.
Feeds |
Contact