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[parent] Jordan triple product (Definition)

Given a Jordan algebra $ J$ we define a ternary product on $ J$ by

$\displaystyle \{xyz\}=(x.y).z+(y.z).x - (z.x).y.$

When $ J$ is a special Jordan algebra of characteristic not 2, we know the product $ x.y=\frac{1}{2}(xy+yx)$. In this context a computation shows

$\displaystyle \{xyz\}=\frac{1}{2}(xyz+zyx).$
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 $ x=z$ (setting $ x^{.2}=x.x$ for notation). Here we get

$\displaystyle \{xyx\}=2 (x.y).x-x^{.2}.y.$
In a special Jordan algebra this becomes $ \{xyx\}=xyx$ 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: $ U_y:y\mapsto \{xyx\}$. The other is the usual squaring product $ y\mapsto y^{.2}$.

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: $ x^{.2}$ and the triple product $ \{xyz\}$. The connection between the triple product and the quadratic unary product is found in the Jordan identity:

$\displaystyle (a^{.2}.b).a=a^{.2}.(b.a)$
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

$\displaystyle \char93 xyz\char93 =[[x,y],z]+[[y,z],x]+[[z,x],y].$
To make a closer parallel use the typical assumption that $ [z,x]=-[x,z]$ (outside of characteristic 0) then we can write:
$\displaystyle \char93 xyz\char93 =[[x,y],z]+[[y,z],x]-[[x,z],y].$
Since Jordan algebras are commutative we can also write
$\displaystyle \{xyz\}=(x.y).z+(y.z).x - (x.z).y.$
However, unlike Lie algebras where $ \char93 xyz\char93 =0$, in Jordan algebras the triple product is almost never 0.



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Cross-references: Lie algebras, commutative, parallel, multiplication, algebra, Jordan identity, connection, unary, exceptional Jordan algebras, proof, operations, quadratic Jordan algebra, associative, characteristic, special Jordan algebra, product, Jordan algebra
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This is version 4 of Jordan triple product, born on 2006-12-11, modified 2006-12-14.
Object id is 8615, canonical name is JordanTripleProduct.
Accessed 1290 times total.

Classification:
AMS MSC: 17C05 (Nonassociative rings and algebras :: Jordan algebras :: Identities and free Jordan structures)
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