Jordan triple product
Given a Jordan algebra we define a ternary product on by
This gives a simple method by which to compute the triple product in a special Jordan algebra.
A key instance of the Jordan triple product is the case when (setting for notation). Here we get
In a special Jordan algebra this becomes in the associative product. To treat a Jordan algebra as a quadratic Jordan algebra this product plays the role of one of the two quadratic operations: . The other is the usual squaring product .
To establish uniform proof for Jordan algebra in characteristic 2 and also include exceptional Jordan algebras it is often preferable to encode computations using the unary product: and the triple product . The connection between the triple product and the quadratic unary product is found in the Jordan identity:
This idea was exploited by McCrimmon to establish quadratic Jordan algebras and many uniform and previously unknown results on Jordan algebras.
The triple product can be compared to the Jacobi product on an algebra with a multiplication
To make a closer parallel use the typical assumption that (outside of characteristic 0) then we can write:
Since Jordan algebras are commutative we can also write
However, unlike Lie algebras where , in Jordan algebras the triple product is almost never 0.
|Title||Jordan triple product|
|Date of creation||2013-03-22 16:27:27|
|Last modified on||2013-03-22 16:27:27|
|Last modified by||Algeboy (12884)|
|Defines||Jordan triple product|